Arthur Alt

Herve Andre

Christiane Anselme

Alain Audry

Stéphane Auffret

Matthieu Au- Thier

Denis Bardot

Chrys Barthe

Thierry Beauvais

Matthieu Becot

Vickie Beduneau

Rahmona Belgaid

Fabrice Bernard

Bruno Bertrand

Antoine Besnier

Philippe Bibes

Danielle Bihoes

Benoît Bilheude

Beverlie Billerey

Philippe Birac

Armel Bonneron

Cé- Dric Bonzi

Guillaume Borragini

Jean-Jacques Boubert

Colin Bouchard

Gilles Bourhis

David Bourles

Vincent Bournel

Olivier Briard

Julen Bridier

Gaëtan Brindejonc

Mick- Aël Buanic

Eric Bujard

Matthieu Cabaussel

Jérôme Cabelguen

Didier Cadiou

Philippe Cales

Anna Capietto

Marie Capoulade

Kristell Carric

Alexandre Cauchy

Serge Caule

Antoine Chabrolle

Romuald Chaigneau

Laurent Chailloux

Carine Char- Ronneau

Philippe Clavier

Jean Philippe Coeffet

Bernadette Cottalorda

Monique Croze

Willy Dabin

Cécile Dars

Frédéric Dechaine

Nathalie Delliou

Fabien Demaret

Ronan Deredec

Laurent Desbrugeres

Alexandre Dewez

Philippe Diraison

Stéphane Dixneuf

Ghislain Doremus

Hugo Dubois

Pascal Ducasse

Perrine Dulac

Christine Dumas

Benoit Dumeau

Charles Dupe

Fabrice Dupont

Jean-Bernard Etchebarne

Jacques Farre

Justine Fontaine

Matthieu Fortin

Pascale Fossecave

Loralie Francois

Hélène Gadenne

Olivier Gallet

Susan Gallon

Gilles Garnier

Marcel Gaudin

Françoise Gaudin

Sébastien Gautier

Daniel Gerard

Cécile Gicquel

Céline Girardeau-Guevara

Florent Goulo

Nolwenn Grandjean-Lefevre

Matthias Grandpierre

Joris Grenon

Anaïde Grit

Annick Groisard

Myriam Gueguen

Benjamin Guichard

Stéphane Guyot

Gaétan Guyot

Sylvain Haie

Sami Hassani

Thomas Herault

Sylvain Hunault

Mathilde Huon

Julien Huteau

Xabi Indaburu

Yann Jacob

Florence Jehenne

Loïc Jomat

Thierry Josse

Emmanuel Joyeux

Jean-Luc Jung

Marie Kerdavid

Sylvie Labatut

Sophie Labrut

Sébastien Labulle

Camille Lacroix

Jean-Pierre Lafond

Philippe Lamy

Alexandra Lang- Ford

Patrick Lapouyade

Sophie Laran

Evelyne Large

Claire Laspougeas

Arnaud Le Cras

Anthony Le Doze

Thomas Le Gall

Christophe Le Pimpec

Claude Lecointe

Cécile Lefeuvre

Marine Leicher

Kévin Leleu

Marie Lelong

Audrey Lemoine

Thierry Lucas

Hélène Maheo

Jacques Marquis

Bernard Martin

Frank Mathe

Eléonore Meheust

Mathieu Mendes

Paula Mendez-Fernandez

Jeremie Mener

Jean-Roch Meslin

Sylvain Michel

Victor Millour

Christian Mirtain

Gaël Moal

Emilie Morauxbulteau

Alizée Morin-Repincay

Annabelle Morisset

Benoit Nauguet

Marine Nodjoumi

Lu- Dovic Noisette

Sébastien Palier

Eric Passalaqua

Yves Paturel

Hélène Peltier

Myriam Perriere-Rumebe

Enrique Petit

Daniel Pilvin

Laurent Plantier

Yann Planque

Alexan- Dre Portmann

Jean-André Prat

Sébastien Prouet

Erwan Quemeneur

Yann Quillivic

Mélanie Racine

Noriane Rhouy

Yamin Ridoux

Catherine Robert

David Robert

Jean Guy

Virginie Rog

Anne Sophie Rosier

Mélina Roth

Pierre Rousseau

Jacques Ruatti

Yaouenn Sabot

Flore Samaran

Livier Schweyer

Laurent Soulier

Jérôme Spitz

Elodie Stolear

Antoine Tallec

Isabel Tardieu

Manon Tissidre

Hélène Tomowiak

Aurore Toulot

Karin Tournemille

Jean-Charles Trainaud

Olivier Trevidic

Yann Turgis

Yan- Nis Turpin

Olivier Van Canneyt

Samuel Veillard

Baptiste Verdoux

Philippe Vignaud

Jean Vimpere

Christophe Vuillemot

Frederic Walch

Agnès Wigdorowicz-Dupont

Gregory Ziebacz

Etienne Rouby
email: etienne.rouby@univ-lr.fr

Vincent Ridoux

Matthieu Authier
email: mauthier@univ-lr.fr

Lola Gilbert
email: lola.gilbert@univ-lr.fr

Emilie Tew-Kaï

Victor Quilfen

Devin S Johnson

PTB -Bottom pair trawlers RLA -Removal Limit Algorithm

Ce travail de thèse est le fruit d'un effort collaboratif.

En premier lieu je tiens à remercier mes directeurs, Vincent Ridoux et Matthieu Authier, qui m'ont fait confiance. Vous avez été des mentors, chacun à votre manière. Travailler à vos côtés a été une expérience enrichissante à la fois sur le plan professionnel mais aussi humain. Pour vos contributions respectives à ce projet (qui ne sont pas des moindres), votre pédagogie et votre investissement, à vous deux, Messieurs, Merci.

Je remercie la direction et toute l'équipe de l'Observatoire Pelagis qui m'a accueilli avec bienveillance, considération et qui m'a permis de participer aux activités de l'Observatoire. Merci pour le partage d'expériences et de connaissances. Merci à celles et ceux qui ont été impliqués dans l'apport de données, l'aide avec la base, les sessions terrains et dissections. Merci à certains membres de l'équipe pour leurs conseils avisés quant à la manière de mener le projet ou bien de rédiger le manuscrit. Merci également aux stagiaires et contractuels de passage avec qui j'ai partagé des moments de vie. Je remercie également la direction, les équipes, doctorantes, doctorants et stagiaires du CEBC pour leur accueil au Centre.

Merci aux personnes qui ont contribué de prêt à ce projet. Audrey sans toi les lectures d'âge auraient été très difficiles. Pour ne pas dire impossibles. Merci pour ton implication dans ce projet et ta bonne humeur. Merci Paula pour le coup de main qui a apporté un second souffle salvateur. Merci Willy pour ton expérience en la matière et la passion pour la transmission dont tu fais preuve. Merci également à Cécile, Fabien et Eleonore pour m'avoir aidé à intégrer les échantillons quand cela était nécessaire. Lola merci pour ton implication lors de ton stage. Tu as été un exemple d'investissement, de justesse scientifique et de persévérance. Tu as mené ce stage d'une main de Maître et cela a abouti de la plus belle des manières. Je te souhaite de la réussite pour ta thèse. Merci à Justine pour ton sérieux dans les lectures de lames histologiques de Gonades. Mathieu, merci pour ta disponibilité, ta justesse analytique implacable et ton efficacité à toute épreuve. C'est un réel plaisir de collaborer activement avec toi.

Merci aux membres du Shom pour leur accueil et leur contribution à la partie la plus exploratoire de ce projet, Emilie, Marie et Victor. Merci à l'Equipe d'Ifremer, Laurent, Thomas et Sébastien pour leur collaboration active qui fût fructueuse. A Kelly aussi pour ton expertise dans chacun de nos travaux communs.

Pour leur accueil et leur professionnalisme je remercie les membres du département de biologie de La Rochelle Université, grâce à qui j'ai pu découvrir l'enseignement à l'Université.

Je tiens à remercier Emmanuelle Cam, Liz Slooten, Anita Gilles et Christophe Guinet pour leur intérêt vis à vis de ce travail ainsi que pour leur disponibilité à le parcourir.

Merci à Ken Arroyo Ohori et Federico Marotta pour la mise à disposition de leurs fichiers LaTeX avec lesquels j'ai pu concevoir ce manuscrit. Une mention spéciale aussi pour les équipes de Rstudio et de RStan dont les travaux permettent d'aller toujours plus loin pour faire parler les données. D'un point de vue plus personnel maintenant je tiens à remercier ma Famille qui m'a toujours soutenu et encouragé dans la direction que j'ai pour l'instant choisie.

Merci aux personnes qui m'ont accompagné pour repousser bien des limites au cours de ces trois ans: PB, AK et SG. Merci spécialement à G pour ton exigence et tes transmissions.

Merci à mes amies et amis de La Rochelle avec qui c'est toujours un plaisir d'aller partager un verre ou quelques vagues. Merci à mes collègues de promotion à Nice et à Paris pour le parcours que nous avons accompli tous ensemble. Une considération éminente pour mes amis, le noyau du Sud-Est de la France. Nous sommes une équipe.

Je remercie le Ministère de la Transition Ecologique pour avoir accordé un intérêt tout particulier à la problématique du dauphin commun et avoir financé ce projet.

Enfin je tiens à remercier sincèrement les membres du Réseau National Echouage sur la côte Atlantique impliqués dans la collecte d'échantillons ayant servi au court de ce projet. Votre investissement bénévole est à la base de toute analyse scientifique. Grâce à votre dévouement, il est possible de produire des études permettant de nourrir des politiques de gestion de la mégafaune, ô combien importantes de nos jours.

The term megafauna comes from the combination of the Greek word "megalos", which means large, and the Latin word "fauna", which means the total number of animals living in a certain area at a certain time. The term megafauna is broad and ambiguous. It can refer to large animals, large vertebrates, large mammals or even giant mammals. In the scientific literature, the term can also lead to confusion. Moleón et al. ( ) recently proposed a critique of the use of the term megafauna through a literature review and proposed a goal-oriented framework for megafauna research. Megafauna can be defined in several ways. The most common definition is based on the size of organisms: mass for vertebrates and length for invertebrates. Another possibility is to define megafauna according to its functional role in the ecosystem and in particular in the food chain. The megafauna can be described as "apex megafauna" in the case of top predators or "keystone megafauna" in the case of keystone species.

The marine megafauna is a compartment of the ecosystem that includes long-lived marine species such as reptiles, birds, sharks, turtles and mammals. The species diversity of the marine megafauna is marked by the presence of top predators within it. These top predators are so called because they are at the top of the food web. They directly regulate the abundance and biomass of their prey, which may themselves be predators of other species. Within an ecosystem, each species is structured into populations. A population of marine megafauna can thus be defined as all the individuals of the same megafauna species living together in the same environment at the same time. The predatory action of top predators causes a trophic cascade of regulatory effects on the other populations that compose the ecosystems. This regulatory action of lower trophic levels in terms of abundance and biomass is theorised as top-down control (Hairston, F. E. Smith, and Slobodkin ; Paine ). The presence of top predator populations is therefore essential to maintain the balance of energy flows within the ecosystem. These populations therefore structure Based on (Paine ).

the entire ecosystem despite their proportionally lower abundance than other populations. Their presence contributes to structuring the communities (set of populations) that make up the biotic part of the ecosystem (Odum and Barrett ). Populations of top predators can sometimes be considered as keystone species, a concept developed from the observation of the ecological importance of the purple starfish (Pisaster ochraceus, Brandt ; Paine ; L. S. Mills, Soulé, and Doak ; Paine ).

Definition of the term "population" used in this project

I will use here the following definition of a population, provided by Berryman (Berryman ): "A group of individuals of the same species that live together in an area of sufficient size to permit normal dispersal and/or migration behaviour and which numerical changes are largely determined by birth and death processes."

The presence of certain species is essential for maintaining the diversity of ecosystems. Some species are considered keystone species, a concept that includes, but is not limited to, predatory species and species associated with megafauna (Figure . ). For example, elephant populations (genus Loxodonta, Anonymous ) control the tree population in the savannah and allow other herbivores access to the plains for grazing (Western ; Ishida et al. ). But top predator species are often key species for the ecosystem. For example, the grey wolf (Canis lupus, Linnaeus

) was an important top predator in Yellowstone National Park, preying mainly on the abundant elk (Cervus canadensis, Erxleben

) and bison (Bison bison, Linnaeus ) herds. The disappearance of the grey wolf from Yellowstone Park around led to a proliferation of elk that consumed primary producers in large numbers. Other populations dependent on primary producers were affected and the biodiversity of the ecosystem was greatly reduced (Ripple and Larsen ). The reintroduction of wolves in the s helped to regulate the elk population and promote greater biological diversity (Ripple and Beschta ). The concept of keystone species should not be confused with the concept of umbrella species. Confusion is possible because both terms refer to a species on which other populations depend. One of the main differences is that umbrella species cover a wide geographical area and are linked to a wide variety of habitats and ecosystems. This difference is important from a conservation point of view. The umbrella species concept is linked to ecosystem conservation. It is recognised that the conservation of umbrella species is a means to preserve one or more ecosystems and their associated communities. An example of an top predator umbrella species is the Amur tiger (Panthera tigris altaica, Temminck ; Miquelle et al. ). Populations of this predator extend over a vast area covering part of Southeast Asia. They are at the top of the food chain in many ecosystems with different environments (Carroll and Miquelle ). The conservation of keystone and umbrella species is an interesting conservation approach, as it helps to maintain the diversity and balance of one or more ecosystems. From a marine ecosystem management (and conservation) perspective, it can be effective to focus on populations of top predatory marine species (Roberge and Angelstam ). There is a challenge to correctly define the size-effect relationship. b. Two-dimensional qualitative disruption of animal species. The twodimensional space is defined by body size and ecosystem effect. Keystone species have an high effect on ecosystems and only the largest keystone species are defined as megafauna.

Figure from Moleón et al. . Humans are currently impacting marine megafauna in a variety of ways. These impacts can be described as indirect if the human action affects an intermediate factor that has an impact on the population (e.g. human-induced climate variations). Conversely, they can be defined as direct if the human action directly affects the population (e.g. hunting or accidental capture in fishing gear). Recently, Authier et al. ( ) has proposed a new classification of impacts using the example of marine megafauna populations, these impacts are classified into three categories. Each is associated with a pressure that generates a different response within the population (Figure . ). The pressure is classified as tertiary if it generates a response in the behaviour of individuals in the population. Pressure is classified as secondary if it generates a response in the health of individuals in the population. Pressure is classified as primary if it generates a response in the demographic rates of the population or individuals within the population. In my opinion, this classification is relevant because it allows for more nuance. For example, it has been shown that pile driving during the construction of offshore wind farms can impact marine mammals such as the harbour porpoise (Phocoena phocoena, Linnaeus

) by causing displacement of individuals (Nabe-Nielsen et al.

; Pirotta et al. ; Graham et al. ). This displacement can generate an energy cost which is then passed on to other levels of response (Authier et al. ). These impacts constitute tertiary pressures in the proposed new form of classification, which allows for better mapping of their effects on population status. Demography is considered to be the response level with the most direct effect on population status, although populations are rarely exposed to only one type of pressure, particularly marine megafauna (Authier et al. ). It is mainly this level of response that interests us here and in particular the primary pressure associated with it, especially for marine mammal species. .

Marine mammal demography

Vital rates definition Vital rates are refereed as demographic parameters such as agespecific survival and fecundity distributions (Reilly and Barlow ).

The life histories of marine mammal species are typical of long-lived species. The life history of an organism is its survival and reproductive pattern, as well as characteristics that directly affect survival and the timing or amount of reproduction. As long-lived species, marine mammals have low reproductive rates, high survival rates and a focus on caring for young individuals. They are iteroparous with several reproductive episodes during their lifetime (Cole ). They can also be described as species, according to the definition of MacArthur and Wilson ( ), as they exhibit stable strategies with low growth rate values, high care for young survival and low fecundity in stable environments. If we take into account the continuous reference framework proposed by Pianka ( ) and Stearns ( ) which described the strategies as a continuum, they would tend towards the extreme of the slow strategy. The slow strategy is characterised by a long life expectancy with low reproductivity (i.e. high investment in survival and growth over reproduction). The life history of marine mammals is characterised by a trade-off between survival and reproductive vital rates that appears to favour survival over reproduction. Although life histories within a species, or even a population, may appear to be uniform, they may in fact differ between individuals.

Differences

Some concepts

Fitness: Ability of an individual of a given genotype to reproduce. Also called selective value or reproductive success. Cohort: A group of individuals who have experienced the same event during the same period. The term cohort is usually used to refer to birth cohorts which refers to all individuals born in the same year. Phenotype: The set of apparent characteristics of an individual conferred by its genotype, its development and its environment. in life history between individuals in the same population can be explained by observable factors (e.g. age or sex) but some inter-individual variability is not accounted for by these factors (Camus and Lima ). This variability is defined as individual heterogeneity or frailty (Cam, Aubry, and Authier ). Heterogeneity can be defined as time-invariant if we consider the differences between individuals acquired at birth and inherent to the individual's phenotype (Van Noordwijk and de Jong

). Time-invariant differences between individuals are considered as the basis of the quality of individuals (Camus and Lima ; Vaupel, Manton, and Stallard ; Coulson and C. Thomas ). The quality of an individual can be seen as the latent probability of surviving and reproducing in a time interval. High quality individuals are therefore more likely to survive and reproduce than lower quality individuals who will tend to die younger. These differences between individuals due to time-invariant effects, acquired at birth, can lead to a selection process. The best phenotypes are thus selected by mortality selection within the birth cohorts (Curio ; Endler ). Lower quality individuals may not contribute to population renewal to the same extent as higher quality individuals, which would lead to an age-related decrease in the proportion of lower quality individuals in cohorts and this would lead to an increase in population-level survival (C. S. Thomas and Coulson ; Vaupel and Yashin ). The proportion of individuals with a low probability of survival, tends to decrease over time while the probability of survival of the remaining (i.e. better quality) individuals should increase (Camus and Lima ). Thus, the apparent probability of survival of the population would appear to increase while in reality the probability of survival of individuals would not change. This could be the source of surprising correlations between the probability of survival of an individual and its age, since this probability of survival would be linked to the population (Camus and Lima ). The probability of survival and viability of a population is also influenced by human activities. For marine mammals, one of the activities that has the greatest impact on them is fishing. This is particularly the case for odontocetes. ).

The link between Frailty Hazard and Survival

In demography, frailty can be referred to as susceptibility to disease or death (Kannisto ). Individuals in a population do not have the same life histories, particularly with regard to hazard and survival. Differences between these components of the life history of individuals that are not explained by observable factors are called heterogeneity or frailty (Aalen ). Susceptibility to death or disease can vary from one individual to another but this variation cannot be explained directly (i.e. by observing the phenotype of individuals). It is impossible to take these variations into account as covariates in survival analyses because they are hidden. However, it is possible to statistically account for this inter-individual frailty in order to model its effect on the survival and hazard of a population. These models are called frailty models (R. Henderson and Oman ). Frailty models include the notion of individual frailty in demographic models (Kannisto ). Individual frailty is then taken into account statistically as a random effect having a multiplicative action on hazard (Hougaard ). This random effect can be individual or common to several individuals. In case of individual random effect, modelling is called univariate and frailty describes heterogeneity. In this thesis project, we have developed a new approach to survival analysis, based on taking into account the effect of individual frailty on hazard (Reed ).

The case of odontocetes, especially delphinids

Among the marine megafauna, odontocetes are particularly threatened because they are often subject to bycatch pressure. Of the human activities that can have an impact on odontocete populations, fishing has the greatest impact on populations worldwide (A. J. Read, Drinker, and Northridge ). Odontocetes are cetaceans with teeth, as opposed to mysticetes, which are baleen whales. Because of their top predator position, odontocetes compete with humans for food resources (i.e. fish). These interactions can sometimes be positive for odontocete populations by favouring their survival. One example is the phenomenon of depredation, which is defined for marine mammals as the removal of fish caught on commercial fishing gear. Recently, it has been shown that among the orcas of the Crozet Island population (Orcinus orca, Linnaeus ), individuals with depredation behaviour have a better survival than those that do not engage in depredation (Margin-Figure . ). From a more general point of view, these interactions are deleterious to odontocetes as they can reduce the survival of pop- ).

Picture from https: //cseweb.ucsd.edu/~awilby/ Brink-of-Extinction _ NGS.html ulations (and by extension their growth and viability). One of the most worrying phenomena associated with competition for food resources is the phenomenon of bycatch. It is defined as the accidental capture of a non-commercial or even protected species (M. A. Hall

). Populations, even species, of odontocetes have already disappeared or are on the brink of extinction because of this additional mortality pressure, which they cannot withstand. This is the case of a Chinese river dolphin named Baiji (Lipotes vexillifer, Miller ). This species is now considered extinct. Intensive human activity in the Yangtze River, including shipping traffic, water pollution and bycatch, is suspected to have caused its extinction (Dudgeon ; D. Wang et al. ). Another example is the vaquita (Phocoena sinus, Norris and McFarland ), whose population is now almost extinct. The vaquita is in fact the most critically endangered marine mammal, with very few individuals remaining in the wild. This species is endemic to the shallow, turbid and highly productive habitat of the upper Gulf of California, between Baja California and mainland Mexico (Rojas-Bracho, Reeves, and Jaramillo-Legorreta ). The Vaquita population has been subject to increasing bycatch pressures associated with fisheries that target totoaba (Totoaba macdonaldi, Charles Henry Gilbert

) and has been severely depleted. This fish, which is about the same size as the vaquita, is targeted for the black market trade in its swim bladders in China (Rojas-Bracho et al.

). The implementation of management measures is one way to prevent a population from becoming too depleted. It is based in part on determining the viability of delphinid populations. This viability can be determined from population dynamics models (Wade ; Mannocci et al. ).

aspect ; Harwood ). The concept behind managing exploited animal populations is intrinsically linked to growth rate. A population that is reduced from its maximum level of abundance given the environmental conditions (i.e. carrying capacity) will have a positive growth rate. It is on this positive growth rate that the removal of the population is carried out (Harwood ). More specifically, it is assumed that density-dependence influences population recruitment in a stock-recruitment relationship that is at the heart of population harvesting (Sale ). This foundation is inherited from fish stock management studies (Beverton and Holt , ). The use of population growth models is therefore particularly interesting in the context of the management of exploited populations and in the event of accidental catches, the bycaught population can theoretically be considered as an exploited population (Barbraud et al. ; Lebreton ). Theoretical bases related to the science of fisheries management can therefore be applied to implement bycatch management measures. The basic premise changes. We no longer seek to exploit the population for a continuous yield, but we want to reduce bycatch to a sustainable level that is relative to the dynamics of the population concerned. To do this, we use models that take into account either abundance (Wade ) or abundance and vital rates by age (Hammond, Paradinas, and Smout ; Genu et al. ) within or outside of a Management Strategy Evaluation procedure (Punt et al. ). European waters are fishing grounds where many different gears are used at different times of the year, with complex and structured ecosystems, including marine megafauna (Peltier et al. ; Corrales et al. ). Among the cetacean species frequently observed in these waters (and in particular in the Bay of Biscay), the common dolphin (Delphinus delphis, Linnaeus ) currently presents strong conservation challenges and is possibly a Keystone species (Peltier et al. ; Corrales et al. ; Murphy et al. ).

Population growth models

Population

Here, the margin notes will include the mathematical expressions of what is said in the text. This is done so as not to interfere with the reading, but to provide a more mathematical background and insight.

dynamics is the scientific field of ecology concerned with the changes in abundance that populations experience over time and the factors that influence these changes (Gotelli ). The main objective of the study of the dynamics of a population is to carry out an evaluation of the population which can be considered as the evaluation of both the status of the population and its vitality (or viability, Skalski, Ryding, and Millspaugh ). The population status corresponds to the current state of the population with regard to certain characteristics such as age and sex structure or abundance in relation to a certain reference. Status using abundance can be assessed through time series of abundance estimates, but it is also possible to study the underlying mechanisms that shape these changes in abundance (J. D. Baker, A. Westgate, and Eguchi ). To do this, it is possible to assess the viability of the population through the evaluation of its demographic health and its capacity to maintain itself from one year to the next. Several models (i.e. representation of reality. In our case, representation of the dynamics of a population from a mathematical point of view) have been developed in order to represent the annual dynamics of populations.

Geometric population growth models

Basic

The abundance at time + 1 ( +1 ) is obtained from the previous abundance at time ( ), additions through births and immigrants ( + ) and losses through deaths and emigrants ( + ):

+1 = + ( + ) -( + )
dynamic changes within a population are due to the losses of individuals through processes of death/emigration and additions through processes of births/immigration (J. D. Baker, A. Westgate, and Eguchi ). Considering a closed population (without immigration and emigration) it is possible to imagine its growth dynamic geometrically (i.e. in way that each abundance value is determined by the multiplication of its previous value by a constant factor) according to a continuous or a discrete time stamp (depending on population dynamic to be modelled).

The rate of change in population size through time is expressed given instantaneous birth rate , instantaneous death rate and intrinsic rate of increase :

= - = ( -) =
Considering a continuous time model (i.e. birth and deaths occurs continuously) associated with a continuous population growth, the rate of change in population size through a continuous time period could be explained as the difference between births and deaths. They depends on the total abundance, instantaneous births and death rates. In other words, the rate of change in population size through a continuous time period is equal to the intrinsic rate of increase effect on population abundance (Gotelli ). The rate of increase value is determinant to understand the viability of the population and define if it is stationary ( = 0), decreasing ( < 0) or increasing ( > 0).

The abundance at any time ( ) is obtained from the initial abundance ( 0 ), time , the intrinsic rate of increase and the exponential function : = 0

It can also be used to model the abundance of the population at any time given the initial abundance estimate. Depending on the value of the rate of increase, the population can grow exponentially. One example of exponential growth on large mammal was demonstrated (Scheffer ) with the reindeer (Rangifer tarandus, Linné

). The intrinsic rate of increase of the population can also be expressed when no resource is limiting and can be referred as compared to the observed intrinsic rate of increase which can be defined as and the intrinsic rate of increase under a stable age distribution (Skalski, Ryding, and Millspaugh ; Caughley and Birch ).

From is the finite rate of increase expressing the proportional changes in abundance from on year to the next ( +1 ). To project the population size at any time the initial abundance 0 is also used:

= +1 +1 = = 0
the expression of the geometrical continuous time populations model, which admitted that birth and deaths occurs continuously, it is possible to represent the dynamic of populations for which births and deaths are not continuous. This is the case for numerous animal species which present a breeding season and unequal mortality levels through the year. Such models are refereed as geometrical discrete time populations models. The use of as instantaneous rate of increase is replaced by the use of as finite rate of increase. This rate expresses the proportional changes in abundance as a multiplicative factor by which abundance changes from one year to the next (Skalski, Ryding, and Millspaugh ). As a proportional change, the rate expresses the viability of the population relatively to , reference value for a stationary population ( = 1), growing population ( > 1) and declining population ( < 1).

Relationships between growth rates:

= + 1 =
The way time is considered in the choice of representation of population dynamics regarding geometric growth determines the relationship between growth rates as is the intrinsic per capita growth rate, is the net discrete per capita growth rate and is the discrete per capita growth rate (J. D. Baker, A. Westgate, and Eguchi ).

Logistic population growth models

The Logistic models forms:

= × × 1 - +1 = + × ( -1) 1 - +1 = + × ( -1) 1 -
previously seen models admit that the population can grow forever which is not the case in natural population due to several factors such as predation, hunting, starvation or overcrowding (Leopold ). Natural populations can also be limited in growth due to a lack of resources and some environmental effects leading the growth in logistic population growth (Krebs ). The access to resources may be limited among organisms through population density. This factor can be defined as a density-dependent factor. Density-independent factors, on the other hand, have an effect on population growth rates independently of the population size or density. Density-dependent processes are important to take into account, as they directly influence the maximum sustainable yield value regarding fisheries and harvested species. The maximum sustainable yield (MSY) is the highest theoretical equilibrium yield that can be continuously taken from a stock under the average environmental conditions (Schaefer ). The highest catch value allows the population to sustain through somatic growth, spawning and recruitment (Tsikliras and Froese ). We will see that this concept may be adapted to set up management plans for accidentally harvested species. Density-dependence is an important factor shaping the growth of animal populations, in particular considering large mammals (Fowler ). The basis of the concept is the relationship between instantaneous birth rate, instantaneous death rate and abundance at a given time. When the population size tends to increase, the instantaneous death rate increases too. Assuming that resource are limited, the instantaneous birth rate tends to decrease as abundance increases. The equilibrium between both rates is obtained when the population attains the carrying capacity ( ) level which can be viewed as the maximum population size that a population can reach given its environment its resource limits and associated pressures. : Relation between per capita growth rate, abundance and carrying capacity. When abundance (N) is low, the per capita growth rate is high enough to allow the population to produce individuals and increase abundance to the carrying capacity which is the stable equilibrium point. In the case of a population subject to the Allee effect, a too low abundance below a critical threshold is associated with a decline in growth rate. It is no longer high enough to support mortality and the population disappears.

From the link between those parameters, (Verhulst ) created the familiar logistic population growth model assuming a continuous time. From this model the discrete one was obtained and allows to represent the dynamic of a population given densitydependence with a discrete time stamp (Pearl ). Then, the generalized logistic model was introduced taking into account the rate ( ) at which the population reaches its carrying capacity (J. D. Baker, A. Westgate, and Eguchi ). Depending on the value of this rate, the population can grow at a rate equal to that of a standard logistic growth ( = 1), can growth more rapidly in an exponential way close to the carrying capacity ( > 1), can fluctuates about the carrying capacity ( ≥ 10) and can grow at a rate lower than that of the standard logistic due to an anticipated effect of density dependency on growth ( < 1) (Pella and Tomlinson ).

While Baseline Allee Effect model with * as the Allee threshold (Courchamp, Clutton-Brock, and Grenfell ):

+1 = × 1 + 1 - × * -1
density dependence is commonly seen as a limiting factor for population growth, Warder Clyde Allee proposed in another view, that can have strong consequences for modelling population dynamics, which he developed in his book: Allee et al.

. Its premise is that some populations are affected by a positive relationship between population growth rate and density when they reach low levels of abundance which can lead to their extinction if the density of individuals is too low (Courchamp, Clutton-Brock, and Grenfell ). If abundance is not high enough, the reproduction or survival of the population may be impacted in a negative way which leads to a decrease in the per capita growth rate that can reaches negative values as it is depicted in Figure . . The Allee effect can also be refereed as the depensation, positive density dependence (W. F. Morris et al. ) or Allee's principle (Odum and Barrett ). Factors that may lead to this inverse density dependence at low density can be classified into three types according to (Courchamp, Clutton-Brock, and Grenfell ) which has led to the recent emphasis on the importance of this idea in modelling population dynamics (Skalski, Ryding, and Millspaugh ). The first type of factor is genetic (i.e. genetic inbreeding). It is characterized by a loss of heterozygoty that lead to a decrease in fitness. The second type of factor is demographic (i.e. demographic stochasticity). It is characterized for example by sex-ratio fluctuations that lead to a low reproductive output (e.g. Kakapo Strigops habroptilus, Gray

). The third type of factor is interactive (i.e. facilitation or behavioural). It is characterized by less interactions that are important for the survival of the population '(e.g. bluefin tunas Thunnus thynnus, Linné

) or the shortage of receptive mates (e.g. Kakapo) and cooperative breeders (e.g. African wild dogs Lycaon pictus, Temminck ) (Courchamp, Clutton-Brock, and Grenfell ). As it is illustrated in Figure . Allee effect produces unstable equilibrium at low densities and increases the probability of extinction. A strong Allee effect is characterized when there is population density below which the per capita growth rate. It is therefore dangerous to harvest a population that may be sensitive to the Allee effect. It is also possible that population growth models that do not consider Allee effects underestimate the risk of extinction (Skalski, Ryding, and Millspaugh ).

Age-Structured population growth models

These

Matrix notations and models for the Leslie matrix modelling approach:

n +1 = Ln         0 1 2 3         +1 =         0 1 2 3         ×         0 1 2 3 0 0 0 0 1 0 0 0 2         ( . ) 0 +1 = 0 0 + 1 1 + 2 2 + 3 3 1 +1 = 0 0 2 +1 = 1 1 3 +1 = 2 2
population models are limited to study the influence of specific age-specific demographic rates on growth rates and it could be of interest to represent the dynamic of a population given particular age or stage classification (Leslie ; Lefkovitch ). The use of matrix population models allows to project the population state given initial abundance or in terms of percentages which is particularly useful for studying viability. Here only the Leslie matrix will be discussed as this concept is part of two aspects of the project. This matrix model reflects the dynamics of a unisex population (usually females) independently of the density dependency processes discussed earlier (Skalski, Ryding, and Millspaugh ; Leslie ). The conceptual model of this population dynamics matrix model can be represented as follows: This population model involves two components to project the number of females of a population through time that are the number of females within each age-class and the probability for female each age-class to both survive and produce a new female in the population. These matrices are associated with eigenvectors and eigenvalues which will not be discussed in detail as it is not the purpose of this chapter but for more mathematical information see Caswell . They are key properties of populations that can be represented by the matrix populations models. The asymptotic population growth rate corresponds to the dominant eigenvalue and the stable age distribution is represented by the eigenvector. Another interesting component of matrix population models are obtained through perturbation analyses. Caswell explains that the "results of perturbation analyses are sometimes more interesting, more robust and more useful than the parameter estimates themselves". The two common perturbation analyses are the elasticity and the sensitivity analyses. They provide both the proportional response of the growth rate to a proportional perturbation in the vital rates and the magnitude of changes in the growth rate with respect to each element of the matrix (J. D. Baker, A. Westgate, and Eguchi ). These information are particularly valuable in case of population management when it is necessary to have information about the age-classes the most important to manage in terms of the value of the growth rate.

Table . summarise important information about the models introduced here. Only those most relevant to the project are presented. These models are of importance to assess the viability of a population and have also been developed in the context of exploitation of some populations (e.g. population of fishes or whales). They therefore serve as a reference for the implementation of management strategies. Both for populations that are exploited voluntarily and for populations that are accidentally exploited.

Population matrix models that have an age-dependent structure need to be filled with vital rates. To obtain these age-dependent vital rates (survival and fecundity), it is necessary to monitor mammal populations. Two categories of monitoring exist and only one General Introduction is applicable to elusive populations. Why "elusive" species ?

The term elusive comes from the Latin elus, and is an adjective meaning "difficult to find, define or achieve" (Oxford Dictionary).

The term is used in this manuscript to refer to species that may have one or more of the following characteristics: difficult to track individually, sparsely distributed, for wich it is difficult to estimate anthropogenic mortality and for which it is difficult to implement management policies.

. Estimation of life-history traits

Longitudinal monitoring to estimate vital rates

Quantifying the life history traits of a population is nowadays mainly carried out using longitudinal monitoring and capturerecapture protocols (CMR; B. K. Williams, Nichols, and Conroy ). The study of life history traits (which can be described as life history parameters) has become important since the end of the 20 th century (Lebreton et al. ). This importance has materialised in the desire and need to detect differences in life history traits between populations in order to study their regulation, structure and viability (Stearns ; Noble and Slatyer ; Crawley and May ). These differences are thought to be the source of changes in life history, through a modification of fitness in relation to survival and fecundity (Manly ). While it is easier to estimate fecundity in wild populations, survival remains a major component of life history that is difficult to quantify (J. Clobert, Lebreton, and Allaine ). The importance of being able to follow each individual over the long term in order to deduce the value of vital rates in relation to survival seems to be an essential condition for achieving these objectives (Lebreton et al. ). The history of survival analysis modelling will not be detailed here. This will be done in the introduction to Chapter , which is a chapter specifically dedicated to survival analysis.

Long-term individual monitoring can use different forms of natural or artificial recognition methods. Longitudinal tracking protocols involve both the recognition of living individuals (Lebreton et al.

; Sandercock ) and dead individuals (Brownie ). For example, it is possible to follow individuals on the basis of physical marks that distinguish them from one another in the population (Castro and Rosa ; Wirsiig and Jefferson ). This is the case for giraffes, which can be recognised individually on the basis of their coat patterns (Berry and Bercovitch ). Where individuals cannot be observed directly, camera traps can be used to record the passage of individuals at points of interest (O'Connell, Nichols, and Karanth ). These camera traps are used to identify individuals of cryptic feline species such as the snow leopard, lynx or Siberian tiger (Jackson et al. ; Can and Togan ; T. M. Wang et al. ). Individual recognition can also be performed in marine mammals. The recognition of individuals is based on the shape of the tail fin and its patterns (e.g. humpback whale and sperm whale), the shape of the dorsal fin (e.g. bottlenose dolphin and killer whale) or by the recognition of specific callus patterns (Right whales; Pace, Corkeron, and Kraus ). These identifications of individuals in marine mammal populations are made possible by longitudinal tracking protocols called "photo identification". Sometimes it is not possible to identify a species on the basis of its natural markings, but it is possible to easily access individuals in a population and place artificial markings on them. These artificial markings can be rings, collars, ear tags, radio transmitters or bands. Ringing is widely used in population dynamics studies of birds that may sometimes be accessible in colonies (Busse and Meissner ). Tagging is also deployed in pinniped species that are accessible in breeding colonies (Costa et al. ; A. Henderson et al. ). Collars are preferred for small mammals or certain ungulates. For the latter, the use of earmarks is common (Swenson et al. ). Collars can also be designed as radio transmitters that are used to study both dispersal and survival (Millspaugh and Marzluff ; G. C. White and Garrott ). Finally, tags can also be used to collect demographic data when individuals are dead. This is particularly the case for rings used in monitoring programmes for hunted populations. These are known as ringing recovery protocols: (Brownie ).

Longitudinal monitoring

Longitudinal monitoring involves repeated observations of individuals through a period of time. It can be perform using Capture-Mark-Recapture, band recovery and tagging protocols.

For cetaceans, the most common longitudinal monitoring protocol is photo-identification (Wirsiig and Jefferson ). This monitoring allows the vital rates of the population to be obtained (Lebreton et al.

) along with variations in vital rates and individualheterogeneity quantification (Camus and Lima ; Gimenez, Cam, and Gaillard ).

This monitoring makes it possible to determine multiple demographic parameters of the population (i.e. vital rates or lifehistory traits) such as survival (Lebreton et al. ), dispersal rates (Brownie et al. ), recruitment rate of new breeders (Clobert ), breeding frequency and transitions between breeding states (Cam et al. ). The expansion of the use of CMR monitoring has made a considerable contribution to the quantification of many life history traits (Lebreton et al. ). However, it is not possible to use this monitoring on all species. For elusive species, it is necessary to use the old approach, combined with difficult assumptions, less sophisticated analysis methods and basic estimators of population life-history traits.

Cross-sectional monitoring to estimate vital rates

Basically, cross-sectional monitoring has been used to describe the age distribution from which life tables can be constructed to estimate age-specific survival and mortality rates (Skalski, Ryding, and Millspaugh ; Gompertz ). The use of this type of monitoring is common in fisheries studies as it allows the agefrequency distribution to be described. The age distribution of a fish population is an essential piece of data that can provide information on changes over time, how a stock responds to exploitation and how it may recover from external disturbance (Jennings, Reynolds, and S. C. Mills ; Greenstreet, Spence, and McMillan ). The age composition of a fish population can be determined from otoliths, backbone and fin radii (Quist, Pegg, and DeVries ; Goldman et al. ; Carbonara and Follesa ), but also from allometric age-length relationships (Westrheim and Ricker ; Ailloud and Hoenig ) using the age-length key which may be subject to bias (Kimura ). The use of this cross-sectional tracking scheme, especially from otolith ageing, is still common in fisheries studies to obtain age composition and demographic parameters (Durant et al. ).

Cross-sectional monitoring has historically been widely used to monitor fish stocks. It is still used today to monitor fish populations and exploited stocks.

Cross-sectional monitoring

In contrast to long-term monitoring of individuals, a population can be monitored by means of snapshots over a given period. This type of monitoring is called cross-sectional or transversal. For example, the characteristics of a few individuals in a population will be collected at a given time.

From the data collected, an extrapolation will be made to the whole population. It is from this type of monitoring that the life tables are constructed (Caughley ).

In the case of tetrapod populations, age composition (or age distribution) is obtained by analysis of morphometric features or even samples of biological material (Caughley et al. ). The determination of the age of amphibians is generally done using osteological techniques (Peabody ) whereas sauropsids are generally aged according to their size (i.e. allometric relationship) (Caughley et al. ). For some bird species (Fulmar, Shag, Redshank and Great Skua), it is possible to count the endosteal layers of the tibia to determine the age of individuals (Klomp and Furness ). For turtles, age can be determined by the annuli method on the carapace (Sexton ; Gibbons and Semlitsch ). For mammals, the methods of age determination are varied. The eruption of the teeth can be an indicator, particularly for bovine species such as mouflons or equids (Garel et al. ), but also for sea otters (Nicholson et al. ). Body size and size frequency classes could also be an indicator of age (Klomp and Furness ). The most widely used and effective method for obtaining a population age distribution and creating the associated life table is the use of biological material taken from dead animals (Caughley et al. ). The biological material used consists of teeth (Laws ; Benjaminsen ), claws (R. B. Thomas et al.

) and horns (Geist ). The method (e.g. odontochronology) consists of counting the number of growth rings (or groups of growth layers) that make up the biological tissue. The age data obtained by this method is called age-at-death data and the age distribution can be called the age-at-death distribution. Classically, these data were obtained by harvesting and recovering dead carcasses (Caughley ). The process of obtaining vital rates associated with survival and mortality from these data therefore constitutes a cross-sectional monitoring of wild animal populations (Caughley ). Since Deevey Jr (

), this type of monitoring has been used to estimate vital rate parameters and construct associated life tables. From these life tables, it is possible to provide age-specific vital rates to inform population dynamics matrix models such as Leslie's matrix model (Leslie ). It is also possible to explore the covariation of age-specific life-history traits to test certain life history theories (Purvis and Harvey ). ). L. Endosteal layers in tibia for birds from Klomp and Furness ( ). M. Claws growth layer groups from E. O. Ferreira, Loseto, and Ferguson ( ).

Preference for longitudinal monitoring

Longitudinal monitoring is preferred for species that offer adequate opportunities for longitudinal studies.

However, the derivation of these rates is based on crosssectional monitoring, which itself is based on basic assumptions that can be problematic (Caughley ). The most important assumption to take into account in this type of monitoring is that the age-at-death distribution obtained from the age-at-death data is representative of the stationary age distribution of the population (Caughley ). In other words, the sampled population must be in a stationary state (i.e. neither increasing nor decreasing) and there must be no selection bias of individuals at the time the data set is constructed (Caughley ). One problem is that mammal populations are rarely stationary in nature as they are in a con-stant state of flux influenced by biotic and abiotic factors, which are themselves in a state of flux (Szuwalski and Hollowed ). Furthermore, when population dynamics studies are conducted, it is precisely to determine whether the population is viable because there is reason to believe that it is not (i.e. it is not stationary). The impetus for conducting a population dynamics study is often at odds with the basic assumption that must be made to conduct the study. Cross-sectional monitoring is associated with strong assumptions, unrepresentative data and the difficulty of trusting life history trait estimates. It is mainly for these reasons that cross-sectional monitoring has been abandoned in favour of longitudinal monitoring in species that offer adequate opportunities for longitudinal studies. Although cross-sectional monitoring is less expensive in terms of data collection (Seber ). In order to provide the best possible information on the demographic rates of these populations, it is necessary to carry out methodological developments associated with their monitoring. In particular, statistical developments through models integrating essential demographic parameters (e.g. frailty). This work would allow more accurate assessments of the viability of elusive mammal populations, taking into account various covariates and random effects.

Stationary and Stable age distributions

The concepts of stationary and stable age distribution are both related and different. If the population growth rate is constant over time, as are its fertility and survival rates, its age distribution will eventually take a stable form. Age frequencies will remain constant over time. When a population is stationary, the population growth rate is equal to . Therefore, the stationary age distribution is the stable age distribution for which the population remains constant over time. A stationary population necessarily has a stationary and stable age distribution, but a population with a stable age distribution does not necessarily have a stationary age distribution. The stationary age distribution is a special case of the stable age distribution. (Caughley ).

. The case of the common dolphin population in the North-East Atlantic ocean

The common dolphin is the most abundant and widespread odontocete cetacean species in the North-East Atlantic Ocean and is associated with significant management issues (Peltier et al. ; Perrin ). It is a top predatory species in the North-East Atlantic ecosystems, with a possible (i.e. no formal recognition ) keystone role in the Bay of Biscay (Corrales et al. ; Lassalle et al.

). The Bay of Biscay (Margin-Figure . ) is a biologically productive area of the North-East Atlantic associated with high fishing activity, involving for example pair trawlers and larger scale driftnets (between the Azores and Ireland) since (Peltier et al. ). The history of fisheries management in this area is closely linked to the common dolphin. Several legislative initiatives related to common dolphin mortality events in the area took place between and (Figure . ). As with other odontocete populations worldwide, the main threat to the common dolphin population in the waters of the North-East Atlantic, and in particular in the Bay of Biscay, is bycatch (M. A. Hall ; Peltier et al.

; ICES ). .

Issues of the thesis project

It is necessary to assess the viability of the common dolphin population and to quantify the main human pressures (i.e. bycatch) on it in order to propose appropriate management objectives with regard to the conservation objectives of European waters. The common dolphin is defined here as an elusive species for which it is almost impossible to track individuals longitudinally through CMR or radiotagging protocols. Furthermore, despite the programmes for observing incidental catches at sea, it is difficult to quantify the actual number of common dolphins bycaughts in the Bay of Biscay. It is therefore difficult to know whether the number of dolphins removed is sustainable for the population, given its demographic viability. A study should be designed to make the best use of the cross-sectional analysis protocol to obtain the population's vital rates. This study should also make it possible to use data from at-sea observers to quantify bycatch. Finally, it should also allow the proposal of sustainable removal thresholds that do not threaten the viability of the population. The challenges of this thesis project are multiple and fall into two main categories. From an applied research point of view, it is now necessary to produce demographic indicators for small cetaceans in the North-East Atlantic and in particular for the common dolphin. These demographic rates are of interest from the point of view of the management of this population as they will provide information on its status in the context of the OSPAR Convention and the Marine Strategy Framework Directive. Some life history traits have been partially estimated and biological characteristics are already known but the need is still there. Secondly, given the apparent impact of fishing on the population in terms of additional mortality, it is essential to quantify this pressure. Finally, given the management issues related to the population, it is necessary to determine the maximum number of removal that the population can sustain in the current state of knowledge and by applying a precautionary principle. This limit constitutes a support point for public policies, in close collaboration with the European Commission. This work is therefore closely linked to the production of the ICES scientific expert groups' opinions and is sometimes cited in the reports of the working groups focused on the case of the North-East Atlantic common dolphin population. This thesis project therefore has a very applied basis. In my opinion, it is important to place this applied basis in a more fundamental context. This recontextualisation is part of a desire to place the present work in the scientific history of the study of population dynamics. I believe that this work has its place as a reflection on the problem of studying the dynamics of elusive animal populations. It was therefore necessary to identify a methodological gap in terms of cross-sectional analyses and to partially fill it. A new approach to sample collection and vital rate modelling is proposed. A general discussion linking the work is also proposed. The present manuscript summarises this three-year work in seven parts. The first part was the general introduction.

. Project outlines

Chapter is a description of the important aspects related to the common dolphin. This chapter is not intended for publication. It was written to provide an overview of the current state of knowledge on common dolphin demography and associated monitoring and conservation programmes.

Chapter includes a description of survival analysis and a history of associated methodological developments. I then discuss the main problem identified and how we responded to it by developing a new method for estimating survival that takes into account covariates and random effects. This part was carried out during the first year of the project and was published: Rouby, Ridoux, and Authier (

). I presented this work at the poster session of the World Marine Mammal Conference * .

Chapter is the central part of the manuscript, from the point of view of writing and organisation of the project. It includes the acquisition of demographic data (age-at-death, reproductivestatus-at-death) after the definition of a sampling plan respecting carefully chosen criteria. It also includes the analysis of these data using methods commonly accepted in the literature as well as those developed in the framework of the project. The highlight of this work is the estimation of the influence of several factors on the demographic rates of the North-East Atlantic common dolphin population. I will present this work orally at the Society for Marine Mammal Conference † . This part was carried out during the years of the project and is currently being prepared for publication.

Chapter corresponds to the quantification of the human impact on the dolphin population through bycatch and to the explanation of bycatch by studying oceanography. The first part required a methodological development but my greatest contribution was the use of this method with an applied case. The strength of this approach is that it allows an estimate from biased bycatch observation data. This work led to the publication of the development of the method: Authier a and my contribution was essentially related to part of the writing of this manuscript. Concerning the application, I was able to use this method on the dolphin case study. This work is published: Rouby et al. (n.d.). This work was carried out in close collaboration with Ifremer (Institut Français de Recherche pour l'Exploitation de la Mer), which provided the data and participated in the writing of the article. This work was carried out during the second and third years of the project. I presented it in the ICES working groups WGBYC and WGMME. I also presented it at the ICES workshop WKMOMA.

The second part of the development of this chapter, concerning the influence of oceanography on bycatch, is for me the most collaborative part. It was carried out as part of a Master internship that I co-supervised with the support of Matthieu Authier and Emilie Tew-Kaí from the Shom (Service Hydrographique de la Marine Nationale) in Brest. The Shom provided us with oceanographic model outputs allowing us to link the occurrence of mesoscale oceanographic structures to dolphin mortality at sea. This work has been published as: Gilbert et al. ( ). I have chosen not to include this section in its entirety as I was not at the heart of the data analysis. This part will serve as a basis for introduction in Chapter . This work was carried out during the second year of the project.

Chapter , deals with the management and conservation aspects of the common dolphin population. It includes the practical and conceptual use of the preliminary draft (including demographic rates and numbers of dolphins caught) to produce tolerable bycatch thresholds on a European scale. This part still needs to be refined in the analysis but is in preparation to be proposed for publication. This work was carried out during the last months of the thesis project. It is not currently planned to valorise this work in the form of a conference. The common dolphin is a long-lived delphinid species that presents important conservation challenges in European waters. Individuals of this species appear to live on average up to years and produce between and pups per female during their lifetime (Perrin ). In European waters, there is a resident population that appears to number around , individuals (Hammond et al. ). Vital rates are already available for this population, including sexual maturity profiles and survival rates (Mannocci et al. ; Murphy et al. ). These rates appear to show that the population is not viable and will become extinct within years. Since , the number of dolphins found stranded on European shores has increased significantly (Murphy, Pinn, and Jepson ). In , eighty percent of stranded dolphins showed signs of bycatch (Dars et al. ) and it is estimated that between , and , dolphins are bycaught each year (Peltier et al. ). The state of this population is worrying with regard to the conservation objectives set by the Habitats Directive and the MSFD (Peltier et al. ; Murphy et al. ). The common dolphin population is monitored both by aerial surveys and by collecting biological data on stranded individuals. This chapter presents the current state of knowledge on the North-East Atlantic common dolphin population. It also provides a description of each entity or instrument involved in the conservation of this population. Finally, it presents the different monitoring schemes carried out on the population. ; Margin-Figure . ).

T

The subspecies studied here

Throughout the manuscript and subsequent chapters, when the term "common dolphin" is used, it will refer to the current subspecies of common dolphin Delphinus delphis delphis and to the former species considered in the classification as the shortbeaked common dolphin Delphinus delphis. The name common dolphin (and its Latin name Delphinus delphis in figures) will be used to designate this subspecies only and it will be the only one studied in this project.

. Distribution and biology

On a global scale

The common dolphin is a widely distributed odontocete cetacean species with a known life cycle. Its distribution ranges from tropical to temperate waters across the globe (Perrin ; Figure . ). This species occupies both oceanic and continental shelf habitats. It appears to exhibit seasonal distributional movements, sometimes linked to changes in water temperature (Perrin ; E. E. Henderson et al. ). Individuals may form groups ranging in size from a few hundred to a thousand. The common dolphin is a long-lived species, with a maximum recorded lifespan of years (Perrin ). It is an iteroparous species with a calving interval of approximately -years. The gestation period is estimated to be -months. Maximum length is about cm for males, which are slightly longer than females. Calves at birth measure -cm. The life history traits of the species have common characteristics but are population dependent. In Table . , characteristics (including age at sexual maturity) are presented for several common dolphin populations.

In the North East Atlantic and European waters

In European waters, the population appears to be abundant and absent from some areas. The common dolphin population appears to be distributed in both oceanic and continental shelf waters. However, common dolphins are rarely recorded in the Eastern Channel and North Sea. The highest densities and abundances of animals are found on the continental shelf (Hammond et al. , ; Laran et al. ). The population in European waters is currently estimated at , individuals ( , -, , ) while , ( , -, ) individuals are estimated in the Bay of Biscay (Hammond et al. ; high density area: Figure . ). The history of abundance estimates is provided in the Supplementary-Table . : whole NE Atlantic population.

Since the s, an increasing number of common dolphins have been found stranded on European seashores. Among the European countries, France is the country with the highest number of stranded common dolphins on its beaches (Figure . ). In France, the number of individuals found is increasing almost constantly, with a stranding record of individuals for the year (Margin-Figure . ; data provided by Observatoire Pelagis). This phenomenon is notably characterised by Unusual Mortality Events (UME) that are seasonal (Peltier et al. ). The majority of dolphins stranded during the year are stranded in winter, from January to April (Dars et al. ). A significant proportion of stranded dolphins show evidence of capture in fishing gear. ). Examples of evidence of bycatch on stranded common dolphins are provided in Figure . at the end of this chapter. It is estimated that around , dolphins are incidentally caught in fishing gear each year in the Bay of Biscay (Peltier et al. ), which could pose a major threat to the viability of the population. Bycatch pressure appears to have increased since the late s and is now a recurring pressure on the population (Peltier et al.

). From to , the number of incidental captures of common dolphins has been estimated at between , and , (Peltier et al. ). This recurrent incidental capture pressure is a point of concern for the viability of the population. The mortality suffered by the population is in the order of . % to . % of its total abundance each year (Peltier et al. ). Mannocci et al. ( ) projected its abundance over the next years (after ) using the vital rates depicted in Figure . panel A. It appears that the population would not be viable. Data collected for population projections, and to quantify the number of incidental catches in absolute terms and relative to the total population abundance, are obtained from population monitoring programmes. The main types of monitoring associated with the common dolphin, as well as the main legislations and its current conservation status, are presented in the following section.

Figure .

: Common dolphin bycatch estimates from strandings. Estimation was done was made to cover the time series from to . It was done using direct drift modelling (black points, associated with the confidence interval in grey bars), and using reverse drift modelling (grey polygon). These estimates use the dolphin's drift trajectory corrected by the probability of being buoyant ( %), stranded (depending on drift conditions) and discovered ( %). Drift conditions are weather dependent and are modelled by the meteo France drift model MOTHY. For more information on the method see Peltier et al. ( ) from which the figure is taken.

. Conservation and monitoring

In the North East Atlantic and European waters

The conservation status of the common dolphin differs according to the instrument or regulation used. At the international level and independently of populations, the species is currently classified as being of least concern on the IUCN Red List of Threatened Species (Perrin ). Its international conservation status is that of a protected species under the Convention on International Trade in Endangered Species of Wild Fauna and Flora (CITES), which has listed it in Appendix II (Perrin ; CITES

). On a global scale, the species is therefore not listed as threatened but as a protected species.

The conservation status of the population in European waters is less favourable and conservation objectives associated with monitoring programmes have been set up on a European scale. The Habitat Directive (European Directive on the conservation of natural habitats and of wild fauna and flora, / /EEC) is an instrument of European environmental legislation. The objective of this instrument is to achieve a favourable conservation status for listed species and habitats, including the common dolphin (listed as a species in need of strict protection under Annex IV). In , the conservation status of the common dolphin population was assessed as "Unknown". Since , it is now assessed as "Unfavourable-Inadequate" in European Atlantic waters and "Unfavourable-Bad" within French waters (Murphy et al. ). : The MSFD sets the goal of good environmental status for European waters. This status is described by descriptors whose state is informed by criteria. Criteria of relevance to the population will be detailed in Chapter : The ICES Scientific Expert Group advises governments on how to achieve the conservation objectives of the Habitats Directive and the MSFD.

In order to promote the conservation status of the common dolphin in European waters, the European Commission adopted in the Bycatch regulation (Regulation no. / ) which encompasses requirements for bycatch monitoring schemes with on-board observers schemes to be set up for vessels ≥ m (length), and the implementation of pilot studies for vessels less than this size. It also provides for the application of bycatch mitigation measures using acoustic deterrent devices in areas and fisheries that are either known or foreseen to have high levels of bycatch (vessels ≥ m). This regulation is no longer in use and has been replaced by the new bycatch regulation (Regulation no. / ). It is within this framework of regulating catches and estimating their value that the Observer programmes, such as "Obsmer" , have been set up. However, they are not dedicated to bycatch and are carried out under the Direction Framework of the Common Fishery Policy . More details on this dataset and its use will be provided in Chapter . Data acquisition to achieve conservation objectives is also part of the Marine Strategy Framework Directive(EU MSFD ; Directive / /EC). It establishes a framework within which Member States shall take the necessary measures to achieve or maintain "Good Environmental Status" (GES) of marine environment. This European directive defines descriptors of good environmental status of ecosystems. Within these descriptors, criteria are also defined to enable the achievement of this good environmental status. Among these criteria, we find, for example, the need for biological indicators linked to the demography of the common dolphin. All the descriptors and criteria important for the common dolphin in terms of demography are detailed in Chapter . In support of established conservation policies and conservation objectives, international expert groups or agreements can operate. The International Council for the Exploration of the Sea (ICES ) is an Intergovernmental marine science organization which aims to synthesise and coordinate scientific work to achieve the legislative objectives set by the European Commission and provides science advice to environmental ministries and international agencies (such as OSPAR). ICES defines ecoregions (based on ecosystems) and statistical areas (based on important fishing areas) (Figure . ). With regard to the common dolphin population, ICES works on its status mainly in two working groups. The first is the Working Group on Marine Mammal Ecology (WGMME). The second is the Working Group on Bycatch of Protected Species (WGBYC). The Agreement on the Conservation of Small Cetaceans of the Baltic, North East Atlantic, Irish and North Seas (ASCOBANS ) aims to promote the achievement of European conservation objectives, in particular with regard to cetaceans, under the auspices of the United Nations. It focuses on the achievement of cetaceanrelated conservation objectives and provides Member States with recommendations for achieving these objectives. The agreement takes place within a defined area in European waters (Figure . ). It is within the framework of this agreement that the limit of . % for total anthropogenic removal has been defined, using the harbour porpoise (Phocoena phocoena, Linnaeus ) as a case study . On August , ASCOBANS proposed the Species Action Plan (SAP) for the North-East Atlantic population of common dolphins. This action plan sets targets to address the threats facing the population. It includes a series of research, mitigation and monitoring actions to improve the conservation status of the population. The SAP seeks to involve a variety of stakeholders, including national agencies, : ASCOBANS is an intergovernmental agreement focused on pressure mitigation and monitoring of small cetacean populations in European waters.

: https://www.ascobans.org/ en/species/threats/bycatch intergovernmental organisations, non-governmental organisations and regional bodies.

The Convention for the Protection of the Marine Environment of the North-East Atlantic (OSPAR Convention) aims to periodically assess the state of the marine environment, partly in European waters. This convention originated in with the Oslo convention.It has been ratified by governments and the European Union. The Convention entered into force on March . The OSPAR Commission provides a forum through which the contracting parties coordinate international cooperation on the protection of the marine environment of the North-East Atlantic. It shares common regulation and management of human activities (except fisheries). It facilitates the coordinated implementation of the MSFD, with the aim of achieving "Good Environmental Status" in European waters. It sets out recommendations for achieving the objectives of the Habitats Directive and the MSFD. In , OSPAR implemented the "OSPAR Intermediate Assessment " . This assessment aims at developing OSPAR's knowledg of the marine environment of the North-East Atlantic and its current status. It provides an update of the assessment and introduces some new indicators and assessment methodology. Within this framework, five regions have been defined within a maritime area (Margin-Figure . ).

In order to inform on the conservation status of small cetaceans and to assess their status in the light of the objectives of the Habitat Directive and the MSFD, a series of abundance surveys have been carried out. These abundance surveys are referred to as "Small Cetaceans in European Atlantic waters and the North Sea", (SCANS surveys and CODA which is a complement to SCANS II). These programmes take into account both aerial and ship-based monitoring. The first one was carried out in under the name SCANS (Hammond et al. ). Sampling was carried out mainly by ship from June to August (summer) and did not involve the Bay of Biscay. The next one was SCANS II in (Hammond et al. ). Sampling included the Bay of Biscay continental shelf and more extensive aerial monitoring from June to August (Summer). In addition to SCANS II, the CODA programme was carried out in to cover the non-continental part of European waters. The latest monitoring programme is SCANS III, conducted in (Hammond et al. ). This time, the continental shelf was sampled almost exclusively by aircraft, while the oceanic part was sampled by boat. Sightings of common dolphins within the framework of this programme are presented in Margin-Figure .  and are the data allowing to estimate their abundance and density as presented in Figure . above. Sampling took place between June and August (Figure . ).

Current management unit for the common dolphin in the North-East Atlantic

To manage the common dolphin population, several management units have been proposed in the North-East Atlantic Ocean (ICES a). On the basis of ecological tracers (stable isotopes, fatty acids, metal tracers, stomach contains), two management areas should be considered for the management of the common dolphin with oceanic and neritic ecological stocks (Caurant et al. ; Lahaye et al. ). Based on the low genetic differentiation throughout the North-East Atlantic, it is commonly admitted that common dolphins can be managed as a single management area (Murphy, Pinn, and Jepson ). In order to support the study and conservation of the common dolphin in the area covered, ICES WGBYC and OSPAR have defined the population as a single management unit based on genetic analyses (Murphy et

In France

The monitoring of the common dolphin population in French waters is mainly carried out in two ways. The first approach is a monitoring based on abundance surveys. In addition to the SCANS campaigns presented above, the seasonal SAMM abundance campaigns (Laran et al. ) were conducted (only in French waters) between November and February for the winter part and between May and August for the summer part (Figure . ). The objective of these campaigns is to explore seasonal changes in the abundance of small cetaceans in French waters to inform the Habitat Directive and the MSFD. These campaigns have highlighted a seasonal change in abundance of common dolphins in the Bay of Biscay. In winter, their abundance (estimated jointly with the stripped dolphins) was estimated at , ( % CI: , -, ) individuals compared to , in summer ( % CI: , -,

). These surveys provide information on biological indicators of abundance for common dolphins within the MSFD. ). When a dolphin washes up on the beach, it is reported to the Pelagis observatory (often by citizens). The observatory sends a volunteer to the site to establish an initial diagnosis on the cause of death and to take biological samples, which vary according to the animal's state of decomposition (Figure . and Figure . ). Stranding reporting rates handled by the RNE are considered stable since the s. The observed increases in strandings are therefore not due to a difference in reporting over this period but to other factors such as increased bycatch (Authier et al. ). The biological samples collected in this project were collected by the RNE.

We have seen that the common dolphin population is subject to major conservation issues in European waters. In particular, there is a need for demographic data within the framework of the MSFD, which must complement the abundance surveys carried out. Demographic data on this population are already available, including vital rates such as survivorship. However, survivorship is obtained using survival analysis methods that are not very flexible. In the next chapter, we will see a recent methodological development, which allows us to obtain survivorship curves by taking into account the effects of covariates and inter-individual frailty. : Unknown cause with scoliosis on calf (Unknown). For information purposes, in % of stranded dolphins are due to bycatch (according to external reviews). % are stranded without the cause being known. % from pathological reasons and % for unknown reasons. These percentages vary slightly after internal review. To see more detailed data: rapportecouhage . ©Observatoire Pelagis. Survival analyses are essential for estimating life-history parameters and producing life tables which are then used in the implementation of management strategies if required by the population status. Where longitudinal studies (e.g. capture-markrecapture monitoring) are not feasible, the only data available may be cross-sectional, for example in the case of marine mammal strandings. Survival analysis deals with data on age-at-death (i.e. time-to-event) and allows for the estimation of survival and hazard rates assuming that the cross-sectional sample is representative. Accounting for a bathtub-shaped hazard, as expected in wild populations, has historically been difficult and has required specific models. We identified a simple linear regression model with individual frailty that can fit bathtub-shaped hazard, account for covariates, allow for goodness-of-fit assessments, and give accurate survival estimates in realistic settings. We first performed a Monte Carlo study and simulated age-at-death data to assess the precision of the estimates in relation to the sample size. We then applied the framework to a handful of case studies drawn from published studies of marine mammals, a group with many threatened and data-deficient species. We found that our framework is flexible and accurate in estimating survival with a sample size of . This approach holds promise for obtaining important demographic information on data-poor species.

. Introduction: About survival analyses

Minimising

Conservation needs demography which is logistically challenging. human impacts on wildlife is a major challenge in the Anthropocene, where human-induced pressures are both geographically and temporally far-reaching, and have an acute impact on biodiversity (Bongaarts ). Long-lived species are particularly sensitive to these pressures, especially from a demographic point of view. Their demographic trajectory (e.g. extinction) can also affect the functioning of ecosystems, especially in the case of top predators (Beschta and Ripple ; Ritchie et al. ) since they can act as keystone species for entire ecosystem (Heithaus et al. ). Assessing the demographic viability of long-lived species in the face of current and future pressures is necessary for their proactive conservation and is sometimes used to complement indicators of abundance status; but such an undertaking can be hampered in practice by the logistical challenges of monitoring and collecting relevant data at appropriate spatio-temporal scales.

The Cross-sectional monitoring may be logistically easier than longitudinal monitoring, depending on the species. classical method of assessing the viability of a population consists firstly of constructing a life table in which the mortality and fertility rates of a cohort of individuals are given at each (st)age of their life cycle (Caughley ). Knowledge of mortality at each (st)age allows the study of associated hazards, the estimation of vital rates, such as (cumulative) survival, and the projection of the population trajectory over time using age or stage matrix models (Leslie ; Lefkovitch ; Caswell ). Historically, the data on age-at-death used to construct life tables were obtained by cross-sectional monitoring (e.g. Gompertz

): a sample of the population is taken at a specific time and the observed age structure is assumed to be similar to that of a cohort of individuals if we had been able to monitor them individually from birth to death. This cross-sectional monitoring allows for a quick demographic assessment (Margules and Austin ; Boyd, Bowen, and Iverson , pp. -) whose accuracy, however, relies on the assumption that the cross-sectional monitoring does approximate a longitudinal monitoring. An essential process to observe when studying demography, regardless of the type of monitoring design, is survival. Survival is an essential component of population demography as it contributes strongly to the growth of a population and its life history, particularly for large mammals such as cetaceans (Heppell, Caswell, and Crowder ). Annual survival has been identified as more important than annual fecundity in shaping the fitness of species with life spans greater than one year (Crone ). The study of survival in a given population can be called "survival analysis". Historically, survival analysis ranged from the collection of demographic data to statistical methodologies applied to characterise the time-toevent process (e.g. time to death) and has been transposed into ecology from epidemiology and industry (Kleinbaum and Klein ). The Survival is the essential component that determines the value of the growth rate of a population. The analysis and modelling of survival is very important in demography.

time to events can also be considered as sequential while characterising the time between several events (Skalski, Ryding, and Millspaugh ). The two main frameworks for measuring survival are annual survival (i.e. year-to-year frame) and survivorship (i.e. birth-to-death frame). The latter is the most informative and can be applied as a survivorship curve, or function, to an entire cohort (Ferguson ). To measure survival, several modelling or estimation methodologies have been developed, depending on the monitoring design. In this chapter, I briefly review each modelling and estimation approach that has been developed to measure the survival of wildlife populations. In the case of longitudinal monitoring, developments allow the effect of various variables on vital rates to be quantified at the population and individual level. Developments associated with cross-sectional monitoring do not currently allow such flexibility. The review begins with the approaches associated with longitudinal monitoring models, as these are the preferred approaches for obtaining population vital rates because of the less stringent underlying assumptions and greater flexibility. Next, Several methods of survival analysis exist. They depend mainly on the type of monitoring (longitudinal or cross-sectional). In the case of crosssectional monitoring, they depend on the inclusion of modelling parameters (non-parametric and parametric).

I review the estimation and modelling of survival inherent in cross-sectional monitoring, approaches that require more restrictive and sometimes unrealistic assumptions (e.g. a stationary age distribution of the sampled population; Caughley

). However, they are of interest for rapid demographic assessment, for studies of populations in crisis (i.e. which paradoxically may not satisfy the stationary age assumption) and are often the only applicable methodologies for studying the demography of mammalian populations.

The

We have developed a parametric approach based on the Reed ( ) model which has the following advantages: inclusion of covariates, inclusion of random effects, evaluation of goodness of fit and individual heterogeneity parameter. methodological developments associated with estimating mammalian population survival from age-at-death data are limited. They do not allow for easy estimation of covariates or random effects and do not allow for easy comparison of the resulting vital rate values. It would be interesting to develop a new approach to survival estimation that overcomes these limitations. There are two ways of estimating survival from age-at-death data (for the sake of clarity, I do not discuss the Cox proportional hazard model which allows the use of truncated data). The first estimation method is called non-parametric. This method is mainly limited in the case where the number of data is small. The second estimation method is called parametric and is mainly limited by its implementation and the impossibility of testing the factors and comparing the quality of the estimates. It is in this context that we have developed a new parametric flexible linear regression approach to facilitate the estimation of survival in cross-sectionally monitored populations. Our work is based on Reed's model (Reed ) which admits a bathtub-shaped hazard rate curve (i.e. representative of the long-lived mammalian mortality pattern). We believe that this approach is an important and innovative contribution to the literature of cross-sectional survival analyses, and thus to the study of associated population dynamics. Its strengths are the consideration of the effect of covariates, the facilitation of goodness-of-fit measurement, and the consideration of the effect of random variables that can highlight key temporal processes in relation to cohorts, as I will show in the Chapter .

. Methodology: Designs and modelling

Survival models based on longitudinal monitoring

Longitudinal

Longitudinal monitoring was essentially a means of estimating abundance. They were then used to estimate vital rates. studies, in which individuals and cohorts are monitored from birth to death (Figure . for the associated lexical diagram) are, in wildlife ecology, a by-product of capture-recapture studies. The initial motivation for the latter was the accurate estimation of abundance in open populations when detectability is less than perfect (Seber ; Cormack ; Jolly ). Models (e.g. the Cormack-Jolly-Seber model) fitted to these life history data can provide accurate estimates of survival that can be fed into population matrix models (e.g. Fujiwara and Caswell

). Since the s, the focus of capture-recapture studies in wildlife ecology shifted from abundance to survival estimation (Lebreton et al. ) and the inclusion of individual-level covariates, which has paved the way for a better understanding of life history evolution and natural selection in nature (Cam ). Models for estimating survival (initially abundance) use either the recovery of live animals ("live recapture") or dead animals ("band recovery"). Some models may also combine the two data sources (J. D. Baker, A. Westgate, and Eguchi

).

The objective of capture-recapture modelling is to develop probability-based models characterising the biological processes that shape the capture history data (B. K. Williams, Nichols, and Conroy ). The common approach to each modelling mode is to model the subsequent entries of an individual, after the initial capture, as a function of parameters associated with both sampling and actual population change (B. K. Williams, Nichols, and Conroy ). Survival probabilities are then estimated by focusing on the loss of individuals within the population (Lebreton et al. ). There (Pollock ). The multi-state modelling approach allows some stochasticity to be incorporated between transition states. The states can be considered for example as different locations or phenotypes. This modelling approach was developed to study transition states. It was first introduced by Arnason (Arnason ). The inverse time modelling approach developed by (Pollock, Hines, and Nichols ) allows the recruitment process to be deduced by considering the recaptures in an inverse temporal order. All these approaches use capture-recapture data from live animals. Approaches have also been developed to use capture-recapture monitoring data incorporating the recapture of dead individuals. This is the case, for example, with survival modelling based on band recoveries, which is of interest for species that are harvested (Burnham ). A common feature of all these modelling approaches is that they are based on the use of data obtained through longitudinal monitoring. They require individuals and cohorts to be followed throughout their lives. For species where this may be difficult, other approaches have been developed. Approaches using the same data but taking into account only certain life stages or even using only abundance data have been developed to avoid the cost of long-term monitoring.

Survival models with longitudinal monitoring and abundance data

From an applied conservation perspective, the obvious drawback of capture-recapture studies is the time and man-power required to collect the data, especially for long-lived species. E. R. White, Nagy, and Gruber (

) recently proposed a new development that allows to avoid birth-to-death monitoring at the individual level. This Survival models have been developed to relay partly on longitudinal data or to get rid of it and use abundance data.

is a physiologically structured model with age class as the structuring variable. Although the model is agestructured, only the juvenile population is observed. From juvenile mortality and adult fecundity, it is possible to infer adult mortality. However, this requires monitoring the juvenile part of the whole population in order to model the overall population dynamics. An alternative to long-term individual monitoring for estimating vital rates is the use of count data. This approach may involve building an N-mixed model based on knowledge of individual states. Zipkin et al. ( ) has proposed a development that allows the data to be taken into account even if the individual stage is unknown. Another way to obtain vital rates with count data, in the form of time series, is to conduct an inverse modelling approach: this method also has the advantage of considering state data at the individual level (E. J. González, Martorell, and B. ). Although these approaches are promising for elusive species, they remain difficult to apply to highly mobile ones. The time lag between data acquisition and the urgency of mitigation on conservation decisions can be acute, suggesting an interest in cross-sectional data which have the added advantage of being applicable to species where individual identification is difficult (B. K. Williams, Nichols, and Conroy

).

Survival models with cross-sectional monitoring

Long-lived

Elusive species are impossible to monitor on an individual basis and abundance data can be very scarce. It is therefore necessary to use age-atdeath data. In the case of marine mammals, these are collected from stranded individuals.

, mobile and elusive (weakly marked) species are difficult to monitor longitudinally. This is particularly true for species that live in a large and sometimes three-dimensional space in the case of the marine environment. Instead, data on these species can be collected through cross-sectional monitoring. One or more samples are taken during a time series (snapshots; Figure . ). Sampling of the study population is carried out as described in . . Once age-at-death data have been collected, they can be used to calculate the survival of the population. This calculation relies heavily on modelling choices. Assuming that cross-sectional ageat-death data are available and representative, ecologists are now faced with other choices, such as non-parametric or parametric modelling.

Non parametric modelling

The Non-parametric survival modelling requires a large amount of data. It does not allow to obtain survivorship in the case of elusive species. non-parametric modelling approach does not assume any functional form of the survival curve. This approach allows the raw observations (i.e. data) to dictate the shape of the survival curve. Two non-parametric approaches are commonly used in wildlife demography. The first was developed by E. L. Kaplan ) and is referred to as the Kaplan-Meier estimator, or product limit estimator. It consists of monitoring a cohort of individuals, whether or not they entered the study at the same time, and reporting the remaining number of individuals at each age. A fictitious cohort can be reconstructed from the ages-at-death. The first study to transpose this approach from epidemiology to wildlife demography was conducted by Pollock et al. ( ). They conclude that this estimator is flexible and useful in wildlife studies and have optimised it to allow successive entries of individuals in the survival estimate. This approach considers the maximum survival ( ) at time (discrete time approach): (0) = 1.00. At each age-at-death interval, the survivorship decreases to until the last animal of age die: ( ) = 0.00. The survivorship at each time can be estimated as follows with = number of animals alive (at risk) at time and = number of animals that died in the interval between time and + 1:

( ) = - ( . )
A limitation of this approach is that it performs poorly with small amounts of data because the number of points on the survival curve depends directly on the amount of data. If the number of data points is small, the curve is not smooth and has an unreliable stepwise appearance. An alternative was developed by Nelson ( ) and Aalen ( ). This approach is known as the Nelson-Aalen estimator. The survivorship is modified based on the cumulative hazard function. The approach has better properties regarding small sample size. The cumulative hazard function is constructed as detailed here with = number of animals alive (at risk) at age and = number of animals that died in the interval between age and + 1:

ℎ( ) = =0 ( . )
The corresponding survivorship function at time is:

( ) = -ℎ( ) ( . )
Despite its ability to accommodate smaller datasets, it is also limited for the types of datasets we are interested in studying the survival of animal populations. The use of these methods, in particular the Kaplan-Meier method, appears to be more suitable for describing "observed" survival and not estimated by modelling. The use of a parametric modelling approach is more suitable with small samples to estimate survival and characterise the mortality process of a fictitious cohort in a population over time (Skalski, Ryding, and Millspaugh ). It has the advantage of providing smoothing and summarising the data into a small number of parameters. Even with a small sample size, parametric modelling allows smooth survival to be estimated and parameters to be compared (Kleinbaum and Klein ).

Parametric modelling

The Parametric survival modelling allows the use of small data sets. But it requires the definition of a type of mortality curve.

parametric modelling approach requires the choice of a shape for the mortality curve or hazard rate. The accuracy of the approach depends primarily on the suitability of the shape of the mortality curve for the species whose survival and mortality are to be modelled. The challenge is to find a parsimonious model that takes into account the available data without fitting the analysis into a convenient but not necessarily realistic curve shape. Each parametric model is associated with a probability density function (i.e. pdf), a cumulative distribution function (i.e. cdf) and a survivorship function (i.e. ( )) associated with a hazard rate function as follows : ℎ( ) = long-lived animal species, a realistic model should provide the so-called "bathtub curve", which is characterised by high juvenile mortality, followed by lower and rather constant adult mortality and finally a late increase due to senescence (Siler ; Choquet et al. a). The hazard should decrease as the animal adapts to its environment and becomes stronger, remain constant in mid-life when the animal is in its prime, and increase with senescence (Emlen ). Although there are models to accommodate this bathtub pattern (Siler ; Heligman and Pollard ), they can be difficult to fit (Saavedra ). Moreover, their goodness of fit is hard to measure and compare. Finally, they do not allow for testing the effect of various factors on survival and mortality.

Our parametric modelling approach

Our

Reed's model can fit this hazard curve. It can do so by including a parameter for individual heterogeneity. We believe that this model can be integrated into a parametric approach.

aim is to develop a parametric approach for the construction of life tables in a data-poor context, with long-lived vertebrate species in mind. We will first present a simple regression modelling framework for the analysis of age-at-death data (with or without right censoring) that can accommodate mortality patterns such as constant, increasing/decreasing, unimodal or bathtub-shaped. Importantly, this framework allows for the transparent inclusion of individual-level covariates and random effects. We then perform a Monte Carlo simulation study with five biological scenarios to evaluate our framework, focusing on the accurate estimation of survivorship and mortality. Finally, we illustrate our methodology on a handful of real-life case studies, focusing on marine mammals. This group includes many elusive and long-lived species that are threatened (Avila, Kaschner, and Dormann ), data-deficient (Parsons ) and difficult to study with a longitudinal approach. The most common age-at-death data for marine mammals are obtained from strandings (i.e. crosssectional monitoring). Teeth are collected from individuals found dead on the shore and are then processed to allow the reading of growth layer groups that are indicative of the age of the stranded animal (Mannocci et al. ). The stranding filter may introduce a selection bias of individuals between the natural population and the age-at-death dataset (Barlow and Hohn ). However, stranding samples are a source of age data from which vital rates can be obtained (J. D. Baker, A. Westgate, and Eguchi ). In this framework, several underlying assumptions are associated with the developed approach:

Baseline assumptions (use of this parametric approach)

. Each age-at-death is known and correctly recorded . The cohort is representative of the overall population . The fate of each individual is independent . The shape of the survivorship and mortality distribution are known and well chosen to represent the life history of the species

Definitions

Survival analysis deals with the analysis of the timing of death (E. L. Kaplan

Pr( ≤ < + d ; > ) d ( . )
With parametric models, the hazard rate can be expressed in terms of the probability density and survivorship functions:

ℎ( ; ) = ( ; ) ( ; )
, with ( ; ) > 0 ( . )

Data simulation scenarios

We considered five biological scenarios (i.e. five different life histories) corresponding to different patterns in survivorship and the underlying hazard (Margin-Figure . ):

. a unimodal distribution of age-at-death corresponding to a unimodal hazard, whereby mortality risk increases in early ages, peaks and decreases in late life; . a mixture of two hazards corresponding to either a unimodal (a) or bimodal (b) distribution of age-at-death; . a bathtub-shaped hazard due to individual frailty, that is individual-specific risk of mortality; and . a bathtub-shaped hazard with an additional bump in early life due, for example, to an additional source of mortality.

Each scenario is associated to survivorship and hazard functions (Figure . ). : low juvenile mortality, a high adult mortality and no senescence. a: low juvenile mortality and a higher adult mortality plateau.

b: high juvenile mortality and a lower adult mortality that decreases in a linear fashion.

: high juvenile mortality, a lower adult mortality and senescence (bathtub shaped hazard).

: high juvenile mortality and a high adult mortality, with a transient dip in mortality risk between these two life-stages. The next supplementary material that will be referred in margin notes will always be available at this URL. To find the supplementary material mentioned, see this URL

Statistical Analysis of age-at-death data

We used parametric models M of age-at-death data to estimate hazard and survivorship rates. We assumed that the exact timing of death is available, but our framework can easily accommodate censoring (that is cases when death is known to have occurred before or after measurement). Our framework consists in modelling the logarithmic transform of in a regression framework (location-scale model The choice of the statistical distribution for the residuals determines the shape of the underlying hazard rate. We considered three different choices, each corresponding to a model .

M 1 : ∼ N(0, 1) and 1 = 0 ( = +∞)
This model assumes a log-normal distribution for , which corresponds to a unimodal hazard curve. M 2 : ∼ G(0, 1) and 1

= 0 ( = +∞)
This model assumes a Gumbel distribution for the residuals , which corresponds to a Weibull distribution for . It is known as the Accelerated Failure Model. The hazard rate is monotonic: it can be constant, increasing or decreasing depending on the value of . M 3 : ∼ N(0, 1) and 1 > 0 This choice leads to assume a normal-Laplace distribution for log , which induces a flexible hazard curve depending on the value of (Reed ). In particular, the hazard can be a bathtub-shaped, as expected for example for long-lived species of vertebrates in the wild (Choquet et al. a).

: See SI Text .

Model/Scenario a b ) with a handful of parameters = ( , , ) to accommodate a large diversity of survivorship and hazard curves . One attractive feature of this framework is the seamless incorporation of individual-level covariates in equation . (Reed ):

M 1 + - - -- M 2 - - - -- M 3 - - - + -
log = + =1 + × - ( . )
An important restriction of our approach is that only timeinvariant individual-level covariates can be included. It is also impossible to take into account changes in states within the life history trajectory at the individual level. However this approach covers interesting cases such as sex-differences in survival, differences due to geography or cause of mortality (that is comparing different populations of the same species).

Monte Carlo study

Our aim is to carry a Monte Carlo study (e.g. T. P. Morris, I. R.

White, and Crowther

) to investigate whether our modelling framework can provide accurate survivorship and hazard rate estimates from age-at-death data. We considered 5 biological scenarios to cover a diversity of realistic mortality patterns. For each scenario, we simulated 100 datasets of sample size 100, 200, 300, 400, 500 and 1000 to provide recommendations on the minimum sample size required for accurate estimation. Data simulation was carried out in R version . . (R Core Team

) using base functions such as rnorm and rexp. Each simulated dataset was then analyzed with our framework that considered 3 parametric models. Crucially, we considered scenarios for which the true model was not among the set (Table . ). In other words, we assessed the performance of our framework under the possibility that none of the candidate models is correctly specified with respect to the data at hand. For example M 2 is a Weibull model, which is widely used in survival analysis (Kleinbaum and Klein ) it is biologically unreasonable for marine megafauna and top predators as a model over their entire lifetime because it cannot accommodate the expected bathtub-shape hazard. Model fitting was done with software Stan version . (Carpenter et al. a) called from R via the library rstan (Stan Development Team

). Three chains were run with a warm-up of 500 iterations, followed by an additional 1000 iterations. No thinning was performed but the delta parameter of the NUTS algorithm was increased from 0.80 to 0.95 to avoid divergent transitions and the maximum tree depth increased to 15. Parameter convergence was assumed when its ˆ statistics was lower than 1.10. Upon convergence, the 3 chains were pooled to obtain a sample of ≈ 1000 values from the posterior distribution.

Model fit was assessed with the Widely Applicable Information Criterion (WAIC, (

)), computed with R package loo (Vehtari, Gelman, and Gabry ). Survivorship estimates from each model were computed from the posterior distribution of parameters, and visually compared to the true survivorship curve and a non-parametric (Kaplan-Meier) estimate. Because we carried out a simulation study, the true survivorship curve was known and could have been used to compute the Root Mean Squared Error (RMSE). However, in practice, this is not the case and we chose instead to compute RMSE with respect to the non-parametric Kaplan-Meier survivorship estimates. The latter were thus assumed to represent the best estimates available to researchers, and the aim was to assess whether a parametric model could provide a fit as good as that from a non-parametric approach.

RMSE M = ( ˆ ( ) -ˆ M ( )) 2 ( . )
In order to assess model selection and model check, it is necessary to determine if the models represents well the data. (Conn et al. ) recently reviewed some ways to proceed. We choose to do a prior predictive check to test for adequacy between models and time to event data . We also did a posterior predictive check to see whereas data simulated through the fitted models are similar to that observed from the (Barlow and Hohn ) dataset. The comparison is done with both the true Kaplan-Meier survivorship curve and the ones from posterior simulated datasets.

Our focus was on accurate estimation of survivorship, and thus we assessed goodness-of-fit by comparing the expected mean survivorship under each model to the observed Kaplan-Meier estimates. This focus was in line with downstream use of such estimates in matrix population models: here the salient statistics we want our model to reproduce (Gelman ) is the survivorship function.

Our study design is summarized in Table and

Applications

We analyzed real datasets from published case studies (Table ) within our framework, and compared the estimated parametric survivorship curves with the non-parametric Kaplan-Meier one. All species are marine mammals except the spur-thighed tortoise (Testudo graeca, Mertens

) which was included to compare our approach with that of (Rodriguez-Caro et al.

) in a data-poor context for conservation. It is important to precise that data from (Rodriguez-Caro et al.

) were obtained from live animals, still alive at the time of measurement (Sanz-Aguilar et al.

; Rodrıguez-Caro et al.

). However, we used these data as if they were age-at-death data and ignored right-censoring.

. Results: Simulation study

Across all scenarios and sample size, parameter convergence (assessed with ˆ ) was not equivalently reached depending on the model and scenario . M 1 always converged very easily. In contrast, M 2 and M 3 were not as good as M 1 to converge with 1000 iterations (500 as warm up). More precisely, convergence for M 3 was difficult for given this configuration . An increased in the number of iterations per chains (here from to ) solved the problem.

Hazard rate estimation

Estimated hazard curves are shown in Figure . for each combination of scenario, model and sample size. Models M 1 and M 3 were the best fitting ones in the first and third scenarios respectively (i.e. positive control, Table . ): estimates were accurate and precision increased with sample size. For scenarios a, b and , all estimates were biased, and precision increased with sample size. In contrast, model M 2 never provided accurate hazard estimates (i.e negative control). The same results were obtained when covariates were included.

Survivorship estimation

Estimated cumulative survival curves are shown in Figure . for each combination of scenario, model and sample size. Across all scenarios and sample sizes, estimates from model M 3 were the most accurate . Precision increased with sample size. In particular, survivorship rates estimated with M 3 were very close to Kaplan-Meier estimates for sample size ≥ 300. This sampling size also provided a good confidence interval precision with a maximum width of . % while estimating survivorship . Predictive ability, as measured with WAIC, was the greatest for model M 3 : it was consistently ranked first across each combination of scenario and sample size, even for scenario where model M 1 was the true data-generating mechanism .

Covariate Effects

Covariate effect estimation is summarised on Figure .  as a difference between survivorship rates. Model M 2 could not estimate difference in survivorship. Model M 1 is only effective for both scenario and a. M 3 could estimate accurately covariate effects when the difference is expressed as an unimodal pattern (i.e. scenario , a and ). The precision of the estimated effect increased with sample size for models M 1 and M 3 . With small sample size, sign errors on the effect of covariate was possible but disappeared with sample size ≥ 300.

Case studies

Estimated survivorship curves from published dataset are plotted against the Kaplan-Meier curves in Figure . for each model. Model M 3 was the most flexible: it provided the most accurate estimates for each dataset. Uncertainty, as measured with 80% credible intervals were narrower with model M 3 , and overlapped most with Kaplan-Meier estimates. Model M 3 consistently had the lowest WAIC. In the handful of case studies where covariates were available, a similar pattern arose. Including covariate can be expected to account for more variation in the data, and a lower WAIC. For both M 1 and M 2 , this was indeed the case. For M 3 it was only true with the (Murphy et al.

) dataset . The estimated covariate effect for both (Barlow and Hohn ) and (Kesselring et al.

) case studies is summarised on Figure . . Models M 1 and M 3 both estimated a survivorship difference between each covariate. As it is the case without covariates (Figure . ), the M 3 curve fits the Kaplan-Meier estimate better. However, there is a discrepancy between both, for each covariate. M 3 is able to spot a difference depending on covariate, but is not able to perfectly fit the Kaplan-Meier estimate. . Discussion: Implications of this new cross-sectional parametric approach

We assessed the ability of a simple linear mixed model to estimate hazard and survivorship rates with cross-sectional age-at-death data. We used Monte Carlo simulations to investigate the accuracy and precision of estimates across a variety of mortality patterns and several sample sizes. We compared the performance of three different models and found one model, the model M 3 , to be consistently better at predicting survival, even when misspecified.

Age-at-death data and sampling bias

A

To use age-at-death data, we assume that the population is under a stationary age distribution which is rarely the case in nature... crucial but implicit assumption of the cross-sectional approach to survival analysis is that the sample is representative of the population as a whole, particularly with regard to age structure (Caughley ). This assumption relates to the dataset, not to the modelling. The way in which the age-at-death data are collected is therefore crucial in supporting this assumption. Approximate protocols involving, for example, the recording of dead carcasses, hunting bags or population censuses must meet the representativity premise. However, it is possible that an implicit process (e.g. bycatch) shapes the observed age frequency (e.g. from stranded animals). In this case, it is necessary to explore some potential age-dependent selection bias in the population (Barlow and Hohn ). With stranding data, the assumption of a stationary age distribution needs to be supported by ancillary data, but ultimately it is likely to remain a working hypothesis on which any cross-sectional methods will be based. In the case of cetaceans, many species are not amenable to study with a longitudinal design, and strandings remain an important source of information (e.g. Murphy et al.

), particularly demographic information (Saavedra ; Ferguson, Stirling, and McLoughlin ). When longitudinal studies are not possible, there are very limited options to obtain demographic information. In some cases, a comparative approach may be possible where information on species with similar life histories can be leveraged (Caswell et al. ; Hashimoto et al. ). This choice is essentially based on the assumption of similarity between species.

... so we should use auxiliary data to interpret results. But if age-at-death data is the only source of data available, so it should be used whenever possible.

Even if comparative data are available, the ecological context may be too different to justify this approach in some instances. Strandings may be the only available source of data, and even if they are suspected to suffer from some selection bias, a pragmatic approach to conservation requires to use them (Boyd, Bowen, and Iverson ), keeping in

: SI Text .
mind the inherent limitations of these data.

Many

We can inform conservation policies and management strategies with demographic parameters obtained from age-at-death data.

conservation instruments specifically require the use of the "best available science" (according to the Marine Strategy Framework Directive EC / in Europe). The operative expression "the best available science" can be broadly understood as what lies at the intersection of state-of-the-art methods, good data and accurate knowledge. In practice, there may be a hiatus between the canonical approach that should be, and the pragmatic one that can be pursued at the time conservation actions need to be decided. A conclusion of a species being data-deficient often leads to the doldrums with respect to conservation decisions (Parsons ). Likewise, the many uncertainties that can affect any scientific studies can easily lead to inaction regarding the implementation of conservation measures (Ascher ). It is because we are starkly aware of these limitations that we carried out this study to identify a pragmatic approach to estimate survivorship rates from age-atdeath data. Crucially, all the models we used are conditional of the underlying sample being representative of the population it is taken from. Granting this assumption, we identified a simple model from (Reed ) to obtain accurate estimates.

Prediction Accuracy and Models' Goodness of fit

Traditionally,

The most commonly used models for obtaining marine mammal survival from age-at-death data are the Siler and Heligman Pollard models.

estimating survivorship rates with crosssectional data was done with parametric models such as the Siler or Heligman-Pollard models, and this is still the case for some marine mammals (Mannocci et al. ; Siler ; Heligman and Pollard ; Huang et al. ). These models aim at reproducing a bathtub-shaped hazard curve, but need several parameters to do so. Furthermore, they can be difficult to fit although new tools have been developed to use these models (Saavedra ). A remaining challenge with these models is to assess goodness of fit, to incorporate individual-level covariates and perform model selection. The simple parametric form of (Reed ) for analyzing the logarithm of age-at-death data provides a bathtub-shaped hazard with a linear mixed model, the current workhorse of ecologists (Bolker et al. ).

These models can be used to obtain vital rates but their use is limited. Their construction is not simple and they do not allow for the estimation of vital rates as a function of various factors.

This linear mixed modelling framework allows for a seamless incorporation of covariates, and to use standard tools for model selection and goodness-of-fit assessments. In other words, model M 3 brings back survival analysis within the comfort zone of ecologists. We harnessed the simplicity of (Reed ) to conduct our Monte Carlo study, and found that the model suggested by (Reed ), our model M 3 , was very accurate in predicting survivorship (i.e. it consistently had the lowest RMSE), even in cases when it was not the true model behind the data . This result : SI Figure .   is important as it suggests to start building model of increasing complexity from M 3 (as it is done after Chapter ) and use tools such as WAIC (Gelman, Hwang, and Vehtari ) to balance model complexity with prediction accuracy. From

Our approach correctly estimates survivorship with age-at-death data and correctly approximates hazard with age-at-death data.

our simulations, we can recommend a sample size of at least 300 individuals to obtain accurate and precise estimates of survivorship, from which age-specific survival estimates can be derived This sample size recommendation is practical and realistic (e.g. Mannocci et al. ; Murphy et al. ; Kesselring et al. ). This recommendation is mostly to obtain precise estimates, but may be relaxed in some cases where only sparse data may be available (data-poor context hereafter). Estimates from a simple linear model with an individual frailty term, were accurate, if imprecise, with a sample size as small as 100. The possibility to use sparse data is critical as it can help conservation of elusive species, such as marine mammals, many of which being classified as data-deficient (Parsons ; Schipper et al. ). Some of the case studies presented in this paper support this statement, although, with so few data, including covariates in the model will be difficult or will require great care (see for example Cox et al. ). Rodriguez-Caro et al. (

) recently provided an approach to estimate survival in data-poor settings using inverse modelling, also to obtain accurate estimates of demographic rates. With the latter, population matrix models can then be used to assess population dynamics and the fate of populations over time (Caswell ).

Hazard and Frailty

A

The inter-individual frailty parameter that allows the mortality curve to have a bathtub shape should not be interpreted alone. It is not yet a biological indication in its own right.

linear mixed model can fit age-at-death data very well: this ability comes for the individual frailty term, which corresponds to an individual random effect in the mixed modelling framework. Individual frailty in statistical models translate the empirical observation that two similar individuals (with respect to observable features of their phenotypes) can nevertheless differ markedly in their longevity (Cam, Aubry, and Authier ; Kannisto ). The parametric form (i.e. exponential) for individual frailty gives extra flexibility to the model, and can accommodate a bathtub-shaped hazard curve (Reed ). However, in our simulations, we found that estimating hazard rates was more difficult than estimating survivorship rates (Figures . & . ). Thus, even though we identified a model (M 3 ) for reliable and accurate estimation of survivorship, the same model was less reliable with respect to hazard. In other words, the individual frailty term in our model M 3 should not be over-interpreted, and is probably best seen as a statistical device for robust estimation. Hazard estimation is a difficult statistical problem (Watson and Leadbetter ), for which there are better tools available, especially non-parametric ones (see for example Hanson and Jara ), than the simple parametric approach we considered in this study. Non-parametric approaches to infer the shape of the hazard curve are data-hungry: Hanson and Jara ( ) use Bayesian non parametrics, which is better described as a model with a massive number of parameters (Hoff ). The traditional Kaplan-Meier approach is truly a non parametric approach but it does not give access to the underlying hazard, and give rough (that is, non-smooth; Figure . ) survivorship rates with small sample size. In data-poor settings, parametric modelling remains attractive because it has interpretable parameters (e.g. individual frailty) and because these parameters can smooth out noise in data, yielding more precise estimates if the model is at least approximately correct, or more pragmatically, if it is grounded in theory (e.g. bathtub-shaped hazard for natural populations) and cannot be rejected from a goodness-of-fit test. It is precisely in this data-poor setting that we envision our parametric modelling approach to be most useful. Because the approach boils down to linear mixed effects modelling, great flexibility in model specification of additional random effects (e.g. year effects, sex-specific frailties) is possible provided there are enough data to offset the increase in complexity.

Benefits and limits

We Our model performs well in case of unimodal age-at-death distribution... believe our approach is very valuable to estimate survivorship from sparse data but may be inadequate for bimodal age-at-death data distribution. A bimodal distribution of age-atdeath data may be a sign of selection in data collection (Barlow and Hohn ). Rather than consider the sampling as biased, our approach can accommodate the selection phenomenon through two ways. Firstly, it is possible to consider the selection bias as a covariate in the study if one such covariate is available (e.g. Bycatch index). If no such covariate is available, it may be possible to build a mixture of two models (e.g. M 1 and M 3 ) in order to take into account an additional mortality on some age classes. However, such a development requires to conduct its own simulation study which is beyond the scope of this study. Since our approach is based on linear regression, it is straightforward to expand the model (e.g. random year effects, mixtures) while using familiar and well established methods for model selection (e.g. WAIC) and assessment of model fit (e.g. posterior predictive checks, 2 statistics). These features are assets, and suggest that our approach is complementary to existing ones (e.g. Saavedra ; Siler ; Heligman and Pollard ) to estimate vital rates in some data-poor species. On the basis of the size and morphology of the animals, the largest can be defined as "megafauna" (Estes et al. ). Large animals, usually long-lived vertebrates, refereed as "megafauna" are generally elusive. When associated with the marine environment (referred as "marine megafauna") it may be more difficult to obtain data and study these animals given their living environment. This is particularly true for demographic data and studies. Given their functional importance (Pimiento et al. ), there is a need to inform the biology and the demography of these species.

... and can allow joint estimation of fertility in a joint modelling approach.

The approach presented here may be of interest to study the demography of marine megafauna, in particularly marine mamals such as the common dolphin in the North-East Atlantic. And this is what we will see in the next Chapter. The common dolphin is one of the most abundant and widespread odontocetes in the world. This species is very common in the Atlantic Ocean as it is found from the centre to the north of the Atlantic. The North-East Atlantic population, distributed from Gibraltar to Norway, is considered a management unit by the OSPAR Commission. This species is a top predator in European waters, supposedly a keystone species. The current status of this population (and its future trend) is a key issue in the MSFD for descriptors and . Since , the number of common dolphins found stranded on European coasts, notably in France, has been steadily increasing. Among these individuals, there is a high proportion of individuals showing signs of capture in fishing nets. Previous demographic studies have shown that this population is not viable. To study its viability, sampling was carried out in the conventional way (collection of all available data on age-atdeath and reproductive-status-at-death). The analysis of these data was carried out using standard survival analysis methods. In the previous chapter, we saw that it is possible to use a new approach to survival analysis that allows more flexibility than conventional methods. In this chapter, we will see how it is possible to combine this new approach with a sampling procedure that has never been used before to study the demography of a marine mammal species from a cross-sectional monitoring. By combining these two methods, we will see that it is possible to quantify more precisely the vital rates of the North-East Atlantic common dolphin population. It is also possible to highlight the effect of by-catch pressure on this population from a demographic point of view. Finally, it is also possible to highlight the presence of a cohort effect showing a modification of the value of the vital rates during the temporal period considered. The population does not seem to be viable and its condition seems to have deteriorated over the time period considered. Anthropogenic mortality pressure from bycatch appears to have a negative effect on the survival of the population. As the individuals used in this study are specific to the Bay of Biscay, it is difficult to generalise the results to the OSPAR management unit. However, the vital rates produced can inform the MSFD descriptors.

North-East

. Introduction: A need for demographic knowledge

The common dolphin is one of the marine mammal species assessed under the MSFD, in particular under descriptors and which include marine mammals (Table . for details of the descriptors and their relevance to marine mammals; M. B. Santos and Pierce ). This marine mammal species is found in all temperate and tropical parts of the oceans (Perrin ). Individuals are organised into distinct populations, or stocks, which may be associated with different management issues (Natoli et al. ; Amaral et al. ). The North-East Atlantic population is a population with high conservation stakes today, particularly with regard to the MSFD (European Parliament

), since it is the most abundant cetacean species in European waters (Hammond et al. ). It is important to conserve this population for the purposes of the MSFD, but it is currently subject to high levels of extrinsic mortality through bycatch (Murphy et al. ; Peltier et al. ) which can impact on its viability (Mannocci et al. ). The MSFD descriptors are used to assess the status of the population and are based on sets of criteria. Under the MSFD, descriptor requires the production of indicators of demographic status to assess the viability of the common dolphin population in the face of bycatch pressure (Benjamins et al. ). The aevaluation of the achievement of the objectives set by the descriptors is carried out on the basis of a set of criteria which constitute a : See Table . for a schematic presentation of each criteria within the descriptors relevant to marine mammals and their need for demography : See Appendix descriptor . Within descriptor in particular, the demography of marine mammals (and therefore the common dolphin) occupies an important place (European Parliament

). The only biological indicators recognised to define the status of the population in the North-East Atlantic (i.e. increasing, stabilising or decreasing) are the abundance estimates obtained from aerial and boat surveys (Hammond et al. , ). Seasonal abundance estimates have been collected on a smaller scale (Laran et al. ; Authier et al. ) but their purpose is rather to provide information on the abundance of the common dolphin population on a seasonal basis. Such abundance estimates are not sufficient to provide information on the status and viability of the population, and the impact of bycatch on its viability (Murphy et al. ; M. B. Santos and Pierce ). There is a need to produce demographic indicators (vital rates) to better inform the management policy of the MSFD and describe the status of the population (Murphy et al. ).

Demography is a more refined indicator than abundance for determining the status of a population and an essential complement in terms of management. The use of abundance data to indicate the viability of a population is mainly based on observed trends and the determination of depletion levels (B. L. Taylor et al.

). In the context of bycatch pressures, demography can be a supplementary indicator to abundance surveys to study the anthropogenic demographic impact on the common dolphin population (see A. J. Read, Drinker, and Northridge (

) for a broader discussion regarding marine mammals). Before detecting a change in abundance, it is possible that the values of the life history traits will change because their evolution depends on the extrinsic pressures (e.g. bycatch) that the population is undergoing (Stearns ). In the case of the common dolphin, the primary pressure on the population seems to be strong, given the number of individuals found stranded on the French seashore (Dars et al. ). Obtaining demographic data on this population appears both essential from an applied point of view and interesting from a more fundamental point of view in order to highlight possible variations in the values of important vital rates associated with survival and reproduction. The vital rates of the common dolphin population have already been estimated, as well as the life tables, and the population has been considered non-viable under current bycatch pressures (Mannocci et al. ) but it is necessary to complete these analyses.

Obtaining vital rates can be improved by using a more representative sampling method and more flexible statistical methods. Previous demographic work has provided age-at-death data using all available biological material, as is usually done to study the demography of odontocetes (Barlow and Hohn ). From these ) for a less schematic description and the detailed elements. The primary and secondary aspects are not shown here for the sake of clarity. But the demographic issues appear as secondary.

Descriptor

Criteria Need for demography

. Maintain biological di- versity D C
The mortality rate per species due to bycatch is below the level that would pose a threat to the species, so the long-term viability of the species is assured

Yes through the use of population growth models taking into account removals and quantification of the byctach effect on vital rates D C Anthropogenic pressures do not adversely affect the abundance of populations of the species concerned, so that the long-term viability of these populations is guaranteed Yes through the use of population growth models taking into account anthropogenic pressures data, vital rates were estimated using standard survival models from the literature (Siler ). Sampling bias is poorly controlled and it is difficult to identify variations in the estimated vital rates and the effect of variables (e.g. cause of death, area or sex). We provide here a complementary and updated demographic estimates which helps to fuel the debate on how to consider and study an elusive marine mammal population such as the common dolphin in the North-East Atlantic. In order to better control the bias in the representativeness of the strandings, we performed a stratified random sampling to select individuals (females and males). For each individual, where possible, we obtained age-at-death data using odontochronological methods on dental tissue. We then constructed another dataset of reproductive status from the analysis of all available gonad samples for both males and females. The data https: //www.observatoire-pelagis. cnrs.fr/?lang=en on age and reproductive-status-at-death range from to . From these data, we obtained vital rates associated with survival and reproduction using a joint model (including survival and reproduction). The survival part is based on the model construction presented in Chapter . The reproduction part is based on an Accelerated Failure Time model construction. Vital rates were estimated in three ways: without effect, with covariates (sex and cause of death) and with random effect (years of death). Finally, we filled a Leslie matrix model with the obtained vital rates to obtain the population growth rate and to geometrically project its evolution.

. Methodology: Stratified random sampling and Joint modelling

The methodology developed in this study implies five major actions (or steps):

. Define the sampling plan . Acquire data from biological samples . Estimate vital rates (with covariates and random effects) . Build life-tables . Project the fate of the population

Sampling of a stranded population

A major problem with randomly selecting individuals for analysis from a stranded population is that representativeness bias is not controlled. Random selection can add representativeness bias to the already existing bias associated with the stranding filter. Ideally, age and reproductive-status-at-death distributions should be exactly the same between the stranded population and our sample. This would allow us to avoid adding bias and to be sure that the only bias associated with our results is due to the stranding filter. One way to control for this selection bias is to implement a stratified random sampling design. This is based on the choice of important criteria defining strata within which random sampling is carried out (Figure . ). In order to maximise representativeness, a sampling design must be defined that takes into account several criteria that can influence the representativeness of demographic data. We used a statistical sampling method used in various scientific fields (e.g forestry ecology, polls surveys or psychology) that has never been applied to study marine mammal demography from stranded individuals. The objective is to define a sampling plan that will allow the selection of the most representative individuals of the stranded population, from which biological samples will be obtained to provide data on age and reproductive-status-at-death. The main steps (Figure . ) are as follows:

. The first step is to define the need for representativeness.

On what basis to select individuals. To select the individuals, we chose criteria that can influence the distributions of the demographic parameters. This selection is made in a non-hierarchical way: The first is length since length is an indicator of age and reproductive status. The second is year, since year can provide information on a possible cohort effect or even a change in the value of vital rates over time. The third is sex since it can be an element influencing age-atdeath and it is essential to respect the proportions observed in the stranded population. The fourth is the cause of death because it seems necessary to transcribe the mortality signal as observed in the field. Examples of external examinations of stranded individuals to determine the cause of death of the animal are described Figure . . The fifth is the location that is not very important and can be removed, but it is intended to select individuals stranded all along the coast. Having defined the criteria to be considered, it is necessary to calculate their proportion within the stranded population. The proportion of each criterion for the age-at-death dataset can be seen in Figure . .

. Next comes the stratification step. For each stranded individual recorded in the stranding database, a value for each of the criteria was assigned. Using the example in diagram . , some individuals will have the following criteria values:

-cm, , Male, Natural death, Department while others will have:

cm, , Female, Bycatch death, French Department . These individuals share common criteria but do not belong to the same stratum because they differ by at least one criterion value. The stratification stage consists of classifying all the individuals in the stranded population database into different strata to ensure inter-stratum representativeness.

. Finally, random selection within each stratum is used to ensure intra-stratum representativeness. Individuals from the stranding database are selected on the basis of their criteria values but also on the availability of samples. Indeed, it is not possible to obtain teeth and gonads from all stranded individuals despite the existence of the National Stranding Network ("Réseau National Echouage" in French). Sample collection is dependent on the state of degradation of the animal (Figure . ).

In summary, the list of individuals in the age-at-death dataset is obtained by stratified random sampling. The list of individuals in the reproductive-status-at-death dataset is obtained by considering all available gonad samples.

Data acquisition through histology Teeth: age-at-death dataset

The number of individuals with an age in the age at death dataset is . The methodology used to determine the age of an individual from its dental tissue involves a technique called odontochronology. This technique involves counting the annual layers in the teeth resulting from the incrementation of cementum and dentin in the dental tissue (Perrin and Myrick ). The deposition of these layers is mainly due to seasonal phenomena but other phenomena linked to the organism (stress) can cause the appearance of accessory growth layers (Hohn ). One year of life corresponds to Growth layer Group (GLG) which consists of an alternating light and dark layer. This technique is considered valid for determining the age of common dolphins (Mannocci et al. ; Murphy et al.

). The objective is to observe these growth layers. To do this, it is necessary to treat the teeth. The first step was to decalcify the teeth.

Here the teeth were placed in a bath of concentrated DC acid. The decalcification time ranged from to hours depending on the size of the tooth. Then teeth have been cut using a freezing microtome in a longitudinal section plan, a cut known as "dolphin cut" (Bowen and Northridge ). For each individual, three teeth were removed, all from the mid-left jaw. Six sections were produced from each tooth for a total of sections. These sections were then stained. The staining was carried out with toluidine blue (a frequently used stain) which allows the growth curves to be highlighted (Bowen and Northridge ). The stained sections were finally placed on slides for microscopic observation. Three slides per individual were produced, each with sections. An example of the final product is available in Figure . , which shows the growth curves at the apex, middle and base of the tooth for three individuals of different ages and without the same proportion of accessory striae. Two age determinations were made to construct the ageat-death dataset. In each case, three readers were required. One of the readers was common to both periods, the trainer with the most experience in age determination. For age determination that occurred during the project, if the difference between readers was less than years, estimates were averaged. If the difference was more or equal to years, readers were discussing to reach a consensus. If it was not possible, only age reading of the trainer was taken into account. All readings were made without prior knowledge of body length or sex.

Gonads: reproductive-status-at-death dataset

The number of individuals with age and reproductive status in the reproductive status dataset is 240. For all the individuals, age was also available. When the state of decomposition of the animal allowed collection of the gonads, the left and right gonads were sampled and preserved in % formalin with volumes of ethanol. Reproductive status was assessed for both males and females. We also chose to consider males because the analytical framework allows allows for the inclusion of covariates (Rouby, Ridoux, and Authier ). he size of the male testicles depends on the season with an increased size during the mating periods (May to September) (Evans and Teilmann ). It is hypothesised that this size adaptation is associated with spermatic competition (Murphy, Collet, and Rogan ). For this reason, macroscopic examination alone does not allow the reproductive status of the male to be determined. We sent the testicular samples to an independent histology laboratory which provided us with testicular histological slides. From these cross-sections, we determined the reproductive status based on the signs of spermatogenesis, including spermatozoa in the tubules and/or mature spermatozoa present in the epididymes (A. J. Westgate and A. J. Read ; Figure . ). For females, maturity was determined from the ovarian structures which are referred as corpus. Only the left ovary was observed as it is usually larger than the right one and contains twice as many corpus. The primary follicle, which contains an oocyte embryo, develops into a secondary and then a tertiary follicle before maturing into a "de Graaf" follicle. The oocyte is expelled into the fallopian tube when the follicle reaches the "de Graaf" stage. The follicle then deteriorates as a corpus luteum and finally into a scar on the ovary (R. J. Harrison and Ridgwa ) which are refereed to as corpus albicans (Figure in Appendix . ). The concept of persistent corpus albicans have been questioned for the common dolphin and it is now assumed that it is not possible to estimate the number of gestation based on the presence of corpus albicans (Dabin et al. ). Table . summarises the determination of the reproductive status of females based on histological observation. The definition of pubescent status is possible for both sexes and for several species of delphinids (Murphy, Collet, and ). For females the age-at-first-reproduction is commonly assumed when there is a corpus albicans (Slooten ). Females are assumed to be pregnant as soon as they reach maturity (Mannocci et al. ) despite of the non-persistence of ovarian scares (corpus albicans) for the common dolphin (Dabin et al. ). Using pubescent status, we did not explore the ageat-first-reproduction (AFR) and age-at-sexual-maturity (ASM) as it is commonly done in the literature. We explored the access to puberty. I think that this parameter is more sensitive to life history changes. However, we still compute the proportion of sexually mature females as it is commonly done in the literature in order to better compare our observations with previous works on this population. The stranding database is shown in yellow. The age-at-death dataset in dark blue. The reproductive-status-at-death dataset in blue-green. Stratified random sampling allows for similar proportions between the age-at-death dataset and the stranding database, including cause of death, years, size and sex. Within the reproductive-status-at-death dataset, there is a large proportion of samples from the Biocet a program conducted in . There is also a greater proportion of females than males, contrary to what is seen in strandings. Individuals of -cm also seem to be over-represented. The causes of death are globally balanced but not representative of the stranding signal. : Parametric model that assumes that covariate accelerate or decelerate the reaching of the event.

Sex

: For technical details of convergence and parameterisation see Appendix subsection . Joint estimation of vital rates with a null effect, covariate effect and random effect

Cohort effect

The cohort effect is the proportion of the observed outcome that is due to the characteristics of the cohort.

Here, the cohorts are annual and have death as a common event.

The cohort effect captures the effect of the common characteristics of individuals who died in the same year on the estimated vital rates.

The previous steps have provided data on age and reproductivestatus-at-death from which we can calculate vital rates. Usually, these vital rates are calculated from separate models. The Siler (

) and Heligman and Pollard ( ) models are used to calculate vital rates associated with survival (Saavedra ). Logistic regression is used to calculate the vital rates associated with reproduction (Mannocci et al. ; Murphy et al. ). In the previous chapter, I described a new approach to survival estimation based on the Reed model (Reed ). We have developed a similar approach for estimating vital rates of reproduction by considering an Accelerated Failure Time model (AFT) model. Rather than considering these models separately, we combined them into a joint model.

Model selection

The estimation of the parameters of the two models is linked by the presence of random effects related to the year of death of the individuals. This random effect is analogous to a cohort effect (see margin-box for a definition). We considered increasingly complex models. The first do not involve the common cohort effect (no year effect). For each model, we computed the Widely Applicable Information Criterion (WAIC) and the Leave-One-Out cross-validation Information Criterion (LOOIC) as they allows to select the better model given the out-of-sample prediction (Vehtari, Gelman, and Gabry ). The better model to use is the one with the lowest WAIC or LOOIC. The selection given each model complexity is detailed in Appendix Table . According to both WAIC and LOOIC, the better models were the and the . From the perspective of statistical parsimony we used the model as it is less complex than the (no correlation between survival and reproduction estimation).

We also used the approach commonly used in the literature to estimate survival and reproduction of common dolphins (i.e. Siler, Heligman-Pollard and logistic regression). I do not detail here the functions related to these approaches, but I describe them in Appendix . . Only the models and functions associated to the joint modelling approach will be detailed. Functions' parameters are estimated using a Bayesian framework. From the previously obtained data, the times to the event (death or puberty) are used to estimate the parameters of the models. These parameters are then entered into each function to calculate vital rates. The details of the estimated vital rates and the associated models are provided in Table . :

Table . : Vital rates information. For each important vital rate here, the mathematical notation, method of calculation and inclusion or not in the Leslie matrix model are detailed. Only age-specific survival and fecundity rate (female offspring) calculated from the joint modelling approach are used to inform the Leslie matrix. The fecundity rate is calculated using the proportion of mature females obtained from the joint modelling approach. The latter takes into account the pubescent status as a reference event.

Vital rate

Yes

Although the approach involves joint modelling, each model is detailed separately. The join is provided by the random variable (cohort effect) which is an individual effect ( ) related to the year of death of the individual ( ).

Survival part

The age-at-death data obtained from the stratified random sampling are detailed in Figure . . Each age within this data distribution is used to inform the parameter . Age-at-death dataset from Mannocci et al. ( ) is also used to compare the effect of sampling on the survivorship value since authors did not perform stratified random sampling but used all available age-at-death values.

The survival model is the following:

log = + =1 + 1 + × - ( . )
It consists in modelling the logarithmic transform of (which is the age-at-death for each individual ) assuming a normal-Laplace distribution for logx i where is a location parameter; and 1 are (positive) scale parameters; and are independent standard exponential deviates. The parameter 1 quantifies individual frailty (Kannisto ; Reed ) or persistent demographic heterogeneity (Cam, Aubry, and Authier ). We specified the following distribution for residuals ∼ N(0, 1) and 1 > 0 which The maximum age class is -years with some females of this age. The data seem to be characterised by a low representation of age groups -, -and sometimes -and -. Over-representation is also possible for ages to . For comparisons, see Chapter with two age distributions on the same population. In total, there are individuals.

: the proportion of the population that survive at least to age .

: the proportion of the population at age that reaches the age + 1.

allows to accommodate a bathtub-shaped hazard rate curve which is the pattern expected for long-lived species of vertebrates in the wild (Choquet et al. b). It is possible to incorpore individuallevel covariates (such as sex or cause of death) and random variables (such as years). The superscript 1 means that this is the random effect associated with survival model.

The vital rates associated with the survival component are:

Survivorship ( ) at age with Φ as the cumulative density function of a standard normal distribution with Φ = 1 -Φ (with Φ as the density function of a normal distribution):

To apply covariates and/or random and/or trend effects it is on the intercept :

1997 = + 1997 ( ) = Φ log - - exp (log -) + 2 2 2 × Φ + log - ( . )
Age-specific survival ( ) :

( ) = ( + 1) ( ) ( . )
: Instantaneous probability of dying at time + given that and individual has survived until time

Hazard rate ℎ( ) with as the Mills ratio ( = Φ ) which is the ratio between the cumulative density function (Φ ) and the probability density function of a standard normal distribution ( ) :

ℎ( ) = × + (log -) (log -) - + (log -) ( . )
This approach is accurate to estimate survivorship with a sample size of while for hazard rate it is better with at least individuals (Rouby, Ridoux, and Authier ). 12 The reproductive model is the following:

Reproductive part

0 5 10 15 0 -
log = + =1 + 2 + × ( . ) It if ∼ Weibull ( , )
log( ) ∼ Gumbel ( , ) with = 1 and = - consists in modelling the logarithmic transform of x it (which is age-at-death associated with a reproductive state) assuming a Weibull distribution for and a standard Gumbel distribution ∼ (0, 1) . The Weibull distribution used is a twoparameter distribution: is the shape parameter which describes the distribution pattern of the risk of becoming sexually mature; and which is the scaling parameter defining the position of the distribution curve. Residuals ∼ (0, 1) which induces a Weibull law. It is possible to include individual-level covariates (such as sex or cause of death) and random variables. The random variable 2 is the cohort effect associated to the reproductive model. Covariates inclusion allows to compute the proportion of mature females, which is the sex usually used in unisex animals dynamic populations, without subseting the dataset and lose information. Three reproductive status (i.e. model censoring states) are used to inform the model parameters estimation: Immature (right-censored), Pubescent (not censored, the most informative) and Mature (left-censored).

Vital rates associated with the reproductive component: Proportion of mature females ( ) at age with as the shape parameter of the Weibull distribution ( = 1 with as the standard deviation of the Gumbel distribution of log(x it )) and as the intercept of the Gumbel distribution:

( ) = 1 - - - 1 1 ( . )
( ) = ( ) × ( ) 2 × -1 ( . )
The proportion of mature individuals through the ages is obtained considering both sexes. To compute fertility rate, we used the proportion of mature females computed with the pubescent status (through the covariate inclusion). It is more accurate to use females for demographic modelling since it is considered as the limiting sex in the population and it is easier to measure the reproductive output than for males.

Leslie matrices and elasticities

The vital rates obtained above allow the construction of life tables. The tables and their construction are detailed in the supplementary material. Once vital rates have been obtained, including age-specific survival ( ) and fecundity rates from pubescents ( ), a Leslie matrix model can be filled in to obtain the population growth rate. This growth rate is then used in a geometric population dynamics model.

The matrix population model is the following:

              0 1 2 3 ... 24-1 24               +1 =               0 1 2 3 ... 24-1 24               ×               0 1 2 3 ... 24-1 24 0 0 0 0 ... 0 0 0 1 0 0 ... 0 0 0 0 2 0 ... 0 0 0 0 0 3 ... 0 0 0 0 0 0 ... 0 0 0 0 0 0 ... 24-1 0               ( . )
Vector gives the number of individuals in each age-class at time and then at time + 1. The previously estimated vital rates were combined into the matrix with the female age-specific survival rate ( ) and the fertility rates from pubescents females ( ). Female vital rates were obtained by including covariates without and with the random effect of year. The matrix model is therefore relative to female only. Before age class , the gestation rate is . No offspring are produced by females strictly under the age of . The population growth rate was computed as the asymptotic growth rate of the Leslie matrix and allowed to project the remaining population at time ( ( )) taking into account the initial abundance at time 0 ( ( 0 )). The population's trajectory was projected through years, from to .

( ) = ( 0 ) × ( . )
We also explored the relative changes in population's growth rate caused by a proportional change in one of the life cycle parameters, referred to as elasticities (de Kroon et al.

). The elasticities analysis may be used to improve and study the management of long-lived species (Crouse, Crowder, and Caswell ).

Baseline assumptions

Stranded dolphin population is representative of the living population (regarding demographic features).

Age and reproductive-status-at-death are representative of the stranded population.

Each age-at-death is known and correctly reported. Each reproductive-status-at-death is known and correctly reported.

Joint model is accurate to estimate vital rates and the survival and fecundity patterns are congruent for the species.

The population growth geometrically and is not subject to migration.

.

Results: Vital rates and population growth rate

The first part of the results is related to the vital rates of the population. In this part, the first result highlighted is the comparison of survival estimates according to the sampling of age-at-death data. The second result highlights the patterns of survivorship and proportion of matures individuals without effect. The third result highlights the survivorship and proportion of matures individuals patterns with covariates. The fourth result highlights the survivorship and proportion of matures individuals patterns with random effects. The second part of the result highlights the properties of the matrix model used. The first result highlights the projection of the evolution of the population (in percentage) without effect and with random effect. There is no covariate because the projection is always made from the sex covariate, to take into account the females only. The second result shows the value of the elasticities for survival and fecundity, by age and by year.

Common dolphin population vital rates

Survivorship difference according to sampling

We estimated survivorship from Mannocci et al. ( ) dataset ("All age-at-death available" Figure . ) and the dataset obtained in this project ("Stratified-random-sampling" Figure . ) using Siler (

) and Reed ( ) models. This was done with the aim of testing the influence of stratified random sampling on the estimation of survivorship. As a reminder, Mannocci et al. ( ) did not use stratified random sampling. For the Mannocci dataset, the survivorship estimated is equal to that estimated in the Mannocci et al. (

) study using the Siler model (Figure . ). Using the Reed model, there is a gap from the to the age-class (without taking into account the uncertainties). Then estimation are consistent and nearly overlapping. For the dataset collected in this study, both models estimated nearly the same survivorship across all age-classes even if there is a limited gap between and years (≈ . ). ), there is a difference in survivorship estimates. This difference is much less marked, or even almost non-existent, for the dataset constructed from the stratified-random-sampling. the Bayesian credibility estimate. The confidence estimate is at its widest value at years admitted a survivorship uncertainty of . . From to years, the survivorship decreases from to . . It is almost equal to at years. There is a gap between the observed and the estimated survivorships from to years with an higher value of estimated survivorship of about . . (black dots). The estimated fitted curve with the Accelerated Failure Time (AFT) Weibull model is expressed as a dashed line. Likewise for survivorship, the % interval uncertainty around the estimation is displayed in grey. The estimated ASM taking into account the pubescent biological information through the AFT approach is about . years. The maturity difference between the two approaches is therefore about years.

Survivorship and Proportion of mature individuals with a null effect

Survivorship and Proportion of mature individuals with Covariates

The effect of the two covariates (Bycatch and sex) on survivorship is presented in 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 A: Effect of covariates on survivorship. The difference in survival between captured and uncaptured individuals increases with age. It is about % at years old and reaches % at years old. Females seem to survive slightly better than males at the end of their lives, particularly from the age of . Their survival is about % better at years. B: Effect of covariates on proportion of mature individuals. The proportion of mature individuals appears to be higher in the younger age classes among the individuals caught. At years of age, there are % more mature individuals. The females also seem to mature faster than the males. At years of age, females are % more mature than males.

profiles is expressed in percentage. For Bycatch (or cause of death), the reference value is "Natural death". For sex, the reference value is "Male". It can be seen that the difference between profiles is more pronounced when the cause of death is taken into account in comparison to sex. By focusing on the effect of bycatch, it appears that the bycaughts individuals seem to survive less well than non-bycaught ones for all ages. The difference between bycaught and non-bycaught individuals in survivorship goes from to % from to years to to % from to years. The effect of the two covariates (Bycatch and sex) on the sexual maturity attainment is presented in Figure . horizontal panel B. The percentage of change between the two estimated sexual maturity profiles are highlighted in the same way as before. It can be seen that bycaught individuals reach sexual maturity more quickly than non-bycaught ones. From the age of to , however, there is no longer any difference. Sex also seems to have an effect on the estimation of the sexual maturity profile. Females seem to mature more quickly than males.

Survivorship and Proportion of mature individuals with Random effects

Figure . : Cohort effect on survivorship. The black dashed line corresponds to the mean survivorship. Each coloured solid line of is a deviation around this mean (i.e. random effect) associated to a cohort.The cohorts share death as a common event. For example, the cohort includes all individuals who died in regardless of age. This cohort is associated with the lowest estimated survivorship. The highest estimated survivorship is for the cohort. Mean survivorship signal (dashed black line) and the deviation from this mean signal due to the random effect of the years (coloured solid lines) are available in Figure . . The year associated with the highest survivorship is whereas the year associated with the lowest survivorship profile is . Years before show a profile with a higher survivorship than the mean signal. After , the survivorship was associated to lower values with the lowest values recorded for recent years after . Mean proportion of mature idnviduals signal (dashed black line) and the deviation from this mean signal due to the random effect of the years (coloured solid lines) are available in Figure . . The year associated with the latest maturity ( % of the population estimated as mature) according to age is . The year associated with the earliest maturity according to age is . Years from show a profile with a latest age-at-maturity than years after that show a profile with an earliest age-at-maturity.

Population growth rate and elasticities

The mean signal of the projected population abundance trajectory (in percent) between and from the estimated vital rates with the covariate "female" is presented in Figure . . The mean estimated growth rate is about . which correspond to a population's extinction within to years. The mean proportional contribution to the growth rate, of each age-class for both survival and reproduction are displayed in Figure . . For both facets, the mean signal is shown as a black dashed line and the deviations from the mean signal due to the random year effect are shown as solid coloured lines. Regarding the survival component, the contribution decreases along the lifetime. The estimated contribution is at his highest from age to and then drops to at age . From age to , cohorts from to are associated to a lower contribution compared to the mean whereas years from to are associated with a higher contribution. From age to cohorts from to are associated to a higher value compared to the mean whereas cohorts from to are associated to a lower value for the same age-class interval.

Regarding the reproductive component, the contribution increases from age to . The age-class which contributes the most to population growth rate is the 8 ℎ . Then the estimated contribution decreases up to the age-class . The contribution of age classes before is higher than the average for the years-of-death cohorts after and lower for cohorts before . Then, from age until age cohorts with an higher value of contribution than the mean are cohort from to with a difference peak at year . These years are also associated to a lower contribution compared to the mean signal from age to to . For cohorts before the contribution is lower than the mean from age to . Then it is higher from age to . Age-specific survival rates are the vital rates that contribute most to the value of the growth rate ( . when adding up each elasticity). age-specific fecundity rates contributes slightly ( . when summing each elasticity). The contribution appears to be unequal across cohorts. The cohort has the highest contribution to population growth from reproduction and survival of young age classes (for reproduction and -for survival). It is also the cohort whose reproduction and survival of adult year-classes ( to for reproduction and to for survival) contribute least to population growth. This pattern is reversed for the cohort.

. Discussion: Important aspects and implication of the study

We obtained age-at-death data for individuals (females and males) selected by stratified random sampling. We obtained reproductive-status-at-death data for all individuals (females and males) for which we had gonad samples available. We used these data to obtain vital rates associated with survival and reproduction. The vital rates were obtained using a joint modelling approach. This approach takes into account a random effect common to both survival and reproduction, relating to the year of death of each individual. The vital rates that will be presented in this section are presented according to the inclusion of: a null effect, a covariate effect and a random year effect. The rates obtained by this approach were used to inform a Leslie matrix model. The values used are those for females (via the sex covariate effect) without and with random year effect. Female fecundity is obtained from modelling the attainment of puberty. The value of the gestation rate induces a female offspring equal to before years.

The growth rate obtained from the Leslie matrix was used to project the evolution of the population without and with random year effect. The elasticity properties of the matrix(es) were calculated.

Representativeness bias and age distribution

The Strandings are probably biased regarding the juvenile and the older senescent age-classes with fewer individuals than expected for a wild population.

approach is based on a cross-sectional monitoring which is fundamentally related to strong assumptions difficult to meet. The most important to meet was addressed by ( Caughley) which is the necessity for the population to be at a stationary age distribution when using age-at-death recorded from strandings. The stationary age distribution is an age distribution that induces a stable population whose abundance does not increase or decrease over time. This stationary age distribution is a special case of the so-called stable age distribution (Caughley ). The stable age distribution emerges when the birth and death rates are fixed and the distribution of ages in a population are stable (Carey and Roach , p. ). Our age-at-death dataset does not appear to be representative of a so-called stable population for a delphinid (Mannocci et al. ; Barlow and Hohn ). Since the stationary age distribution (required as a baseline assumption) is a special case of a stable age distribution, it is likely that our age-at-death sample does not meet the baseline assumption.

The age classes that could be biased are the calves ( -years), adult ( -) and older-senescent (> years) (Barlow and Hohn ; Stolen and Barlow ). The representation of juveniles and older-senescent individuals may be influenced by drifting processes. They may be less buoyant than adults (Peltier et al. ) It is therefore possible that they sink more easily. The underrepresentation of calves in the dataset may also be explained by a more rapid decomposition, greater vulnerability to predation, scavenging, and lower detection probability of small-sized animals (Stolen and Barlow ). The Strandings are probably biased regarding the adults age-classes around to years with more individuals than expected for a wild population.

Over-representation of adults in the age-at-death dataset may be explained by three assumptions (Barlow and Hohn ):

) Non stable age distribution resulting from a disturbance.

) Inconsistent deposition of GLG during life.

) Oversampling resulting from a filter before stranding.

The second assumption is unlikely to be possible. For common dolphin it is commonly admitted that GLGs deposition is constantly per year (Murphy and Rogan ). To validate or invalidate the first assumption, much more age data would be needed. It would be possible to see whether a change in representation for a given age class is repeated in subsequent years as individuals advance in age. In case of disturbance, it should propagates across years regarding age-class distribution (Barlow and Hohn ). The third assumption could be possible if we consider an overmortality of these age classes due to the bycatch. As shown in Figure . , the age classes most likely to have a drop in reduced survivorship (Age > years) when the bycatch covariate is taken into account are also those that would appear to be over-represented in the dataset. This observation could support the validity of the third assumption. To better accommodate the over-representation of adults and the under-representation of calves, it might be interesting to use a mixture model that allows to represent two hazard rate components (Rouby, Ridoux, and Authier ). The first specific to calves. The second specific to adults. Before testing this survival model, a simulation study should be conducted.

Contribution of stratified random sampling approach

This

The sampling plan seems essential to minimise bias in estimating population survival in cross-sectional analysis.

study is the first to employ a stratified random-sampling framework while estimating the vital rates of a marine mammal population from stranding data. Usually, demographic studies from strandings use all age-at-death data available (Mannocci et al. ; Barlow and Hohn ; Stolen and Barlow ). The stratified random sampling is a way to satisfy both ecologist's and statistician's requirements for the representative dataset (Roleček et al. ). The advantage using this approach is that it allows to maximises the probability of best representing the age structure of the stranded population. It is important to match the stranding signal as closely as possible on the basis of criteria (that shape strata) influencing the demography and individual selection (such as cause of death). Such an approach is commonly used in fisheries studies (Holden and Raitt ; Cochran and William ; Lai ), forestry ecology (de Vries ) or human psychology and social sciences for which it is important to correctly represent age conditions in a population (Banerjee and Chaudhury ). As it is shown in Figure . survivorship results can vary a lot depending on the sample collected.

The sampling in the Mannocci et al. (

) study was not controlled and estimated survivorships may vary. The dataset from (Mannocci et al. ) study were built using all the age-atdeath data available. As it was shown in Figure . , the estimated survivorship is not the same depending on the survival model used (Siler or Reed). There is a gap between the two curves from age to about age . The dataset we built in this project was constructed using stratified random sampling. The survivorship estimates from this dataset are almost the same between the two survival models. It is therefore possible that by not controlling for sampling and taking all available age-at-death data, this reflects an additional bias that may influence the estimation of survival models. In the case where sampling is controlled, the estimated values are the same regardless of the models used. The stratified random sampling may help to reduce the sensitivity of the estimate to the age distribution.

We think that it is essential to perform a stratified random sampling (if sufficient data) in order to better represent the stranding signal. We propose a three-steps framework to minimise representativeness bias using strandings (also see Margin- This approach ensures that uncontrolled selection bias is not added to the sample or is at least minimised. The representativeness bias is therefore due solely to strandings. The common dolphin population is currently considered as a single management unit by OSPAR whose population is panmictic (Murphy et al. ). In principle, the vital rates produced in this study should refer to the whole population (Benjamins et al. ). The samples used here (age-at-death and reproductive-status-atdeath) were constructed from individuals stranded on the French coast. For some, it was possible to trace their supposed origin at the time of their death. This is achieved through the use of inverse drift models (Peltier et al. ). Almost all the individuals It is possible that the bycatch pressure on the population is increasing and resulting in poorer survival. Since , the number of stranded dolphins including bycaught individuals has been increasing almost constantly (Dars et al. ). The time series data we have used takes into account the year . This year was marked by a very strong bycatch episode with a record number of stranded individuals (which will be exceeded by and ). This is the year in which survivorship is lowest, sexual maturity is reached most quickly and the population growth rate is lowest. It is therefore possible that the cohort effect observed is mainly due to an increase in bycatch pressure on this population. To compensate for this pressure, the population would tend to mature and pubesce more quickly as it is suggested by the cohort effect. The inclusion of the covariate "cause of death" may reinforce this assumption. As shown in the Figure . , bycaught individuals survive less well that non bycaught ones and seem to mature more quickly. The external mortality force exerted by humans through bycatch could decrease the survival pattern of the common dolphin and speed up sexual maturation at the population level. An other interpretation of the survivorship difference between bycaught individuals and non bycaughts is sampling bias. The survival profile associated with bycaught individuals may be biased by the over-representation of bycaught individuals in the sample. The inclusion of the "cause of death" covariate would therefore limit the bias introduced by bycatch in the survival analyses. By looking at the elasticity properties of the Leslie matrices (Figure . ), reproduction contributes only for % of the growth rate whereas survival contributes up to %. This differential contribution is expected for a long-lived species like the common dolphin (Heppell, Caswell, and Crowder ). See Margin-Figure .  for a comparison with other mammals and delphinids. The low contribution of reproduction to population growth (regardless of cohort) could explain why, despite the reduction in reproductive age, the growth rate is still decreasing.

Viability of the common dolphin population

Predicted but not observed extinction

Mannocci et al. ( )

The most favourable growth rates we have estimated are associated with the earliest years (< ) and are close to Mannocci et al. ( ) estimates.

projected the trajectory of the common dolphin population and predicted that the population would become extinct within years at an average growth rate = 0.945. The average growth rate that we estimated in this study is lower with a value of = 0.868. Considering this rate, the population will go extinct within years. This growth rate seems to vary between cohorts as highlighted by the projections taking into account random effects (Figure . ) but none of the years taken into account is associated with a projection as favourable as that of (Mannocci et al. ). Our time series covered years to whereas (Mannocci et al. ) used a time serie from to . The difference may be due to a different cohort effect between these two time series. Different characteristics or pressures experienced by our cohorts between and . Considering our time serie, the population's state seems worst in the recent years ( to ) with a growth rate that decreases from . in to .

in . According

The population should have been declining for years, but that is not what we are seeing. It is possible that this is a sink population. This hypothesis will be discussed in more detail in the general discussion.

to our estimates and those of (Mannocci et al.

), a decrease should be observed. However, this does not seem to be the case according to the available abundance estimates (Appendix Table ). It is possible that the population from the Bay of Biscay is a sink population receiving individuals from source stocks.

The basic assumptions made in this study may be unrealistic. Six main assumptions were made to conduct the study. The first assumption: "Stranded dolphin population is representative of the living population" may be unrealistic and lead to a mismatch between the growth rate obtained and the trend observed in the abundance surveys. The demographic rates of the stranded population may not be representative of the population at sea. Another baseline assumption that may account for the mismatch between the growth rate obtained and the observed abundance is the last one: "The population growth geometrically and is not subject to migration". This assumption suggests that the population grows or decreases geometrically (here exponentially) depending on the value of the growth rate. But the geometric population growth model used does not take into account migration phenomena which could be the cause of an apparent stationarity of the population in the Bay of Biscay. The Common dolphin plays an important functional role in Bay of Biscay marine ecosystems as a top predator, but is threatened by bycatch, i.e. the incidental capture of non-targeted species in commercial and recreational fisheries. Bycatch has been increasing, and has been associated with a large number of winter strandings of dolphins on the French Atlantic coast since at least . However, uncertainties around the true extent of common dolphin bycatch and the fisheries involved have led to delays in the implementation of mitigation measures. Current data collection on dolphin bycatch in France is with non-dedicated observers deployed on vessels for the purpose of national fisheries sampling programmes. These data cannot be considered as representative of all fisheries bycatch. This feature makes it difficult to use conventional ratio estimators, as they require a truly random sample of the fishery by dedicated observers. We applied a newly developed approach, regularised multilevel regression with post-stratification, to estimate total bycatch from unrepresentative samples and total fishing effort. The latter is required for post-stratification and the former is analysed in a Bayesian framework with multilevel regression to regularize and better predict bycatch risk. We estimated the number of dolphins bycaught for each week and ten ICES divisions from to by jointly estimating the bycatch risk, haul duration and the number of hauls per days at sea. The bycatch risk in pair trawlers flying the French flag was the highest in winter and , and was associated with the longest haul durations. ICES divisons .a and .b (shelf part of the Bay of Biscay), were estimated to have the highest common dolphin bycatch. Our results are consistent with independent estimates of common dolphin bycatch from strandings. Our method shows how non-representative observer data can nevertheless be analysed to estimate fishing duration, bycatch risk and, ultimately, the number of bycaught dolphins. These weekly estimates improve current knowledge on the nature of common dolphin bycatch and can be used to inform management and policy decisions at a finer spatio-temporal scale than has been possible to date. Our results suggest that limiting haul duration, especially in winter, could serve as an effective mitigation strategy.

Published works

This chapter is associated with three publications. The first is included as part of the introduction. The second is not included. The third constitutes the core of the chapter:

. Gilbert, L., Rouby, E., Tew-Kaï, E., Spitz, J., Peltier, H., Quilfen, V. and Authier, M.

. Spatiotemporal models highlight influence of oceanographic conditions on common dolphin bycatch risk in the Bay of Biscay. Marine Ecology Progress Series.

: -. Doi:https://doi.org/10.3354/meps13894

. Authier, M., Rouby, E. and Macleod, K.

. Estimating cetacean bycatch from nonrepresentative samples (I): a simulation study with regularized multilevel regression and post-stratification. Frontiers in Marine Science. :

. Doi: ) also provides information on this subject.

. Introduction: About the bycatch phenomenon Why does bycatch of common dolphins occur?

The structuring of multilevel trophic pelagic ecosystems is influenced by the presence of submesoscale and mesoscale oceanographic structures (Bakun , ). These structures correspond to oceanic fronts (i.e. the frontier between two water masses of different density), upwellings (i.e. upward movement to the ocean surface of deeper cold usually nutrient-rich waters) and eddies (i.e. circular current of water). Upwellings contribute to the enrichment of the water column with nutrients through different processes, but regarding the eddies, it is the Ekman pump that is involved (Margin-Figure . ). In the event of a cyclonic wind (for the North Hemisphere) there will be a transfer of water from the centre to the periphery of the eddy, which will lead to an upwelling of cold water in the centre (Fieux ). Oceanic fronts support the enrichment of water with nutrients by the same process by allowing two different water masses to encounter and promote the retention of nutrients in a given area. If they are sufficiently stable over time, these structures allow the development of a whole trophic chain from phytoplankton, to zooplankton, to fish and finally to top-predators (Bakun ). Top predators, such as delphinids, integrate ecological processes at all levels of the trophic web through their dynamic distribution (Croll et al. ). A key element of delphinids distribution is the availability of fish, which is variable in space and time due to the oceanographic processes discussed above (Hyrenbach, Forney, and Dayton ).

The common dolphin is a top predator of pelagic ecosystems in the Bay of Biscay that feeds on a wide variety of fish species, some of which are targeted by fisheries (Pusineri et ), and whose presence depends on oceanographic conditions. The main fish species involved in dolphin-fisheries interactions are the sardine (Sardina pilchardus), the anchovy (Engraulis encrasicolus) (both are prey to the dolphin) and the seabass (Dicentrarchus labrax) (in trophic competition with the dolphin; Spitz et al.

; Figure . for an overview of the trophic chain). The latter species is targeted by the Pair trawlers (Margin-Figure . ), a métiers (i.e. the combination of gear, target species and fishing area) historically associated with bycatch of common dolphin in the Bay of Biscay (Peltier et al. ; as well as other fishing métiers). The occurrence of sea bass, and the fish on which the dolphin feeds, is indirectly conditioned by the availability of nutrients (such as phosphorus, nitrogen, potassium and iron). : Service Hydrographique de la Marine Nationale: https://www.shom. fr/ (Bakun ). The presence of phytoplankton (and the diversity of species associated with it) influences the presence of zooplankton that feed on phytoplankton (Lampert et al. ). If the persistence and retention of nutrients are sufficient, then the development of plankton will allow predators such as fish to feed (Beaugrand and P. C. Reid

Nutrients have an influence on the development of phytoplankton

). Finally, large fish and dolphins feed on smaller fish (Spitz et al. ) and pair trawlers target fish of commercial interest (Peltier et al. ). The proximal relationship between fish, dolphins and pair trawlers could drive the spatio-temporal variability of the bycatch pattern: fish distribution is notoriously variable in both space and time (Erauskin-Extramiana et al.

) and is governed by many factors from dynamic oceanographic conditions. Changes in local distribution and abundance of prey species might be a substantial driver of their simultaneous presence with common dolphins, commercial predatory fish and fisheries in localized areas. In order to investigate the link between oceanographic factors and dolphin mortality at sea, we coupled biological data which are related to reverse drift trajectories obtained from the bodies of stranded dolphins with incidental capture evidence and environmental data intrinsic to various physical processes at high and low frequency obtained thanks to the modelling services of the Shom (Gilbert et al.

). We built a mortality estimation model to study the influence of sea surface temperature, turbulence force or eddy kinetic energy (i.e. Eddies occurrence) and mean surface temperature gradient (i.e. Frontal structures occurrence and intensity) over the years to .

Two seasons of mortality with different oceanographic processes were highlighted:

Winter -Freshwater frontal structures. Summer -Tidal frontal structures.

The winter and summer oceanographic processes do not appear to be linked to the same mortality processes. In winter, the Bay of Biscay environment is characterized by a seasonal cross-shore (West to East) surface temperature gradient with lowest temperature close to shore and intense frontal activity parallel to the coast (North to South) (Yelekçi et al. ). These frontal structures are freshwater fronts, correlated to the mixing of oceanic ).

: waters and cold freshwater inputs from river plumes (Yelekçi et al. ). Their location and time line concur with observed patterns regarding bycatch mortality. Together with the effects of the associated oceanographic variables on the mortality index, results suggested these seasonal fronts may be targeted by both fisheries and common dolphins as areas where fish aggregate, thereby putting the latter at risk of bycatch by the former. In July and August, the mesoscale dynamic activity of the Bay of Biscay is rather different than in winter. In summer, there are mainly fronts due to tidal flow (Yelekçi et al. ). Summer tidal fronts are quite consistent from one year to the next because they are correlated to a repetitive process (i.e. tides) (Yelekçi et al. ). In summer, the main frontal activity is a seasonal tidal front, called the Ushant front, located off western Brittany (Yelekçi et al. ). Its activity peaks in July and August (Yelekçi et al. ). The location of this typical tidal frontal structure concurred with the location of hot spots of fitted mortality index as well as with the location of bycaught common dolphin strandings, mainly occurring on the coast of the South Finistére region during the summer mortality season.

These results suggest the influence of spatio-temporal oceanographic processes on common dolphin bycatch mortality in the Bay of Biscay. But the relationship should not be overinterpreted as the model indirectly linked oceanographic processes to common dolphin mortality. While this study aimed to explore the "why" question, it is important to answer the "how much" question. Especially in the context of the Habitat Directive and the MSFD.

Bycatch observation and quantification

In France, the monitoring of cetacean bycatch in fisheries is non-dedicated (Cornou et al. ) (observers are non-dedicated to the bycatch observation), and the collected data are described as non representative, which makes it difficult to estimate the bycatch risk. The bycatch risk is the probability of catching at least one dolphin in a haul. From the calculation of the bycatch risk, it is possible to estimate the number of common dolphins bycaughts for a given year in a given ICES division. This estimation is carried out mainly in the Working Group on Bycatch of Protected Species (WGBYC). Traditionally, using the observer data, the way to quantify the bycatch risk per Day at Sea (also refer as "bycatch rate" per day at sea) is to use a ratio-based estimator. A day at sea is defined as any continuous period of hours (or part thereof) during which a vessel is present within a division and absent from port (Anonymous

). The number of bycaught dolphins is estimated as the : http://www.meteorologie.eu.

org/mothy/ product of the ratio of the sum of observed bycaught individuals to observed days at sea, times total fishing days at sea summed over ICES division of interest (ICES , p. ). This

It is impossible to use the classical ratio-estimator approach on onboard observer data to obtain the number of dolphin bycaught at fine spatiotemporal scales.

ratio-estimator is a design-based approach of the estimation of bycatch and is referred as a Bycatch risk Assessment (BRA) by WGBYC. The accuracy of the approach is largely dependent on the representativeness of sampling protocol (notably the observer scheme design). Due to the low observer coverage, the non dedicated-sampling protocol, the non-representativeness of observer dataset and the small number of hauls with few common dolphin bycatch, estimates based on ratios appear unreliable (Babcock, Pikitch, and ). An other way to quantify bycatch is to use strandings (Peltier et al. ).

The

Strandings are used to obtain the number of dolphin bycaught on a monthly basis. The estimation is done by inverse drift modelling. ICES estimates are currently made in this way.

use of strandings can be referred as model-based estimate since it implies a modelling-framework. Here it will be referred to as "stranding-based" estimates to better flow with the rest of the chapter, including the discussion. This approach is based on the drift reconstruction trajectory of the stranded individual using the MOTHY model ("Modéle Océanique de Transport d'HYdrocarbures") from Méteo France. Then the estimated mortality in each year is corrected by the probability of being buoyant. This correction allows to estimate the number of dead dolphins, included those that sank and were not found ashore (Peltier et al. ). Using this approach, it is not possible to discriminate the involvement of each fishing métier in the mortality of common dolphins. To fill this gap, the use of observer data is very important, even if biased. Some model-based recent improvements allows to minimise the impact of the representativeness bias. For Stranding based estimates do not allow the number of catches per métiers to be estimated on a fine spatial and temporal scale.

example, Luck et al. (

) recently used a modelling framework to estimate the corrected bycatch probability and then use the fishing effort to extrapolate to the whole fishing fleet. Recently, Authier, Rouby, and Macleod (

) adapted the use of a the regularized multilevel regression with post-stratification to infer more accurately bycatch rates at finer spatio-temporal scale than it is actually done.

There

We used Multilevel Post-Stratification to obtain the number of dolphins bycaught by PTM on a weekly basis from to in ten ICES divisions. We also estimated the bycatch risk, mean haul duration and number of hauls per Day at Sea on the same basis.

is a need to obtain bycatch estimates at a fine spatial and temporal scale and to use data from at-sea observers. In addition to estimates of the number of dolphins bycaught, estimates must be made for the variables associated with bycatch. The spatial and temporal scale must be fine enough to allow for inter-and intra-annual differences within ICES divisions. In this chapter, I present how we quantified the bycatch risk, haul duration, number of hauls per Days at Sea and number of bycaught dolphins from biased observer data . We quantified these parameters for each week from to within ICES division to better represent variations in bycatch spatio-temporality using the approach of (Authier, Rouby, and Macleod ). This study involves firstly ) on pair trawlers flying the French flag (hereafter called "PTM") which is called Obsmer ("Observation des captures en mer"). This dataset is the one considered as biased. From this dataset we estimated the bycatch risk, the haul duration and the number of hauls per Day at Sea. Then, we used a fishing effort dataset computed using the SACROIS algorithm and the median number of bycaught dolphin by haul to estimate the number of bycaught dolphins per year and ICES divisions. Both dataset were provided by IFREMER ("Institut Français de Recherche pour l'Exploitation de la Mer"). After presenting the involved data and the broad-lines of the applied methodology, I will compare the estimates made in this study with those of the stranding-based estimates.

. Methodology: Quantify the bycatch parameters

In this section, the methodology used to estimate bycatch parameters (i.e. risk, duration and number of hauls) is detailed. First, a review of the materials available to perform the analyses is made. Then, a review of the methods is presented in outline.

Materials ICES divisions, fisheries and study area

The North-East Atlantic ocean can be geographically divided differently depending on the focus (for example focusing on fisheries management or on ecosystems). ICES defines some ecoregions to refer to ecosystems in its ecosystem-based management approach (EBM) and divisions to give an overview of the fisheries related aspects in these ecoregions (ICES b; Figure . in Page Chapter ).

The Bay of Biscay, the English Channel and part of the Celtic seas are ecoregions associated to a high variety of fishing gears, métiers and targeted species. These areas are associated with submesoscale and mesoscale oceanographic processes, such as eddies, river plumes and tidal fronts, that enhance ecosystem productivity and result in high availability of fishes, including commercial species (e.g. European seabass Dicentrarchus labrax, Sardine Sardina pilchardus or Anchovy Engraulis encrasicolus). Pelagic trawl fisheries, as pair-trawlers, are active within all of these areas and mainly target the anchovy, sardine, seabass and hake in divisions a and b (ICES a). The study area for the estimation of common

Datasets

Two main source of data were used. The first source of data that was used is a dataset called Obsmer. This dataset is collected as part of an onboard observer program set up within the Data Collection Framework of the Common Fisheries Policy and carried out by IFREMER, under the supervision of the Directorate of Fisheries and Aquaculture ("Direction des pêches maritimes et de l'aquaculture", DPMA). Onboarding ObsMer observers' primary duty is to register the length and weight composition of catches. Still, they have to report any bycatch if they witness such events. Obsmer data on PTM covers 4, 484 hauls between and , of which 82 were associated with a bycatch event of at least 1 and up to 50 common dolphins.

Obsmer provides the geographic position, timing, and du- : Obsmer provides the micro-scale information while SACROIS provides the macro-scale information. ration of hauls. Although Obsmer is aiming at a coverage of 10% and 5% of fishing effort for (level-métier) PTM for vessels of more than 15 meters and less than 15 meters respectively. Effort is quite low overall, ranging from 0.0 to 11% of Days at Sea (DaS; Table . ). The number of observed hauls with at least one bycatch record is very small: the yearly percentage of observed hauls with a bycatch event never exceeded 4.5% and was 0 in nearly half of the surveyed years. Obsmer data on PTM are an unrepresentative sample of hauls, largely because allowing an observer remains largely at the discretion of skippers (Babcock, Pikitch, and Hudson ; Benoît and Allard

).

The second dataset provides monthly estimates of total fishing effort in each division. This dataset is generated from the algorithm SACROIS developed by Ifremer and integrates data from Vessel Monitoring System, log-books and landing statistics (for boats longer than meters from 1 of January and longer than meters from 1 of January ; (Système d'Information Halieutique

)). The SACROIS algorithm is interesting for two main points: ( ) it allows to correct errors that could exists in the integrated dataset due to recording or collecting errors ; ( ) it reconstitutes métiers during fishing trip as there are not recorded in logbooks nor fish market data (Cornou et al. ). The SACROIS dataset provides the best available estimates of total effort, in Days at Sea (DaS), between and (Table . ). There are also refusals from skippers due to administrative and security reasons. Skippers must apply for permission to take observers on board and even if they decide to apply, permission may be refused for safety reasons (e.g. not enough space or rails not high enough).

These two datasets are complementary to address the following challenge: is it possible to quantify bycatch parameters (such as bycatch risk, haul duration, number of hauls and number of dolphin bycaughts) from a non-representative sample ? The sample Obsmer contains the information on marine mammal bycatch at micro-level (i.e. on a haul resolution basis). This source of data allows to estimate the bycatch parameters (Luck et al. ). On the other hand, the fishing-vessels effort data from SACROIS provides the spatialized effort data at the scale of the whole fishing fleet (macro-level component). These population-level data on effort allows the post-stratification of bycatch risk estimated from observer data to obtain the number of bycaught dolphins at the level of the whole fleet (Authier, Rouby, and Macleod ). Descriptive statistics of both datasets are displayed in Table . . Used in tandem, both datasets allow using regularized multilevel regression with post-stratification to estimate cetacean bycatch from non-representative samples (Authier, Rouby, and Macleod ).

Methods

The modelling framework involved two steps. First, it is necessary to both estimate the bycatch risk and the duration of hauls for each haul , ICES division , week and year . Also, it is possible to estimate the number of hauls per day at sea at the week, division and year scale. This is done with the Obsmer dataset. Then, it is necessary to rescale these estimations at the entire fleet level to estimate the number of bycatch events and number of bycaught dolphins for the same levels. This post-stratification is done with the SACROIS dataset.

Estimations were carried out in a Bayesian framework using programming language Stan (Carpenter et al. b) called from R v. . . (R Core Team

) with library Rstan (Stan Development Team

). Stan uses Hamiltonian dynamics in Markov chain Monte Carlo (MCMC) to sample values from the joint posterior distribution (Carpenter et al. b). Four chains were initialised from diffuse random starting points and run for a total of 2, 000 iterations, discarding the first 1, 000 as warm-up. Default settings for the No-U-Turn Sampler (NUTS) were changed to 0.99 for adapt delta and 15 for max treedepth (Hoffman and Gelman ). We fitted a total of models of differing complexity (see Margin-Table . for specification and article in Appendix . for model selection): we compared models assuming either a gamma or a log-normal likelihood for haul duration, and models assuming additive effects versus interactive effects of week, year and divisions. Model fitting was carried out on the supercomputer facilities of the "Mésocentre de calcul de Poitou Charentes (Université de Poitiers/ISAE-ENSMA/La Rochelle Université)". Codes are available at https: //gitlab.univ-lr.fr/mauthier/cdptmbycatch.

Modelling Bycatch risk, duration of hauls and number of hauls

The objective of this part of the modelling framework is to model the bycatch risk and the haul duration at the haul level and the number of hauls per day at sea using only the Obsmer dataset. The problem with trying to estimate the value of these parameters is that we do not know how to interpolate between the observed points. Lets start with an example. In the Margin-Figure . , we see that we only have data points (for weeks = , , , , and ). This number is reduced to for a given year and is reduced to for a given division in a given year (Figure . to see the data depletion associated with stratification). We do not know how to fit a line between those points. And even less is known about how to extend the estimation of these profiles beyond the weekly scale to estimate by week for each year in each division. We only have some observations at the haul level scale for the bycatch risk and the haul duration and we also have the number of hauls. Obsmer data are sparse (Margin-Table . ) and the stratification of these data increases sparseness (Figure . ). To interpolate between the observed data points, we used firstly a random walk and then Gaussian Process. See Görtler, Kehlbeck, and Deussen (

) for an interactive web-page explanation of Gaussian Processes. The vector aggregates the mean weekly effects (on the linear predictor scale, e.g. logit scale for bycatch risk) which are modelled with a first order random walk to ensure some smoothness in between-week variations (Figure . ; Authier, Rouby, and Macleod ). The first stratification is performed and the value of each risk profile for each year is estimated from a Gaussian process whose mean is the previous weekly risk profile.

The second estimation then takes place and the value of the risk profile for each division within each year is estimated by another Gaussian process whose mean is the previous Gaussian profile.

From the observed data points, it is possible to infer the bycatch risk profile for each strata using a Bernoulli model (i.e. logit scale; Figure . ). As it is depicted bellow on Equations . , the intercept of the fitted curve corresponds to and is equal to . The vector corresponding to the weekly risk profile. This vector is the mean estimate of the first Gaussian process for each year . The estimate provides a risk profile for each year . From this annual vector, a final Gaussian process takes it as an average to estimate the profile of each division within each year :

. The values of this vector are then used to inform the estimation of the bycatch parameters.

The first order random walk then allows to estimate the following weekly value with a normal distribution: +1 ∼ N( , week ) and so on until the 53 week. From this overall profile (i.e. the vector ), a first Gaussian Process allows to estimate the bycatch risk profile for year , . Conceptually, the idea is to model as a deviation from the mean profile . This can be described as modelling a random effect except that curves rather than scalars are being modelled here. The profile is estimated with the Gaussian Process as follows:

∼ GP( , Σ )

with Σ as the covariance matrix.

From this year-specific profile (i.e. the vector ), a second Gaussian Process allows to estimate the profile of which depends on the year and on the division (Figure . ). The profile is estimated with the Gaussian Process as follows:

∼ GP( , Σ ) with Σ as the covariance matrix.

               ∼ GP( , Σ ) ∼ GP( , Σ ) = = 1 +1 ∼ N( , week ) > 1 ( . )
Simpler models without such interactions, and with only additive effects, were also fitted to the data. The simplest model included only additive random (unstructured) effects (dropping the superscript for convenience):

                     = + * + * * * * ∼ N(0, division ) ∀ * * * ∼ N 0, year ∀ = = 1 +1 ∼ N( , week ) > 1 ( . )
To simplify the equations, I did not include the superscripts in the Equations . but there are three alphas 1 1 1 , 2 2 2 , 3 3 3 and three deltas 1 1 1 , 1 1 1 , 1 1 1 . The superscript 1 refers to the bycatch risk, the 2 to the haul duration and the 3 to the number of hauls.

The Bycatch risk p is modelled as follow with corresponding to bycatch event (0 or 1) of the th haul for each ICES statistical division in week of year . The occurrence of a bycatch event is modelled as the logit transform of 1 in a binary outcome, provided by the Bernoulli law. Depending on the bycatch risk, a bycatch event can occur as:

∼ Bernoulli p ∼ Bernoulli logit -1 1 ( . )
The fishing duration at the haul level is modelled for the same levels with a Gamma distribution Gof shape parameter and scale parameter ¯ . The parameter ¯ is associated to the second alpha as: ¯ = 2 . This law allows the representation of continuous non negative value which is appropriate for a fishing duration. :

∼ G , ¯ ∼ G , 2 ( . )
and the number of hauls for the same levels per Days at Sea DaS is modelled assuming a zero-truncated Poisson likelihood (by definition, a DaS implies at least one haul). The parameter is associated to the third alpha as: =

3

. This distribution allows the representation of non negative and non null discrete events number which is appropriate for a number of fishing hauls. :

∼ P + DaS × ∼ P + DaS × 3 ( . )
The smoothness in and is controlled via the covariance matrices Σ and Σ respectively as:

                           Σ = Δ ΩΔ , Δ =         zone 0 0 0 . . . 0 0 0 zone         Σ = Δ ΩΔ , Δ =         year 0 0 0 . . . 0 0 0 year         ( . )
Equations . and . allow to model an interaction between week, year and division. The joint model defined in Equations . , . and . includes a time-varying component at the week-scale with an interaction with year and division.

As it is showed in Equation . , there are two important component to build the covariance matrix. The first is Δ which corresponds to the variance matrix either for (diagonal of zone ) or for (diagonal of year ). These are controlling the magnitude of variations from a mean profile (Margin-Figure . for a diagram).

The second is Ω, which is the correlation function between the variance matrixes. It is the correlation between week and week . This function is a Matérn correlation function of order 3 2 and range parameter fixed to 3 2 as: The methodology is shown in the Directed Acyclic Graph Figure . taking for example the bycatch risk estimation. We have information about bycatch risk from the dataset Obsmer. From the estimations of hyperparameters (notably and ) we can estimate the others. Finally it is possible to fit the bycatch risk profile (such as the haul duration and the number of hauls per Day at Sea) with a Bernoulli model (as in equation . ). . There is a temporal Independence after weeks.

Ω , = 1 + 2 √ 3× ( -) 3 × exp - 2 √ 3× (

Estimating the total number of hauls and bycatch events

The goal of this part of the modelling framework is to scaleup the estimated bycatch risk and the haul duration at the haul level and the number of hauls per day at sea at the fleet level using the fishing effort dataset (Days at Sea). It allows to estimate the total number of unobserved hauls N and the number of bycatch events Bycatch for each week , year and division .

The number of unobserved hauls N that happened in ICES statistical division in week of year can be estimated from the number of observed DaS in Obsmer (DaS Obsmer ) and from total effort DaS tot (and accounting for zero-truncation):

N = ˆ 1 --ˆ × D aS tot -DaS Obsmer ( . )
: While it is possible to spot bycatch evidence as it is shown in Figure .  in Chapter , it is very difficult to identify fishing gear involved. Sometimes it is possible to see the mesh marks of the net and assume the type of net. But this remains very exploratory and qualitative.

The total number of bycatch events in ICES statistical division in week of year is estimated as the sum of events observed in Obsmer (Bycatch Obsmer ) and the number of unobserved hauls multiplied by bycatch risk ( p ):

Bycatch = Bycatch Obsmer + N × p ( . )
Similarly, for each year, the number of common dolphins bycaught in PTM can be estimated using the observed number of bycaught dolphins in Obsmer, the estimated number of unobserved hauls (Eq. . ), bycatch risk and either the median number of dolphins involved in a bycatch event (Table . , or the grand median of = 2 for years with no observed bycatch event). We used the median to attenuate the influence of some bycatch events involving up to 50 dolphins (Table . ). These estimates are thereafter referred to as model-based estimates.

Baseline assumption for post-stratification

The total fishing effort data are accurate.

Comparison with strandings

One difficulty while estimating the number of bycaught dolphins is that it is difficult to ground-truth. The sample provided by Obsmer, a non-dedicated observer scheme of marine mammal bycatch, may not be representative of all bycatch. In addition it provides very sparse data, with less than 100 observed events over 17 years when strandings have reached several hundreds per week in recent years (ICES b) (for all causes of death). The number of stranded common dolphins with evidence of bycatch can be used to estimate the total bycatch mortality with reverse drift modelling (Peltier et al. ) used in international working groups (ICES b). Reverse drift modelling corrects for at-sea drift conditions, but cannot inform on which fishing gears was responsible for bycatch . Hence, strandings-based estimates are considered here as total estimates of bycatch, and can be used to partially ground-truth model-based estimates of bycatch by PTM.

These model-based estimates use data independent from strandings, but they should not exceed stranding-based estimates. Secondly, whether model-estimates correlate with stranding-based ones is of interest to shed light on the increased mortality reported in the Bay of Biscay (Peltier et al. ). For each year we checked the magnitude of model-based estimates against stranding-based ones, and computed Pearson's correlation coefficient between the two time-series at the month level. To account for drift, these correlation were computed with and without a lag of weeks when aggregating model-based estimates at the month level. Haul duration, hauls per DaS and bycatch risk per haul (Eqs. . and . ) were jointly estimated. Their temporal variations are displayed on Figure . for each week between and . Haul duration was the highest in week 1 with a posterior median estimate of 5.8 hours that decreased to 4.0 h in week 16, before dropping to 2 h in week 24. Haul duration increased up to 3 hours in week 32 and plateaued until the end of the year. Remarkable years were and with the lowest haul durations estimated from week to . Other remarkable years were , , and with the longest haul durations estimated from week 1 to 10. In , an increase in haul duration was estimated in week 48 (5 h vs 3 h on average across years). Bycatch risk was maximum in week 1 (around 0.1) and decreased to almost 0 from week 8 onwards.

, and were the years with the highest estimated bycatch risk in the first 8 weeks. In particular, risk was as high as 0.20 in for the first four consecutive weeks of the year. Two years prior to were associated with an increased risk between weeks 30 and 36. Year showed a rise in bycatch risk in week 48. The weeks in the highest risk years were also those with the longest hauls. The estimated number of hauls per DaS varied little from year to year, but varied substantially within a year. The estimated haul duration is the highest from week to . The lowest at week and then it remains stable from to the end. The estimated bycatch risk is highest between weeks and and is highest for years beyond and the year .

There were noteworthy differences between divisions regarding bycatch risk (Figures . and. ). The overall signal was similar to the one observed on Figure . with the highest risk values estimated between weeks 1 and 8. Risk in and was higher by a factor 5 in week 1 compared to other years. After week 8, this difference disappeared. With respect to divisions, division .a. was the one with the highest bycatch risk, with an estimate as high as 0.50 in winter and . For division . .a, estimated bycatch events were more numerous from week to with high peaks for years > 2016. There was also a peak for year in week . The estimated number of common dolphins presented the same pattern as the number of bycatch events. For division . .b, estimated bycatch events were more numerous from week to with a high peak for and . There were two peaks for , one in week and one in week . The estimated number of common dolphins presented the same pattern as the number of bycatch events. The estimated total number of bycaught dolphins for each year is reported in Table . . The study area was further divided into three strata: a neritic stratum in ICES subarea (divisions .defgh) and another in subarea (division .abc); and an oceanic stratum spanning subareas and (divisions .j and .d). Estimates were the lowest in the oceanic stratum of the study area, and the largest in the neritic stratum spanning ICES subarea . The largest bycatch estimate was in , with a posterior median of > 600 common dolphins bycaught in PTM operating in the neritic stratum spanning ICES subarea . There were large between-year variations in estimates, ranging from less than a hundred (in ) to more than one thousand (in ). Uncertainties around model-based estimates were also large. Weekly profiles for the estimated number of common dolphins is also available for both divisions .a and .b in Figures . and. . (Figure . ).

Number of Bycaught dolphins

Comparison and correlations with strandings

Model-based estimates of bycatch by PTM were always below stranding-based estimates (which do not allow to disaggregate by métiers) except for (Figure . ). In , model-based and stranding based estimates were and with a large overlap in credibility interval. Correlations between the two time-series were always positive. The estimated correlation decreases from . in to . in (profile without lag). The estimated correlation decreases from . in to . in (profile with lag; Figure . ). The temporal trend in within year correlation was negative over the study period.

. Discussion

Variations of bycatch risk and number of bycaught dolphins

The estimated bycatch risk pattern shows some variations between and within years. This risk directly influences the estimated number of bycaught dolphins which also displays some variations. Here, a discussion about these variations for both components is proposed.

Inter-Year variations

The variation of bycatch risk, and subsequently of the number of bycaught dolphins, between years may be linked to ecological factors. Bycatch can be viewed as the resultant of two components that are the fishing activity and the dolphin presence (Margin-Figure . ). The oceanographic processes may influence the two components to enhance the encounter probability between fishing engines such as PTM and common dolphins (Gilbert et al.

). The highest estimated bycatch risk values were reported for the eight or ten first weeks of each year within each division of the study area . . We can speculate that the years associated with a high bycatch risk were also those of notable spatial overlap favoured by particular oceanographic processes. Fishing activities and their operational conditions can also influence bycatch risk. For example, the setting of quotas on particular fish may cause operational changes in certain seasons. Market effects can also have an impact on fishing activity. If a particular fish is too devalued, it will no longer be targeted. These aspects will not be discussed in detail here as they are multifactorial and are more a matter of fisheries management. Variations in the number of bycaught dolphins between years can also be explained by other factors than the bycatch risk. They are model-based estimates obtained through the bycatch risk, total fishing effort (Days at sea) and the median number of bycaught dolphins per year. The two latter parameter may be the source of the variation between years of the number of bycaught dolphins. For example, the (posterior median) estimate is of > 600 dolphins in and it downs to < 100 in . Total effort in DaS in the Bay of Biscay (divisions .a and .b) in the first ten weeks of , when bycatch risk was highest, is twice the value of total effort in . The median number of dolphins involved in a bycatch event is was also twice the number in ( and respectively, Table . ). The bycatch risk being equal for the ten first weeks of these two years, the estimate for is expected to be at least four times that of . Variations

The mortality trend obtained by the model based approach seems to be rather constant while the stranding based trend seems to increase in recent years. This suggests that the increase in mortality estimated from the strandings is not due to PTMs.

in the estimated number of bycaught dolphins through the model-based approach can be related to the estimated number of bycaught dolphins obtained from the strandings estimations. Stranding records are an independent source of data for estimating the number of bycaught dolphins (Peltier et al. ). Reverse drift modelling allows to infer the death location of each stranded dolphin showing bycatch evidence for each month between and . Observed stranding numbers for each month can be corrected by both stranding and buoyancy probabilities (Peltier and Ridoux ). Reverse drift modelling cannot disaggregate estimates by métiers or fisheries but it provides an independent estimates of total mortality due to bycatch in the study area. The bycatch mortality estimated for the PTM should be lower than the total estimated from strandings. This was verified for all years save for , but uncertainties were large and credibility intervals had a large overlap. By looking at the correlations between estimated number of bycaught dolphins for the stranding-based and the model-based approaches, it appeared that the correlations were significant between the two estimated series, especially after accounting for a lag due to drift (Figure . ). The

The correlation between the two trends is stronger when considering a -week delay in stranding-based estimates. This correlation decreases over the period considered. The contribution of PTMs to total bycatch mortality may be lower than before. magnitude of the correlation decreased between and . One interpretation is that of a change in the relative contribution of PTM in total dolphin mortality over time, with PTM having a smaller impact on common dolphins in recent years compared to the s.

Within-Year variations

For each year, the bycatch risk was estimated to be higher in the winter part of the years (i.e. week to ). This pattern is largely congruent with the pattern seen in strandings of common dolphins in the Bay of Biscay (Dars et al. ). A limitation of stranding data is that it does not allow to discriminate between fishing gears (Peltier and Ridoux ). The Obsmer data in contrast included geolocalized bycatch events, with a spatial resolution at the level of ICES divisions kept for analysis. For the PTMs flying the French flag, we could identify divisions .a and .b from week to for nearly each years as the ones with the highest risk of bycatch.

For each year, the number of bycaught dolphins can also be estimated at the ICES division level for each year by the ICES Working Group on Bycatch (WGBYC). Using data collected by onboard observers collected between and , bycatch rates for ICES divisions on the continental shelf of the Bay of Biscay were estimated with ratio-estimators (ICES

). These estimates are not produced at the week-level, but (ICES ) also identified divisions .a and .b as the ones with the highest bycatch in mid-water trawls for common dolphins over the period -(Page ).

ICES (

) estimated yearly rates ranging between 0.285 to 0.372 dolphins per DaS and warned against extrapolation given the low observer coverage). Our

The current ICES mortality estimates do not make a fine spatial-temporal distinction. Our approach allows this distinction to be made for PTMs.

model-based approach overcomes this limitation (Authier, Rouby, and Macleod ) and allowed to identify, within each year, that weeks 2 to 6 were the ones with the highest bycatch numbers for both divisions .a and .b. This results were concomitant with the seasonal stranding pattern observed each year on the French seashore (that is, winter strandings; Peltier et al.

): around 80% of all common dolphin strandings on the French Atlantic seashore is observed between the end of January and the beginning of March.

A key feature of our model-based approach is how it leverages correlations between bycatch risk, haul duration and number of hauls per DaS (Figure . ). Some of the correlations are expected, such as the negative correlation between haul duration and the number of hauls per DaS. If more hauls are performed in a Day at Sea, so they are to be shorter and vice versa. However, average haul duration is not constant within a year, with the variations reflecting change in the commercial fish species targeted by PTM at different time of the year. These variations at the week-level were quite substantial, and were taken into account when estimating bycatch in our model. There was a positive correlation between haul duration and bycatch risk, with at least a two-fold increase in the later when haul duration exceeds 5 hours (Figure . ). This was particularly evident in weeks 1 to 5 in and and week 48 in . The

There seems to be a correlation between the fishing duration and the bycatch risk with a threshold effect. Beyond hours of fishing, the bycatch risk is greatly increased. Restricting fishing duration may be a mitigating measure.

latter was due to a single fishing trip with 5 hauls that lasted > 10 hours, each of which resulted in a bycatch event. We recommend, in light of the within-year pattern in haul duration (Figure . ), to investigate management actions and mitigation measures on limiting haul duration in winter to assess whether bycatch may also be reduced. The ability of our approach to propose more refined recommendations is a considerable advantage to reduce the impact of bycatch on the common dolphin population. However, it must be balanced against the fact that it has some

Benefits and limitations of the method developed and used

The case study presented in this chapter is based on the application of the Authier, Rouby, and Macleod (

) approach. This approach was developed to deal with low quality on-board observer data. A crucial assumption associated with this approach is the availability of accurate fishing effort data to perform the post-stratification step. The measure of effort chosen was the DaS, according to the current practice of international working groups. (e.g. ICES

). This guided the joint modelling of haul-level risk, haul duration and average number of hauls per DaS. This modelling choice has proved successful for PTM but cannot be developed to the same extent for other métiers, in particular for passive gears such as gillnets and trammelnets. In the latter case, a better measure of towing effort is soak time which takes into account the length and height of the net. This information can be difficult to collect, making it difficult to obtain an accurate measure of effort. This makes post-stratification difficult, but it is essential for the application of the model-based approach. The model developed for PTM may not necessarily transfer seamlessly to other métiers.

Since

, PTM flying the French flag are required to use deterrent acoustic devices (pingers ). If these devices are efficient to reduce bycatch risk, this may be taken into account in the model, by adding a covariate in Equation . . This requires post-stratification with the covariate using a fishing effort that explicitly takes the covariate into account. Currently, the model can implicitly account for the effect of pingers because it accounts for variations in bycatch risk from year to year and within a year. The large-scale deployment of pingers in , if effective in reducing risk, will result in a lower estimated risk than in previous years. In other words, the model does not necessarily have to take into account all covariates at the haul level, as long as the objective is prediction rather than explanation (Authier, Rouby, and Macleod ). Explicit consideration of the pinger effect is only necessary to make sense of variations in risk between and within years, but not necessarily to estimate these variations. While Authier, Rouby, and Macleod (

) concluded on increased accuracy of using regularized multi-level regression with poststratification to estimate bycatch with observer data, they also found that estimate precision was low. This was also the case in this study (Table . )

Implication for conservation

The large additional mortality due to anthropogenic activities on North-East Atlantic common dolphin population triggered a dedicated working group on emergency measures in : the workshop on fisheries emergencies measures to minimize bycatch of short-beaked common dolphins in the Bay of Biscay and harbour porpoises in the Baltic Sea (WKEMBYC) took place remotely in spring (ICES b) and informed an ICES advice that same year (ICES c). This advice led to an infringement procedure issued in July against France for failing its obligations under the Habitats Directive which lists the common dolphin as a species requiring full protection on its Annex IV. The same day, the Paris Administrative Court of Justice condemned the French government for failing to transpose and apply in a timely manner the dispositions of the Habitats Directive and of Technical Measures regulating fisheries (in French). Following the unprecedented number of strandings in , a national working group with fishermen, their representatives, government officials, Non-Gouvernmental Organizations and academics was initiated to address the bycatch issue (Peltier et al. ). One recommended action was to improve estimates of bycatch due to high-risk métiers, and to develop adequate methodologies to analyse data from non-representative samples (Authier, Rouby, and Macleod ). The present work reports on a case study on PTM and operating for a large part in the Bay of Biscay, and to a lesser extent in the Celtic seas. Model-based estimates can (i) provide information on pressures on common dolphins, as required by the MSFD (EU / ); and (ii) take into account the ICES recommendation to develop estimation methods to make best use of data already collected to inform timely management (ICES c). In the next chapter, we will see how it is possible to combine both estimated number of bycaught dolphins and demographic indicators to inform conservation algorithm and provide adapted tolerable bycatch thresholds. The North-East Atlantic population of common dolphins is affected by bycatch which threatens its viability. The status of the population is of concern in relation to the conservation objectives set by the Habitats Directive and the MSFD. The conservation objectives are the basis for establishing management strategies. These management strategies determine the thresholds beyond which the conservation objectives will not be met. In the case of the common dolphin population, these thresholds are those of bycatch, or removals. The challenge is therefore to define management objectives that correspond to bycatch limits in order to build management strategies that will allow the population to achieve the conservation objectives. ASCOBANS has proposed as an interim conservation objective "to restore and/or maintain stocks/populations at % or more of carrying capacity". Based on this objective, OMMEG proposed the following quantitative conservation objective for marine mammals in European Waters: "a population should be able to recover to or be maintained at % of carrying capacity, with probability . , within a -year period". Appropriate management strategies must therefore be defined to achieve this conservation objective for the common dolphin population. In this chapter we will determine the appropriate management objectives, i.e. the removal threshold. This study is part of a Management Strategy Evaluation approach that takes into account as much uncertainty as possible to provide relevant threshold. This approach involves the implementation of an operating model based on a Pella-Tomlinson population dynamics and a Leslie Matrix fed by estimated the vital rates in Chapter . A control rule based on the removal limit algorithm is applied to define the bycatch threshold at each time step during the years simulated in the management strategy evaluation. A performance statistic is obtained at the end of the simulated years. This statistical indicator, coupled with the use of real data, will ultimately allow to decide on the bycatch threshold to reach the OMMEG conservation objective.

Conservation

Publication in prep.

This chapter is associated with one publication: Rouby, E. and Genu, M. Setting Bycatch thresholds for common dolphin through Management Strategy Evaluation. In prep. ). This approach is not proactive and may lead to viability problems for dolphin populations (Wade ).

: PBR = 0.5 × R max × N min × F r with R max as the maximum theoretical or estimated productivity rate of the population, N min the minimum population estimate in number of animals and F r the recovery factor.

: For more information on the MMPA:

https: //www.fws.gov/international/ laws-treaties-agreements/ us-conservation-laws/ marine-mammal-protection-act.

html . Introduction: Conservation biology and Management procedure

In Chapter , we quantified the bycatch pressure exerted by PTM on the common dolphin population. In Chapter , we have seen that the bycatch pressure (any fishing métier) seems to decrease the survival within the population and to lead to a decline in its growth rate. Bycatch is actually one of the major threats to marine mammals (Avila, Kaschner, and Dormann ) particularly for small-sized cetaceans (Reeves, McClellan, and ) and is currently impacting the common dolphin population (Peltier et al. ) to a point of non-viability (Mannocci et al.

). The conservation of marine mammal populations requires understanding and assessing the consequences of bycatch (Wade et al. ). Two examples of unsuccessful conservation cases associated to bycatch were mentioned in General introduction Page with both the Baiji and the Vaquita and it is necessary to avoid similar situations for other populations. To do so, it is necessary to define conservation actions to promote the viability of marine mammal populations, in our case the common dolphin. Abundance indices are used to detect declines but once a decline is detected it is often too late to prevent the population from going extinct or it takes longer to recover to initial conservation status ( ). This is a reactive approach that is not precautionary (see Margin-Figure . for diagram; A. J. Read

). The development and use of pro-active conservation tools is of interest to prevent the extinction of the North-East Atlantic common dolphin population. An example of a pro-active conservation tool is the Potential Biological Removal (PBR) developed as part of the implementation of the Marine Mammal Protection Act (MMPA) (Wade ).

The MMPA is one of the pioneering texts in terms of managing marine mammal populations from a conservation perspective and it defines clear conservation objective (hereafter CO). The three CO retained by the MMPA are: ) Maintain the population above its Maximum Net Productivity Level (MNPL); ) Allow the population to recover close to its carrying capacity (K); ) Allow the population to recover considering a growth rate close to its maximum possible (Wade ). MMPA defines clear quantitative CO and lays out management objectives and remedial measures to meet the CO. The quantitative CO for the MMPA is the following: "a population will remain at, or recover to, its MNPL (typically % of the population's carrying capacity), with % probability, within a -year period." In the case of European waters, there is : Removals are bycaught dolphins removed from the population.

a lack of both: ) a legally-binding CO for marine mammals and ) management objectives with respect to human-induced mortality (ICES c; Rogan, A. J. Read, and Berggren ). No clear and quantitative CO are defined regarding the impact of removals on marine mammals population in European Waters. To address this, ASCOBANS defined a fixed percentage of the abundance of marine mammals population as a threshold to set the CO regarding bycatch in European waters (as mentioned in Chapter Page ). ASCOBANS passed two resolutions to set removal limit of marine mammal populations:

. "Unacceptable interactions" are, in the short-term, a total anthropogenic removal above . % of the best available abundance estimate (Resolution . passed in ). . Bycatch reduction have to be reduced to less than % of the best available population estimate as an intermediate precautionary objective (Resolution . and Resolution . passed in ).

This

In contrast to the USA, for which the MMPA defines a clear CO, Europe does not have a defined quantitative CO despite the ASCOBANS resolutions. OMMEG has provided a quantitative interpretation of the objectives set by ASCOBANS.

fixed percentage abundance approach have been used by the European Commission (EC) to assess the capacity of European member states to achieve the "Good Environmental Status" required by the MSFD. This fixed percentage approach has the advantage of being very simple in its use and easily understood. To apply it, at least one abundance survey covering the area associated with the population to be conserved and estimates of bycatch removals is required. It remains limited for two main reasons: ) Inability to take into account life-history traits and ) Does not take into account potential errors and biases (Winship ). Abundance estimates may not be synchronised with bycatch estimates, limiting proactive conservation initiatives. A more conservative approach is to define explicit bycatch thresholds as management objectives in relation to quantitative CO. The OSPAR Marine Mammal Expert Group (OMMEG) provided quantitative interpretation of the ASCOBANS interim objective "to restore and/or maintain stocks/populations to % or more of the carrying capacity" (ASCOBANS ). The quantitative CO defined by OMMEG in European waters in the context of the MSFD is the following: "a population should be able to recover to or be maintained at % of carrying capacity, with probability . , within a -year period.". Explicit bycatch thresholds can be produced in relation to this quantitative CO. ). Management strategy defines management objectives in the form of thresholds. Evaluating the relevance of management strategies (Management Strategy Evaluation, MSE) to achieve COs is an issue that can be addressed through modelling and simulations (Cooke ; Hilborn and Mangel ). MSE

The application of a management objective (e.g. removal limit) is a management strategy. The management objective is implemented through a strategy. To assess the adequacy of strategies, the MSE framework can be used. This framework involves the definition of an operational model and a control rule.

requires generative models that generate (synthetic) data that are similar to observed data and it also requires real currently available data. Models associated with MSE are data-generating mechanisms. They can simulate a population dynamic taking into account removals effect (i.e. Bycatch effect) in addition to natural processes such as density dependence. These models allow scientists to assess the performance of management measures in hypothetical scenarios in order to set efficient management objectives. The management objectives will be assessed using real observable and available data (i.e. abundance and bycatch data with their uncertainties). Taking into account uncertainty at several levels (e.g. underlying model, potential biases and observed data) is of importance to measure the robustness of management strategies and choose the better one. A good management strategy should cover all aspects of management in accordance with prespecified objectives. The thresholds regarding conservation of marine mammals are removal limits. A removal limit is an annual maximum number of animals whose removal would not result in excessive depletion of the population. MSE requires several components, including (list from Genu et al. ( )):

. One or Several Unambiguous quantitative CO.

. A data simulator (or operating model) to emulate the dynamics of the marine mammal population and the effects of anthropogenic activities on this population.

. A control rule, whose computation accounts for the expected quantity and quality of observable data, to set a removals limit beyond which the impact of human activities runs the risk of failing the aforementioned CO.

. Performance metrics (or performance statistics), necessarily context-dependent and reflecting the trade-off between the potentially multiple CO defined previously.

These elements are necessary to project forward in time the population dynamics (i.e. the number of animals at each time step according to population models operating within the data simulator). For each management strategy the control rule is applied.

Finally the performance metrics are monitored and assessed with respect to the CO. Items . and . should be agreed by all stakeholders and scientists should not be expected to set the CO (Mangel et al. ). Items . and . are more relevant to scientists whose task is to test a wide range of realistic scenarios in order to protect the management strategy against uncertainties and potential biases in the available data. MSE involves a tune via simulations. Tuning in the MSE context means to: find, with a large number of simulations, parameter values of the control rule that meet the CO (Genu et al.

).

There

We used an MSE framework to conduct exploratory analyses to define a removal limit (i.e. management objective) that meets the conservation objective set by OMMEG. is a need to produce sustainable bycatch thresholds for the common dolphin population through an approach that takes into account uncertainty by involving several types of data. This approach should be part of a management strategy evaluation process to enable the chosen bycatch limit to meet the conservation objectives set by OMMEG in European waters. In this chapter we will see the MSE procedure applied to the case of the common dolphin in the Bay of Biscay. This application is made in order to define bycatch thresholds for this population calculated according to the achievement of the quantitative CO set by OMMEG from the work of ASCOBANS. The work carried out in demography on the common dolphin (Chapter ) makes it possible to inform the population dynamics model constituting the operating model of the procedure. The recent development by Genu et al. ( ) allows for the calculation and redefinition of management objectives during the process. In line with the steps outlined above, we will start by choosing the CO for this population (OMMEG CO). Then we will see the implementation of the operating model (Pella-Tomlinson, General introduction Page ). Then we will detail the control rule (Removal Limit Algorithm, RLA) and finally the choice of a performance statistic (Quantile that define removals). These steps are described in the baseline scenario case (no bias in abundance and bycatch estimates). We will then explain the robustness trials that aim to measure the effectiveness of the management strategies in achieving the CO when bias is introduced (bias in abundance or bycatch estimates). After selecting the most appropriate performance statistic for the common dolphin through simulations, we will use this statistic to define a bycatch threshold. This definition is based on the use of real data in the procedure. This work was carried out in close collaboration with Mathieu Genu whose role was to code all the analyses as part of the development of the RLA package to implement the procedure described in this chapter. This work is a preliminary analysis and does not yet aim to give a limit. The work was carried out taking into account the Bay of Biscay which does not correspond to the extent of the management unit recognised for the common dolphin population. This choice was made to allow the use of current data from the Bay of Biscay and to start exploratory analyses. The following sections present the method and preliminary results. : The constant effect ( ) on estimation is applied on the survival model from Chapter with a logarithmic transform to be consistent with the modelling of age-at-death .

log = ( +log )+ × - ( . )
.

Methodology: Management Strategy Evaluation (MSE) Demographic adjustment to make the population stationary

The vital rates obtained in Chapter do not allow the population to be viable (Figure . with constant effect = ). To enable it to be viable an adjustment on survival was made through the adding of a constant. Several values were tested from . to . and the one that allows the population to be stationary was . (Figure . ). This constant was then added in the joint modelling and allows the estimation of age-specific survival rates (see Figure . ) that allow the population to be stationary. Using age-specific survival rates without adjustment, the population becomes extinct. The adjustment is made to obtain a stationary population. The adjusted age-specific survival rates are assumed to be representative of the natural survival of the population. years with constant effect on survival. The constant effect was applied to found the value for which the population remains stationary. With a survival intercept to which is added a constant effect equal to . , the population remains stationary over time. This effect is then used in the estimation of the survival model to obtain the age-specific survival rates of a stationary population which will be used to inform the Pella-Tomlinson population dynamics model.

MSE base case scenario

Item . Choice of conservation objective

The conservation objective used here for the common dolphin population in the Bay of Biscay is that of OMMEG: "attain or maintain the population at an abundance equal to % of its carrying capacity (i.e. K) over years with a probability of %" (OMMEG ), to be in line with ASCOBANS Resolutions . and . .

Item . Operating model and Pella-Tomlinson population dynamics

The operating model is the functional part of the Management Strategy Evaluation framework. The operating model corresponds to the true-state of Nature, the state of our common dolphin population that changes every year. The population's state is define through its abundance and its depletion level relative to its carrying capacity. Changes are done through the arrival of newborns, the birth rate, the sex ratio, the proportion of the population that is mature, the age-specific survival (and implicitly mortality) that contributes to the removal of individuals and the removals on the population due to bycatch. No migration is assumed and taken into account.

The population model is age-aggregated with a maximum lifespan for the common dolphin population set at years. The population model is based on a Pella-Tomlinson density dependence process (Pella and Tomlinson ):

N t+1 = N t + r max × N t × 1 - N t
Parameters of Equation . : N t as the abundance at time t, r max as the maximum population growth rate, as the carrying capacity and as the shape parameter.

( . )

Here, the MNPL is determined by . In the case of = 1, population dynamics is given by a logistic growth curve with MNPL = 2 . The value of the birth rate at time ( ) is density dependent. Birth rate is the mean number of offspring per female assuming a sex ratio of : . For example, a birth rate of 0.5 signify that each female produce 0.5 newborns per year. So 1 newborn is produced every two years by each female. Birth rate at time is expressed as follows:

birth rate = + ( max -) × 1 - = + ( max -) × 1 -( )
Birth rate ( ) calculation on Equation . depends on the birth rate at carrying capacity , the maximum birth rate , the real simulated population size at time ( ) and the carrying capacity . The depletion level of the population is assumed as:

and corresponds to the abundance of the population expressed in percentage of .

( . )

The number of newborns in the population is the sum of the product of the population birth rate , the age-specific abundance (

) calculated with the age-specific survival and the age-specific proportion of mature females in the population. The population model updates the total abundance and the age-specific abundance within a time-step of Year. Each year the processes on the population dynamic are applied as:

newborns t = × × (
) Anthropogenic mortality is applied through bycatch removals on each age-classes of the population.

) Natural mortality is applied through the age-specific survival rate on the population.

) Animals reproduce through the age-specific fecundity rate and the abundance of age-class is populated.

) Increment the time by one year.

Processes on the population are shown in Figure . in the operating model block. The conceptual diagram of population dynamics from the demography chapter (Chapter ) is taken up and adapted for the current study.

Since the initial depletion level of the population is unknown, initial depletion levels were simulated simultaneously. The aim is to cover the population dynamics given several initial depletion intervals The population model runs for years. After years, the first control rule (Removal Limit Algorithm) is applied.

Item . Control rule with the Removal Limit Algorithm

The control rule corresponds to the Removal Limit Algorithm (RLA) which is expressed as follows:

RLA t = ˆ × × max(0, -IPL) = ˆ × × max(0, -IPL)
Parameters of the Equation . . is the bycatch threshold at time , is the population growth rate, is the depletion level of the population at time and the IPL is the Internal Protection Level.

( . ) RLA t is the outcome, the maximal tolerable threshold removals that must be applied to the population in order to reach the CO. It is a management objective that set the removals applied each year on the operating model. It is reassessed each cycle of years since every years a survey campaign is simulated to fill in the value of ˆ , the estimated abundance of the population (in the base case scenario) . The estimation of RLA t is carried out in a Bayesian framework. Its posterior distribution is obtained from the estimate of the posterior distribution of both and (Genu et al. ). From the posterior distribution of RLA t a value must be chosen. It is the tuning part of the MSE. The tuning is a decision analysis. A quantile value within the posterior distribution is chosen in accordance with the achievement of the CO. A diagram illustrating the effect of quantile choice is shown in Figure . . The choice of quantile allows to set management objectives: the tolerable bycatch threshold is calculated by choosing a quantile from the Equation . . The choice of quantile affects the definition of the management objective. Given the posterior distribution of RLA t , a quantile value of % involves a tolerable bycatch threshold of individuals per year. A quantile value of % involves a tolerable bycatch threshold of individuals per year. The choice of the quantile value therefore defines the management objective. This choice is made in order to achieve the CO.

One important element is the presence of the Internal Protection Limit (IPL). The IPL is set to 0.54 = 54% of . If the abundance of the population at time , N t < IPL, so RLA t = 0. This protection is set in order to limit removals to if the population is too much depleted. One essential assumption of this part is that the removal limit is always respected (in the base case scenario) for the next years.

The steps constituting the items of the operating model can be completed with those of the control rule:

Operating model:

) Anthropogenic mortality is applied through bycatch removals on each age-class of the population.

) Natural mortality is applied through the age-specific survival rate on the population.

) Animals reproduce through the age-specific fecundity rate and the abundance of age-class is populated.

) Increment the time by one year.

Control rule:

) From the estimated abundance and simulations estimate the distribution of possible removals.

) From this distribution, determine the value corresponding to the quantile applied.

Item . Performance Statistic, Quantile for setting Removals

Originally, the Management Objective set by ASCOBANS was translated by the OSPAR Expert Group on Marine Mammals (OMMEG) into the following: After years, the depletion level of the population should be equal to % of its carrying capacity with a probability of %. At the end of the simulated years, the depletion level of the population is established. If it is above %, the management strategy is assessed as viable. But it is necessary to choose one. The choice is performed on quantiles. The management strategy which is retained to be applied is the one associated to the maximum value of quantile that allow to reach the management objective. For example, if quantile . is associated to a depletion of % after years and the quantile . to a depletion of % after years, the retained quantile will be . . Because it allows to reach the management objective set by OMMEG under ASCOBANS. The selected quantile can then be applied in the Management Strategy Application as it will be depicted latter. Figure . summarises all the steps needed to apply the MSE for the base case scenario which is the case where we suppose that we can fully trust our abundance and bycatch estimation data. However, it is not always necessarily the case. This is why we can take into account some bias in these estimations.

Robustness trials

We Two robustness trials were performed:

) Over-estimation of dolphin estimates by a half ) Underestimation if bycatch estimates by a factor of These robustness trials are preliminary. Further trials need to be carried out. The values have been set to explore the impact of bias on the final result.

considered an abundance estimate bias and a bycatch estimate bias in order to represent some cases when we can not trust the data (abundance and bycatch) that we have. The interest of using these bias scenarios in addition to the base case scenario is that it allows to be more cautious while choosing a quantile to perform the real case Application and finally obtain removal limit threshold. The first bias scenarios are based on the abundance estimate. As depicted in Figure . and Diagram . , an abundance estimate is done every six years (simulating a SCANS campaign) and from this abundance estimate, the new removal limit is set and and revised six years later. We studied the case when abundance is underestimated by half (e.g. , dolphins estimated instead of ,

) and the case when abundance is overestimated by half (e.g. , dolphins estimated instead of , ). We choose this values since they cover the confidence interval of SCANS II + CODA and SCANS III surveys which is approximately equal to . . The second bias scenario is based on an underestimate of the bycatch. The population model updates the abundance of each age-class depending partly on bycatch removals which are carried out on each age-class. If the management strategy is conducted with the total abundance obtained from the population dynamic model given a set of removals without bias (base case scenario), the removal limit set during the management strategy will be set according to the bycatch removals assumed as true. But it is possible that we underestimate these bycatch removals and set inappropriate removal limit. To cover this scenario we set an underestimation ratio of bycaught individuals equal to . In that scenario, the estimated number of dolphins bycaught per year is three times lower than the actual number.

Each of the two robustness trials considered here is intended to illustrate the difference in the choice of quantiles at the end of the analysis. They need to be complemented by other robustness trials to cover several possible situations. The definition of robustness trials is the last part of the MSE framework. Figure . shows the MSE steps given the base case scenario but the same scheme is applicable for the bias scenarios. After running the simulations over years, given several types of scenarios, we move on to the final stage of this study. The interest of having carried out an MSE approach is that a quantile value has been defined which is adapted to the achievement of the Management objective and which respects a precautionary principle without being totally conservative.

Application on common dolphins in the Bay of Biscay

The The practical application is based on the use of real historical data in addition to the demographic data used previously. The data used are abundance estimates and bycatch estimates.

final step of the whole procedure is to apply the MSE approach using real historical abundance and bycatch data. The operating model is used once with the historical data. Then the control rule is applied to obtain the posterior distribution of possible removals. The performance statistic (quantile) selected from the tuning in MSE base case scenario and robustness trials is applied on the removals distributions to choose a bycatch threshold (see Real case Application on Figure . ).

The abundance data that are used are related to the Bay of Biscay. This scale is not representative of the common dolphin management unit. We made this choice in order to start exploring the analyses and setting up the approach. We also made this choice on the basis of the available data for two reasons. First the demographic rates that were obtained in the Chapter were more linked to individuals that are associated to the Bay of Biscay than to the whole North-East Atlantic. It seems to me more accurate to use a similar spatial scale. Secondly, bycatch estimates are obtained from the analysis of drifting trajectories of stranded individuals. Here again, the individuals are mainly associated with the Bay of Biscay area. Thirdly, the management and conservation issues for the common dolphin in the North-East Atlantic are mainly linked to bycatch pressure in the Bay of Biscay. It is for these reasons that I believe it is currently more appropriate to use the estimated abundances in relation to the Bay of Biscay only. The tolerable bycatch thresholds that will be obtained subsequently will therefore be related to the Bay of Biscay "population". The abundance estimates : ObSERVE is an aerial campaign conducted in the Ireland's EEZ whose abundance surveys are not useful for application in the Bay of Biscay. More information on ObSERVE on https: //www.gov.ie/en/publication/ 12374-observe-programme/ : The abundance data used are not representative of the entire common dolphin management unit. The results obtained at the end will not be applicable to the management unit. This choice was made to facilitate the exploratory approach.

used for the applied case are obtained from three surveys that represent two time points of abundance assessment separated by about years: SCANS II + CODA and SCANS III . Abundances were estimated from observation by plane and boat. Where Scans III directly combines the two, SCANS II does not cover the whole area and the CODA ship survey was needed to complete the estimates. Abundance estimates were made depending on areas composed by "blocks" Figure . . The other historical data that was used to perform the Real case Application was the historical bycatch estimate made from the strandings between and . Theoretically, it is possible to use French pair trawlers (PTM) estimates that were produced in Chapter . But the estimation would be made according to one of all the fishing métiers operating in the Bay of Biscay. Other fishing métiers than PTM could have an impact on common dolphin in the Bay of Biscay. Within the French fishing fleet, gillnetters (GTR) and gillnetters (GTR) may be associated with common dolphin mortality in ICES divisions .a and .b. The approach used in Chapter is currently only applicable to PTM. In the meantime, it is possible to use the estimates made from the strandings. To Theoretically it is possible to use the estimates made in Chapter . For the time being they only concern one fishing métier and are therefore unusable. estimate bycatch, only common dolphins with evidences of contact with fishing nets were considered (Kuiken ) as well as the common dolphins found stranded during multiple or unusual mortality events related to fisheries. These multiple events are defined when a high number of strandings occur in a restricted area with a common cause of death. The threshold is defined at cetaceans over consecutive days recorded along a maximal distance of km in the Bay of Biscay and individuals per days per km of coastline along the coast of the western English Channel (Peltier et al. ). From the stranding location of the dolphin, a deterministic drift trajectory is inferred by a reverse drift model (developed by Méteo France) to determine its likely area of death at sea. The area of death is included in a grid cell. The number of dead dolphins in each cell is then corrected by the cell-specific probability that a carcass from that cell washes ashore (Peltier et al. ). The number of animals estimated dead at sea is available in supplementary material.

Baseline Management Strategy Evaluation assumptions

The vital rates estimated in Chapter are representative of those of the managed population. There is one closed and stationary population in the Bay of Biscay. The Conservation Objective produced by OMMEG is accepted by all parties and remains the same over the -year period considered. Abundance surveys are conducted every years within years. The Control rule used to produce the removals estimates remains the same over the -year period considered.

. Results

Results are first presented for Management Strategy Evaluation comparing the depletion trajectories obtained under each quantile and each scenario. Then the population dynamic under the real case application and the removals threshold obtained are presented.

Management Strategy Evaluation

Demographic dynamics of the base case scenario

Figure . presents the simulated estimated abundance and associated applied bycatch (limit set by RLA + stochastic variation) along time-line for the base case scenario depending on each quantile. The value of depletion is estimated every year for years. Results were averaged for all initial depletion levels. Only quantiles . to . are displayed because under . the CO is always reached. For each quantile the estimated abundance increases before reaching a plateau. The plateau is not reached after the same number of years depending on the quantile. All abundance estimation profiles are above the management objective ( % of K) except for the quantile at . . The quantile whose profile is closest to the management objective is the one at . . Only quantiles . , . , and . are depicted here. These are the most interesting to look at because it is at this point that the % threshold defined for K is reached. All quantiles below . meet the objective after years. Here it is the quantile at . that should be retained. Because it is the one that best fits the conservation objective. It is associated with annual removals that fluctuate around at the end of the -year simulation.

Removals are also displayed. Their value increases as the quantile increases with for example a maximum value of at the . quantile and a maximum value of at the . quantile. Likewise for the simulated estimated abundance, the profile shows an increase and a plateau which is reached more quickly when the quantile is high. The estimated bycatch value is about dolphins with the quantile at . after years. When we start the simulation, we do not know the depletion state of the population with respect to K. To cover this uncertainty, initial depletion levels are simulated for each scenario and quantile. In the base case, the quantile to be retained is . . In the case of the underestimation of abundance by a factor of , the quantile to be retained is . . In the case of an underestimation of the bycatch by a factor of , the quantile to be retained is . . The absence of curves for the last quantiles of the robustness trials is due to numerical problems.

Selection of the best quantile and associated removals

As the initial depletion state of the population is not known, several initial depletion states were simulated. For each scenario (base case, underestimated abundance, underestimated bycatch), six initial depletion states (from [ . , . ] to [ . , . ]) were simulated under seven quantiles (from . to . ). At the end of the -year time series, the performance of each management strategy is assessed by comparing the value of each depletion curve to the % carrying capacity limit (Figure . ). The best % depletion values for each initial state are presented over a period of to years. For the baseline scenario, the best quantile to select based on the simulation is . . This quantile value allows each depletion curve to meet the management objective of % K. For the underestimated abundance scenario, the best quantile to select is the quantile at . . For the underestimated bycatch scenario, the best quantile to select is the . quantile.

For each combination of quantile and scenario, it is possible to calculate the value of removals as a percentage of K (Figure . ). The resulting distribution of removal for all time steps (independent of the initial simulated depletion state) is summarised. The higher the quantile, the higher the removal values. These removal values are purely indicative and do not constitute the final result of the approach. The minimum is about . The quantiles that were obtained previously in the MSE approach involving the robustness trials allows to select removal value in the distribution.

Abundance

Application on common dolphin population

The previous exercise (management strategy evaluation) aimed to obtain the quantile values for each scenario to achieve the CO. The application case aims to produce a distribution of removals using the actual data. The quantiles obtained previously are used to define the removals to be applied from this distribution (Figure . ). They range from to approximately given the last abundance estimate. From this distribution (Figure .  management application part to a representation of this step in the framework.) it is possible to choose values given the quantiles selected under the MSE. The quantiles obtained from the previous MSE approach are presented as dashed color lines (Figure . ) and associated removals are given in Table . . The difference between quantile at % and % is about a factor of (approximately and ). The quantile at % is associated to the lowest number of removals which is about considering the last SCANS III abundance survey estimation of common dolphins in the Bay of Biscay.

Table . : Removals to be applied. Here the removals that allows the population to reach the OMMEG CO are depicted. The higher removal value is the one for the base case with removals per year. The mid removal value is the one for the underestimated abundance with removals per year. The lower removal value is the one for the underestimated bycatch with removals per year. In order to be conservative and to be cautious with the population viability, the lowest removal value should be kept. Here the yearly removals applied to the common dolphin population in the Bay of Biscay must not exceed individuals per year in order to reach the CO set by OMMEG.

Quantile Removals % % %

. Discussion

The aim of the study is to define a tolerable removals (i.e. bycatch) threshold (management objective) to achieve the conservation objective defined by OMMEG based on the ASCOBANS recommendations : allow the population to reach or maintain an abundance level equal to % of its carrying capacity (i.e. K) over years with a probability of %. The implementation of this management objective will then constitute the population management strategy until the next abundance surveys. For this purpose, demographic data were used considering an adjustment on survival to obtain a stationary population. Then the MSE procedure was applied involving the selection of the CO, and the combined use of the operating model and the control rule over a period of years to obtain the performance statistic. Actual abundance and bycatch estimates were then used to compute tolerable removal thresholds for the population given the latest abundance estimates. The choice between these thresholds then depends on the quantile selected from the MSE approach and the robustness trials associated with the base case scenario.

Conservation objective, operating model and control rule The choice of an appropriate CO

The definition of a clear and justified CO conditions the subsequent MSE approach. The CO is relative to K, the carrying capacity of the environment for the population. In practice it is impossible to know the exact value of this carrying capacity. But knowing its exact value is not limiting. The order of magnitude of the carrying capacity, however, is much more important. In the case of the common dolphin population, it is more important to decide between a carrying capacity in the tens, hundreds of thousands or millions. It is at this level that the maximum abundance surveys inform the CO. Defining the percentage threshold of this carrying capacity to be achieved is a more general management consideration. This definition directly determines the number of annual tolerable bycatches in the population. The The exact value of K has little influence on the final estimate. It is the order of magnitude of K that matters. The choice of an appropriate CB must take this order of magnitude into account.

higher the percentage, the lower the number of removals and vice versa. If the percentage set is very high, e.g. % of K, then only the lowest quantile values will achieve this CO (Figure . ). In this case, the tolerable number of bycatches will be close to . It may be difficult to implement this threshold in view of the socio-economic stakes of fishing in the Bay of Biscay. If the percentage set is low, e.g. % of K, then the higher quantile values will also achieve the CO. In this case the number of bycatches may be very high. The management objective will do little to limit fishing but may be dangerous to the viability of the population. It is possible that the population will be too depleted before the next abundance survey ( -year cycle) to redefine the management objective. The population could also be subject to an Allee effect that would cause its extinction beyond a given abundance threshold. It could be interesting to implement an Allee effect in the operating model to take this into account. This Allee effect could be expressed as a probability of encounter between males and females as an example. The question of how to define an appropriate CO that takes into account the conservation of populations and the management of human activities is a key issue at the border of several disciplines.

Considerations on the Operating model

The MSE used here is based on multiple steps and fundamental assumptions. The operating model is based on the Leslie population matrix model with a Pella-Tomlinson density dependence for birth rates (Genu et al. ). The age-specific survival rates used to fuel this model are those obtain for the common dolphin population in the Chapter adjusted to allow the population to be stationary. This construction makes it possible to obtain : mPBR is the modified PBR. It allows the estimation of tolerable removals thresholds taking into account the European conservation objective. a population that does not decrease without the application of bycatch. This adjustment is a strong assumption that does not go hand in hand with what is obtained in terms of demography on this population. The Leslie matrix population model used here does not admit explicitly immigration. This assumption is very restrictive since the common dolphin population in the Bay of Biscay seems not closed as the abundance surveys and demography suggest.

Control rule and importance of data provision

The main interest of this work is that it is the first to be done with specific and locally determined demographic parameters on this population. This age-specific structured model allows the use of the RLA which is a possible control rule among others. Others control rules that can be used in the MSE are the PBR and the mPBR . They did not require age-structured models but only the abundance of the population. The PBR can be calculated as it follows:

PBR = 0.5 × R max × N min × F r
Parameters of PBR: R max as the maximum theoretical or estimated productivity rate of the population, N min the minimum population estimate in number of animals and F r the recovery factor.

( . )

From this control rule, a total of individuals per year were estimated as the limit of removals for the population in the whole North-East Atlantic area (ICES b). The number of estimated bycaught dolphins for recent years is higher than this threshold. This thresholds is close to the one calculated here, in the base case scenario, using the RLA for the Bay of Biscay. The calculation of the PBR relies on the management objective of the population reaching % of carrying capacity. This CO is the one implemented by the MMPA. It does not corresponds to the CO proposed by the OMMEG group using the interim objective from ASCOBANS. The mPBR is suited for the ASCOBANS management objective and is associated to much lower bycatch thresholds with a removal limit around individuals. This limit is closer to the estimation done using the RLA considering the scenario of a bycatch underestimation.

The use of RLA is less conservative when the accuracy of the data is higher and when there is more data. The removal limit determined when using the RLA on the Bay of Biscay population is close to that obtained from the PBR in the North East Atlantic. In this case, the use of population-specific demographic data allows for less conservatism. The use of the RLA in the framework of robustness trials also shows this particularly in the case of bycatch. In the base case scenario, bycatch estimates are assume to be accurate. This may look like transparent bycatch declarations from fisheries. In this case, the tolerable removals on the population : As a reminder, % of individuals are considered dead due to capture in fishing nets on the basis of external examinations in and % on the basis of internal examinations.

of the Bay of Biscay is dolphins per year. In the case where the number of catches is underestimated by , this limit is greatly reduced by a factor of almost ( tolerable removals each year in Bay of Biscay). Robustness trials in that case shows that knowledge of bycatches makes it possible to be less conservative. It is therefore in the interest of fisheries to communicate transparently on the number of bycatches made per year.

Application of the Management Strategy

In the case of the common dolphin in the Bay of Biscay, the most efficient management strategy to promote the sustainability of the population (without completely closing the fisheries) seems to be to set a limit of bycatches per year. This threshold is associated with an underestimate of bycaught dolphins by a factor of . This scenario is extreme and other and less extreme robustness trials should be tested in the future. Within this study, this management objective seems to be the most precautionary given the scenarios tested and the current data. The base case scenario with a bycatch threshold of implies full confidence (i.e. unbiased) in the abundance estimates, bycatch estimates and in the demographic data in the operating model. Bycatch estimates are currently made by modelling the reverse drift of carcasses (with bycatch evidence) found stranded on the seashore. It is difficult to be sure that these estimates are accurate. For example, they may be overestimated. The overestimation may be related to a possible misinterpretation of some bycatch evidence in favour of an over-representation of bycaughts individuals . This hypothesis seems unlikely given that the evidence of incidental captures is well characterised for common dolphins (see Figure .  in Chapter ). In France, the examination protocol is well defined and applied by the members of the national stranding network. But the scenario of overestimation of catches will be tested. The scenario of underestimated catches foresees a maximum catch limit. If a catch limit were to be applied, it would be individuals per year. This is the lowest limit associated with the robustness trials. It is the most conservative value. Taking this threshold into account, the estimates we made in Chapter are still below the threshold. Except if we take into account the % limit of the Bayesian uncertainty interval. But our estimates are for a single fishing métiers (Table . ). Bycatch estimates are above this threshold in the Bay of Biscay, taking into account baseline stranding estimates. The current level of mortality reflects the current management strategy for the common dolphin population. This strategy involves the economics of the fishery, the viability of the common dolphin population, the viability of the target fish species, public opinion and the certainty of estimates. In addition to these components, management effectiveness is also taken into account. A qualitative representation of the performance of these components for each management strategy can be imagined ( Figure . ). This representation is adapted from (Punt et al.

). The current management strategy in the Bay of Biscay looks more like A. Despite ICES recommendations to implement time-area closures of fisheries, the government has chosen to favour the fishing economy. Public opinion is negatively influenced by the reports and the action of NGOs on the subject. If bycatch thresholds are defined and respected, then the management strategy could move from format A to format B. The sustainability of the dolphin population would be better taken into account, to the detriment of the fishing economy and in favour of public opinion. This chapter closes the analytical part of the manuscript. At this stage we have explored different aspects:

Chapter on page : Description of the common dolphin, its population in the North-East Atlantic, the associated conservation issues and the monitoring carried out on this population. Chapter on page : Development of a new survival analysis approach that takes into account covariate and random effects to estimate vital rates. Chapter on page : Estimation of vital rates for the common dolphin population, within the framework of the MSFD, which are estimated by taking into account various covariate and random effects. Chapter on page : Explanation and quantification of the bycatch phenomenon involving PTMs on a weekly, annual and ICES divisional basis. Chapter on page : Establishment of a tolerable bycatch threshold given the population's demographic viability, abundance, bycatch pressure and the CO set by OMMEG in European waters.

The following part is a discussion presenting three aspects related to this thesis. The first section deals with the general approach of the project. The second section deals with the North-East Atlantic common dolphin population. The third section deals with the cross-sectional monitoring.

. Project approaches

Aspects related to the demographic study

This thesis project was carried out using demographic approaches. The first approach was to develop a new methodology for estimating survival, based on the Reed model, from age-atdeath data. This approach is a useful contribution to the literature. Its use implies the inclusion of an inter-individual frailty parameter. In addition to allowing the modelling of a bathtub shape hazard curve, the use of the frailty parameter allows heterogeneity within the study population to be taken into account. It is difficult to interpret this parameter on its own and it is more accurate to consider it as a statistical device for robust estimation. However A possible improvement in the estimation of survival is to take into account two frailty parameters. One parameter specific to the calf component and the other specific to the non-calf component. Such an improvement could be made through a mixture modelling approach.

it may be possible to strengthen its biological interpretation. It would be interesting to consider two inter-individual frailty parameters. One associated with the calf component of the life history, the other with the non-calf or juvenile-adult component. This would allow a better representation of survival and mortality in the population, taking into account the different frailties at different periods of life. The survival estimation method is based on the use of age-at-death data and it is difficult to know whether the estimated vital rates (e.g. survivorship) are representative of those in the population. A possible solution would be to have both longitudinal monitoring data and age-at-death data for a population. Comparison

To check the plausibility of the survival rates estimated by the statistical approach, it would be interesting to simulate both longitudinal and age-atdeath data and compare the survival estimates.

of the longitudinal survivorship profile with that obtained through the cross-sectional survival approach would highlight the accuracy of the survival estimates. A limitation of this comparison is that, if there is a difference, the source of the difference cannot be defined. Is it a difference due to sampling or a difference due to the statistical approach or a combination of both? One way to compare would be to simulate longitudinal data as well as age-at-death data. Sampling and bias control would be provided by the simulation study. Any differences would be due solely to the statistical estimation of survival.

To control for sampling bias, stratified random sampling was useful. The use of this sampling framework allowed for greater confidence in the estimates of vital rates. When sampling in this way, the survival profiles estimated by the Siler and Reed models were almost identical. But this sampling scheme can only be used if there is a highly structured, centralised and coast-wide stranding Network. The French national stranding network is one of the most stable and unified stranding network in Europe. In Spain there is no single, centralised stranding network. Spain has many regional strandings networks along the mainland coast as well as separate networks for each of the Canary Islands. Germany has two stranding networks, while the Netherlands has four. Few European countries have a single stranding network. For those countries where this is the case, the network may not always be stable or long established (Figure . . The French national stranding network has the advantage of being very reliable since the s in recording all strandings on the Atlantic coast (Authier et al.

). Of the stranding networks operating in North-Atlantic European waters, routinely determine the reproductive status of females and on ad hoc basis. With regard to age determination, data acquisition is variable since only networks determine age ias part of a routine analysis, while do so on an ad hoc basis (see Margin-Table . ). It may therefore be difficult to generalise this sampling approach to the whole of the European level. The possibility of including covariate and random effects in the analysis can provide a demographic indicator for the MSFD. At present, the criteria associated with obtaining demographic data for the MSFD descriptors are considered secondary. In other words, Member States' efforts to fulfil the descriptors should not focus on these criteria as a priority. Criteria Estimating vital rates taking into account random effects can be a new indicator of demographic status within the MSFD. This would help to identify the possibility of depletion before it occurs. associated with obtaining abundance data, for example, are considered primary criteria that can provide information on the viability of marine mammal populations (including the common dolphin). In Chapter , we have seen that the viability of the population seems to have deteriorated over the period of time considered ( to ). The use of random effects taking into account the cohort effect allowed us to identify possible differences in survivorship and age at sexual maturity for females. These differences could be used to develop an operational demographic indicator to immediately inform the demographic criteria of the MSFD descriptors. This indicator could take the form of a percentage difference in vital rates between cohorts (Figure . ). Differences from a baseline or mean profile could be reassessed every years in each MSFD cycle.

- , the survivorship of adults seems reduced compared to the mean survivorship over the period.

Mean profile

The use of covariates also revealed a possible effect of bycatch on vital rates. This effect is quantifiable and could also be a pressure indicator for the demography of the common dolphin population to inform the MSFD. The effect of various anthropogenic pressures, such as pollutants, can be quantified at the demographic level for this population. For these reasons, I think Demographic criteria are considered secondary. They should be reviewed as primary criteria. Demographic information for elusive marine mammal species can provide more detailed information on impacts and viability.

that the criteria for assessing the "Good Environmental Status" of European Waters based on demography should not be considered secondary. Vital rates are important for knowing the viability of the population and it is now possible to quantify variations in the viability of the population. Vital rates provide earlier information on whether the viability of a population is impaired than abundance surveys. It is possible to identify a risk of depletion before it is observed. These criteria should be reviewed as primary criteria. This demographic indicator could be used to fulfil the MSFD at the sub-regional scale. The MSFD considers the evaluation of water quality on a regional and sub-regional scale. For the common dolphin, the regional scale is the "North-East Atlantic", while the sub-regional scale is the "Bay of Biscay and Iberian Coast" and "Celtic Seas". Using the individuals stranded on the shores of these two sub-regions, an operational indicator could be produced to inform the demography of the common dolphin within the two sub-regions.

Aspects related to the conservation approach and management issues

This

The estimation of the number of animals bycaught and associated variables makes it possible to valorise biased observer data. These data provide fine-scale spatio-temporal estimates. However, the use of this approach is limited by the availability of an accurate fishing effort dataset and sufficient bycatch observation data.

thesis project was carried out with the implementation of a management approach. The management approach is based in part on the use of bycatch estimates. The approach used in Chapter to quantify bycatch from on-board observer data shows promise for using biased data. This approach can be used to add value to data that already exists but are considered biased. The ability to work with data that are considered biased should not prevent the collection of more representative data. Although promising for taking observation bias into account, the use of this approach is currently limited to PTMs. To extend its use, it would be necessary to be able to effectively calculate the fishing effort of passive métiers such as gillnetters. The use of this approach in the context of implementing management strategies currently seems to be limited since it cannot be generalised to all fishing métiers. On the other hand, it is interesting in terms of improving data that are considered to be of little use by the research community and providing information on the phenomenon of bycatch at a fine scale. The use of the joint model also makes it possible to highlight possible correlations or links between the various estimated variables (e.g. fishing duration and bycatch risk). The removal threshold produce through the MSE approach does not include any socio-economic index as it is done for some fisheries evaluation models such as the Atlantis model. The Atlantis model includes all the components of marine ecosystems from physics to human exploitation (Audzijonyte et al.

; Figure . ). The use of these complex models may be limited by their complexity. It can be difficult to define the links between all parts of the model and to understand how these parts interact. Analyses in the context of an MSE incorporating RLA are already a complex task involving many assumptions. For this reason, I think that the development of more complex models may be interesting from a theoretical point of view but not necessarily interesting from an applied point of view. Discussions between stakeholders (fisheries, scientists and decision-makers) are already difficult with regard to the bycatch and removal threshold. In my opinion, we need to keep it simple : Follow this link to see the full procedure of the European Commission: https://ec.europa.eu/ commission/presscorner/detail/ en/INF _ 20 _ 1212 in order to facilitate understanding and decision making on the strategy to be adopted. From Methods for estimating tolerable bycatch rates should be kept simple, otherwise they may not be understood by all stakeholders and may therefore be difficult to apply. a scientific point of view, this should be reflected in the use of methodologies with limited complexity, rapid applicability and low dependence on computing time, as well as in the transparent availability of the methodologies used.

The common dolphin, an elusive species

The notion of elusive species covers both demographic and management aspects. From a demographic point of view, these are species for which it is difficult to obtain data and vital rates. It is difficult to get an idea of the viability of the population on the basis of vital rates. It is also a species whose conservation is important in terms of the ecosystem balance (i.e. keystone) and therefore in terms of international legislation, especially in Europe. In Europe, the population is difficult to manage. Limited demographic knowledge is a contributing factor (Murphy et al. ). But difficult management discussions between fisheries, politicians, scientists and NGOs are also a contributing factor. Data on bycatch of this species in French waters are also very limited. The European Commission has set up an Infringement Procedure against France in July :

"The Commission is asking France, Spain and Sweden to implement the measures required under the Habitats Directive (Council Directive / /EEC) and Common Fisheries Policy to avoid unsustainable by-catches of dolphin and porpoise species by fishing vessels. [...]

Despite well-documented evidence of these species being caught in fishing gear, the problem persists. France, Spain and Sweden have not taken sufficient action to monitor by-catches in their waters and by their fleets, nor made full use of the possibilities that the Common Fisheries Policy provides to comply with their obligation under the Habitats Directive and protect these species. [...] As France, Spain and Sweden have not taken the necessary steps to address these deficiencies, the Commission is sending letters of formal notice to the three countries, which will have three months to address the shortcomings raised. Otherwise, the Commission may decide to send a reasoned opinion" .

One of the difficulties in monitoring this population is that it is difficult to define and relate it to the OSPAR Management Unit. The next section is a discussion of the definition of this population.

. Definition of the North-East Atlantic common dolphin population Population migration and possible distributional changes

Exchanges between stocks and management units

Animal

Stocks can exchange individuals through dynamic source (donor) and sink (receiver) migration processes.

populations can exchange individuals through the phenomenon of migration which may occur between different stocks in the case of the common dolphin. Migration is composed of immigration which is the arrival of new individuals, including breeders, into the population, and emigration which is the final departure of individuals from the population. Sometimes the flow of individuals between two populations is not reciprocal. In this case, we speak of a source population that provides individuals to a sink population that receives them. This migration flux implies that the ecological conditions are met. Firstly, there must be a high production of individuals in the source population. Secondly, the sink population must suffer a high loss of individuals (e.g. through high extrinsic mortality) leaving vacant places to occupy the trophic niche of the species. Thirdly, there must be a migration corridor between the two populations with no geographical barrier.

(

).

The

There is an inconsistency between the population projections and the observed trend in abundance in the Bay of Biscay and in the Management Unit.

vital rates obtained in this project are not consistent with the abundance estimates, which raises the question of the definition of the management unit. The vital rates are associated with a large majority of dolphins presumed dead on the Bay of Biscay continental shelf. Population projections show extinction in years, as do the projections made by Mannocci et al. (

). The population status also appears to have deteriorated since , with an increasingly low growth rate. But these estimates are not consistent with the observed trend in abundance. The population does not appear to be declining at the scale of the Bay of Biscay and the management unit covering the Bay of Biscay, Iberian coast, Celtic seas and North Sea. It

The management unit may not be fragmented enough. Several subunits should be defined as there may be sink-source exchanges between continental shelf stocks.

is possible that the Bay of Biscay shelf is associated with a sink population that receives individuals from other continental shelves (e.g. Iberia and Portugal) and Celtic seas. In this case, it is possible that the currently recognised management unit is too large and needs to be redrawn according to the classification proposed by Caurant et al. ( ). Another possibility is that the resulting vital rates are representative of the management unit recognised by OSPAR but that this management unit receives external individual inputs allowing it to be stationary. It is assumed that the population is not viable given current estimates of bycatch numbers across the management unit (ICES ). But the population abundance seems to be stationary at the scale of the management unit (Hammond et al. ). The

The management unit may not include the majority of the common dolphin population. In this case, it may not be large enough and should be expanded to the North Atlantic scale.

currently recognised management unit may not include the majority of the common dolphin population. Migratory flows between the currently recognised management unit and other stocks could explain the stationarity of abundance. This hypothesis is difficult to verify. But it is possible that the currently recognised management unit is not the most appropriate one (Figure . ). On the one hand, by omitting external migrations. On the other hand, by not being sufficiently fragmented. ). There were more common dolphins observed in summer on the oceanic part of the Bay of Biscay than in winter. Figure from (Laran et al. ) It is difficult to have a precise idea of the distribution of the common dolphin within the management unit but it is possible that seasonal variations in abundance in the Bay of Biscay are linked to source-sink dynamics. Abundance between summer and winter on the slope and the oceanic part of the Bay of Biscay appears to be different (Figure . ). The influx of individuals in summer ). These seasonal changes have been identified by the Seasonal Abundance of Marine Mammals (SAMM) surveys (Laran et al. ). Common dolphin sightings are from winter ( November to February ) and summer ( May to August

). The There seems to be a difference in dolphin abundance between summer and winter in the Bay of Biscay. This difference mainly concerns the continental slope and the oceanic part of the Bay of Biscay.

winter period sampled mixes months of very low density (November-December) and months of very high density (February) (unpublished data). In my opinion, one main highlight of these surveys is that the oceanic part of the Bay of Biscay may be subject to immigration during Summer (reproduction period). The abundance of small-sized delphinids (common dolphins and striped dolphins) in the Bay of Biscay in Summer was estimated at , individuals ( % CI: , -, ) whereas in winter it was estimated at , ( % CI: , -, ). It is possible that the difference in abundance of individuals is due to the contribution of nonresident individuals in the Bay of Biscay that come to breed in summer. It is difficult to say whether some of these individuals then remain in the Bay of Biscay to live there. It is also difficult to know the sex and age of the individuals that might move.

Overall change in distribution

These seasonal movements could be nested within larger spatio-temporal phenomena. Oceanographic processes are known to be structured in space and time. One way to represent this structure is the Stommel diagram generally involves two dimensions (space and time) but may involve a third (e.g. biomass). These representations are initially used to characterise the spatio-temporal phenomena of marine ecosystems (Haury, McGowan, and Wiebe ; Margin-Figure . ). These diagrams are useful for gaining a better understanding of the spatio-temporal phenomena structuring a system and their scales. They have also been applied to characterise the development of the biosphere ( Holling) and also the socio-ecological systems (Holling, Gunderson, and Ludwig ). I here provide a rough example of diagram that can be used as an approximate representation (Figure . ). The advantage of such a representation is that it allows the linking of different potentially related phenomena that do not take place at the same spatiotemporal scales. Here, it allows us to move from changes in the life history of a resident population, which may be driven by migration (when allocating resources for breeding or feeding), to larger scale phenomena such as distributional changes.

The presence and residence of top predators is strongly influenced by the presence of their prey. Low prey availability can This raw diagram presents processes associated with the population at three spatiotemporal scales. The population can be resident of the Oceanic or Shelf parts for about two months. Individuals may migrate within the Bay of Biscay each year. The population may change its distribution on a larger spatial and temporal scale later. This diagram is only a rough and conceptual view and should not be considered as completed.

influence the energy budget of the predator and ultimately influence its demography (MacLeod et al.

). The harbour porpoise (Phocoena phocoena, Linnaeus

) is the smallest odontocete in Europe with a length of about . m and a weight of kg; it is also found in the North-East Atlantic ocean. This species has a shorter lifespan, a younger age at first reproduction and a shorter calving interval. Because

The viability of the predator population depends partly on its prey. The harbour porpoise, for example, is heavily dependent on sandeels in the North Sea.

of its small size, the harbour porpoise cannot survive long periods without feeding and almost dies after days without food (Kastelein et al. ). The harbour porpoise was widespread in the North Sea and fed mainly on sandeels (genus Hyperoplus). These fish species are essential to the ecosystems of the North Sea (Greenstreet ). They are also essential for the survival of harbour porpoises in spring. This season is associated with the coldest water temperatures in the North Sea, which requires a high energy intake to maintain a stable internal body temperature (Watts, Hansen, and Lavigne ). The presence of sandeels is also crucial for the survival of weanlings that are about to take their first food intakes.

This

The bottom-up of sandeels over harbour porpoises, combined with a decline in the sandeels stock, may have caused a change in the distribution of harbour porpoises in the North-East Atlantic.

type of ecological control is called "bottom-up" control. Sandeels may induce bottom-up control on harbour porpoises. The large fishing industry that developed during the s contributed to the decline of the sandeel stock in the North Sea (Furness ). Furthermore, the recruitment of sandeels in the North Sea has decreased since the s, with a possible combined influence of fishing and climate change (ICES ) through an impact on plankton communities (Beaugrand et al. , ; Frederiksen et al.

). The decrease in sandeel stocks and quality has had an impact on harbour porpoises, with greater starvation and lower survival rates in the North Sea (MacLeod et al.

). It is possible that this limited access to the resource has led to a migration of harbour porpoise from the North Sea to adjacent waters (Hammond et al. , , ). It is possible that the common dolphin population is at a stationary abundance throughout the Northeast Atlantic and that the demographic rates estimated in this project are related to part of the population sampled in the Bay of Biscay. There may be more prey of interest available in the Bay of Biscay than in the past. Fishing

The demographic rates obtained in the Bay of Biscay may not be representative of the whole North-East Atlantic. It is possible that bycatch pressure is responsible for the observed values.

mortality pressure in the Bay of Biscay would be higher than in other parts of the North-East Atlantic, but population abundance would be essentially stationary on a large scale. Abundance surveys in the North-East Atlantic of SCANS II + CODA and SCANS III seem to show that the abundance of the population is almost stationary or perhaps increasing very slightly. Recent works by Evans and Waggitt (

) also supports an increasing trend in abundance observed in the North-East Atlantic using Waggitt et al. ( ). Recently, Astarloa et al. ( ) also highlighted a trend of increasing abundance of common dolphin in the Bay of Biscay and the correlation of this trend with environmental and biological variables using the Vector-Autoregressive Spatio-Temporal (VAST) model (Thorson ). The authors point out that the increase in estimated abundance is not necessarily a sign of a growing population but may reflect migration from unsampled areas. The correlation between common dolphin abundance and the North Atlantic Oscillation Index, prey abundance and chlorophyll is studied (Figure . ). Some correlations are highlighted but it is difficult to attribute the possible change in distribution of common dolphins to these variables. It is difficult to link variations in abundance or changes in distribution of common dolphins to environmental and prey data. This is mainly due to the scarcity of data and the rare co-occurrence of the two predator-prey system data in the large scales of the phenomena of interest (Gilbert et al.

; Astarloa et al. ).

Abundance estimates: an incomplete indicator

Abundance is an indicator of interest to assess the "Good Environmental Status" of marine European waters (European Parliament

) but it may be incomplete to assess the viability of megafauna populations. This indicator is considered to fulfil Descriptor (D ) and Descriptor (D ). Abundance assessment at the scale of a particular population is very instructive since it provides an initial assessment of the population's state (Laran et al. ). At the scale of a community (e.g. cetaceans in the Bay of Biscay) it can also inform the biodiversity state of the cetaceans community which is an important aspect of the D criterion (Azzellino et al. ; Lauriano et al. ).

Figure . : Percent of stocks within four ranges of statistical power. Statistical power is the probability of correctly rejecting the null hypothesis that a population is not declining (i.e., one-tailed t-test) when the stock is experiencing a precipitous decline ( % over yr). Results are summarized for six categories of marine mammals: large whales, beaked whales, dolphins and porpoises, pinnipeds on ice, pinnipeds on land, and polar bears and sea otters. The ability to detect population declines in abundance below a critical threshold remains poor for many marine mammal species. Abundance trend cannot be effective in identifying populations at risk because the statistical power to resolve abundance trends is sometimes very low if the time series is sparse (B. L. Taylor et al.

). The statistical power to detect a precipitous decline through abundance studies was calculated by B. L. Taylor et al. ( ) (Figure . ). A Using abundance trends alone to define the status of small cetacean populations can be dangerous due to low statistical power.

precipitous decline is defined by the authors as a "decline of % over a -yr monitoring period". They justify this choice by the fact that such a decline would lead to the classification of a stock as "depleted" under the MMPA and as "vulnerable" or "endangered" under the IUCN Red List guidelines (B. L. Taylor et al.

). In Figure . , we can see that the statistical power (probability of correctly rejecting the null hypothesis that a population is not declining) is very low for dolphins and porpoises population (the common dolphin was not included in their analyses). These results indicate that the use of abundance estimates to detect precipitous declines is not appropriate for small cetacean populations.

Examples of delays between the onset of extrinsic mortality and the implementation of management measures

Uncertainty in abundance indicators may limit the conservation of small cetaceans affected by bycatch (B. L. Taylor et al.

). There may be a delay between the onset of pressure and the implementation of management measures. This delay may be detrimental to the viability of populations. Two examples will follow to support this point.

Harbour porpoise in California

The The harbour porpoise in California and the Tuna-Dolphin problem are two representative examples of the delay between the onset of depletion, its recognition and the implementation of management measures.

first example concerns the harbour porpoise in California. In the mid-s, an increasing number of porpoises were found stranded on the shoreline, as well as an expansion of coastal gillnet fishing in central California. These two observations led to the assumption that the population was threatened by fishing. To monitor this population, mortality was assessed by an observer program and porpoise abundance was estimated by surveys and back-calculation from fishing effort and mortality rates (B. L. Taylor et al.

). Uncertainty in the parameters tended to make it impossible to determine the level of depletion and the MNPL. The MMPA was not able to conserve this species and its preservation was achieved indirectly through fishery closures associated with the capture of sea otters (B. L. Taylor et al.

). This example shows that the mere availability of some abundance and bycatch data does not allow the implementation of conservation measures at the right time.

Tuna-Dolphin problem

The second example involves many steps over years, encompassing many areas of expertise (biology, conservation policy, legislation), media coverage and the creation of a label for consumers (Supplementary-Figure ). It is known as the "Tuna-Dolphin Problem" and has recently been the subject of a review made by Ballance et al. ( ). The history of this problem is extensive and is provided in Appendix . . In the East Pacific Ocean, large-bodied yellowfin tuna (Thunnus albacares, Bonnaterre

) associate with several species of dolphins such as the pantropical spotted (Stenella attenuata, Gray

), the spinner (Stenella longirostris, Gray ) and the short-beaked common dolphin. These dolphins interact together with birds in a beneficial way (Scott et al. ; Ballance, Pitman, and Reilly

). These associations are readily visible at the surface and provide some indication to purse seine fisheries targeting tuna. Purse-seine are deployed near dolphin schools, resulting in high levels of bycatch (Figure . ). All three dolphin species were recognised as bycaught in the purse-seine Tuna fishery (Perrin ; Figure . ). Catches were in the hundreds of thousands between and . Between and , they fell to around a thousand. ).

The observation of bycatch numbers from to , and in particular the outcry over the magnitude of dolphin mortality, prompted the establishment of the MMPA in , which aims to reduce bycatch mortality to a level close to zero. Given the degree of uncertainty regarding the status of the population, numerous data acquisition efforts have been made to justify its depleted status. In total, "nine abundance surveys over years, years of relative trend indices from data collected on the tuna vessels, years of observer data on dolphin mortality rates in the fishery, and years of data on fishing effort" have been required. From the first abundance estimate to the recognition of the depleted status of the population, years of inaction were reported (Wade ). By the late s, it was clear that the dolphin mortality was too high relatively to the populations abundances estimates. However, it was only after sufficient data had been collected and analytical methods developed, i.e. until , that it was possible to prove that the dolphin populations were reduced to % of its total abundance (Wade a,b). At this time, the populations were declared depleted under the MMPA (depletion being defined as stock abundance below % of carrying capacity; Ballance et al.

).

This example is similar to the case of the common dolphin in the North-East Atlantic. There is currently uncertainty about the true status of the population, given the abundance estimates and the poor usability of on-board observer data. EAlthough we have developed new methods and provided new bycatch estimates from a biased observer scheme, it is not possible to estimate the total bycatch mortality of the population due to the lack of adequate data (and not because of a lack of methodology, e.g. see Chapter

). Using only abundance estimates, observer programmes and stranding mortality monitoring, it is possible that the mismatch between the actual time of depletion and the recognised time of depletion is dangerous for the viability of the population.

. Population dynamics of elusive species

Life-history traits variations We found a possible decrease in the age at sexual maturity of females compared to the methodology commonly used in the literature. The age at sexual maturity of females decreases by about one year according to these results ( to years). Using the joint model and random effects approach, it appeared that the age at sexual maturity of females also appeared to decrease over the period considered. It can be assumed that this possible change is part of a life history optimisation (Ferriére and J. Clobert ). Age at sexual maturity should respond differently depending on the mortality experienced by the population and on the presence of density-dependent mortality (Reznick, Bryant, and Bashey ; Figure . ). It

The population may undergo a process of variation in vital rates with a shorter life history than before (shorter life expectancy and advanced reproduction).

is possible that increased adult mortality in the population favours genotypes that reproduce earlier. This assumption has also been made by Perrin, J. R. Henderson, et al.; Barlow; Chivers and Myrick Jr for the North-East Pacific dolphins (Perrin, J. R. Henderson, et al. ; Barlow ; Chivers and Myrick Jr ). The survivorship also seemed to decrease according to the results of the joint modelling approach. There may be a link between age at sexual maturity of females and survivorship of adults. This link could depend on the extrinsic mortality rate (Ricklefs ). It is possible that the extrinsic mortality experienced by adults decreases their maximum potential longevity and thus their age at sexual maturity. Longitudinal monitoring is now favoured to study the demography of long-lived megafauna species. Monitoring of these species traditionaly used age-at-death data or hunting bags to obtain vital rates (cross-sectional monitoring). The most important assumption regarding these data is that the population must have a stationary age-distribution (Caughley ). In the wild, this assumption is rarely met (McCullough et al.

Loss of interest in cross-sectional monitoring

; Menkens and Boyce

) and it is even more difficult to achieve it with the large marine mammals whose strandings are used (Barlow and Hohn ; Stolen and Barlow ). Demographic rates estimated by cross-sectional monitoring may not be reliable in these cases.

Longitudinal monitoring (when feasible) and the associated analyses,have therefore been favoured and have undergone significant development. Since Longitudinal monitoring has been favoured for the study of long-lived megafauna species for which such monitoring is feasible. The associated survival analyses have developed strongly since the s and longitudinal analysis is now a field of statistical analysis in its own right.

the Cormack-Jolly-Seber (Seber ; Cormack ; Jolly ) approach, numerous developments in modelling and sampling protocols have taken place (Lebreton et al. ). The s saw many developments in the use of CMR and radio telemetry protocols to estimate survival rates (Lebreton, Pradel, and J. Clobert ; Gaillard, Festa-Bianchet, and Yoccoz ). Survival analyses based on longitudinal monitoring is now an entire well developed scientific field in its own with a firm statistic basis (Gaillard, Festa-Bianchet, and Yoccoz ) and can be used to feed demographic databases such as COMADRE (Salguero-Gómez et al.

; Figure . ).

Cross-sectional monitoring remains the most suitable approach for some marine megafauna species that may be elusive. It is also the only possible approach for fish populations and exploited stocks. For some species of marine megafauna, there is a lack of demographic information (Heppell, Caswell, and Crowder ; Moore and A. J. Read

) and it may be necessary to conduct population viability analyses (W. F. Morris et al. ). For marine mammals associated with bycatch, population managers are tasked to make decision (e.g. fisheries closures) despite a large amount of uncertainty (Thompson et al. ; Harwood ). Since

Cross-sectional monitoring has not received the same attention from the community. Despite the fact that it is the only feasible monitoring for elusive species and fish stocks.

the development of Siler (

) and Heligman and Pollard ( ) approaches, there has been few methodological development associated with cross-sectional monitoring. A recent improvement has been the development of the package strandCet (Saavedra ) which is interesting for taking into account additional mortality and the treatment of sampling biased age classes (e.g. due to bycatch). A question arises: why is the community so reluctant to produce vital rates from cross-sectional monitoring?

The contribution of this project to cross-sectional approaches

Obtaining vital rates from cross-sectional monitoring of long-lived megafauna species is limited by the lack of sampling control and the limited possibility to measure variability in vital rates (Gaillard, Festa-Bianchet, and Yoccoz ). One of the points of this work was to partially address these limitations. With regard to the first item, the use of stratified random sampling is intended to allow for better control of the sampling. This control takes place prior to the analysis of the data and aims to avoid introducing additional uncontrolled bias. The use of post-stratification to quantify human pressures (e.g. bycatch) is also a type of correction. Here, it is carried out during the analysis, in order to take into account possible biases in the representativeness of the data.This is of interest for elusive species, such as the common dolphin, whose bycatch is not well informed. The inclusion of covariates and random effects in the joint model to quantify vital rates is also an improvement in approaches. With this development it is possible to quantify the effect of different factors on vital rates and to highlight variations at the population level.

Application of the demographic approach to other megafauna species

When available, age-at-death data should be used to their fullest potential (Ferguson ). Here I propose an analytical and a conceptual framework which incorporates important elements of this thesis project (Figure . ). This diagram incorporates both elements of the literature and elements developed within the framework of the project for illustrative purpose.

Odontocete species

This framework is transposable to other odontocete species and can serve different purposes depending on the problems associated with the population. It is preferable that data be collected by a stable and centralised stranding network. The more stable the stranding network is over time and centralised, the longer and more representative the period covered can be. This reduces the bias associated with collecting stranded individuals. The harbour porpoise is also a species for which there are strong conservation issues in European waters (ICES b). The Baltic Sea population is now listed as critically endangered (Benke et al. ). In the North-East Atlantic the Harbour population is distributed from Norway to southern Spain. Two genetically distinct stocks are recognised, the first in Spain, the second between France and Norway (Fontaine et al. ). The population has been divided into several management units based on genetic differentiation but also on morphological differences (Evans and Teilmann ; JNCC ; Margin-Figure . ). However, it appears that the overall population is subject to all types of pressure (Murphy et al. ), notably primary (bycatch) (ICES b), secondary ) and tertiary (noise disturbance) (Brandt et al. ). It is also supposed that climate change influences the physical condition of harbour porpoises (Heide-Jørgensen et al.

Harbour porpoise in North-East Atlantic

). With regard to population trends, declining trends have been observed in the Celtic and Irish Seas (NAMMCO, IMR

). Some biological parameters are well know for the overall population and the different stocks (e.g. age-at-death in Figure . ) but there are still uncertainties about the population dynamics of these stocks (Murphy et al. ). It could be of interest to apply the framework used here to produce stock-specific population rates (via covariates). This will also allow the identification of possible changes in life history traits over time as a function of stock to explore possible exchanges between sinks and sources. his would help to refine and prioritise the management interest of the different units recognised by the MSFD.

Hector's dolphin

Dolphins of the genus Cephalorhynchus are distributed in the Southern hemisphere in South-America, South Africa and New Zealand. The Hectors'dolphin (Cephalorhynchus hectori, van Beneden

) is the only species of this genus tto occur in New Zealand waters (A. N. Baker

) has recently been estimated to be around , individuals (Annex D ). The biological parameters of this species were determined with an average lifespan of years and an age at sexual maturity between and years for females (Slooten ). From the age distribution of the dolphins obtained from standings, the survivorship of the population and its growth rate were determined using the Siler model and a Leslie matrix approach (Slooten and Lad ). The population appears to have a low growth rate and entanglement in gillnet fisheries is suspected to be an important source of additional mortality for this population, as is predation by sharks (Dawson and Slooten ; Slooten, Fletcher, and B. L. Taylor ). A sub-species of Hector's dolphin, the Maui's dolphin (Cephalorhynchus hectori maui, Baker

) lives in the north of New Zealand (Margin-Figure . ). The population of this subspecies has fallen to about individuals (C. S. Baker et al.

). It might be interesting to use a similar methodology to that employed in this project on the Hectors dolphin population. The population size of , individuals could provide enough strandings to achieve vital rates. The advantage could be to use the long age-at-death data time serie (without using only females samples) and discriminate the impact of predation and bycatch on the value of survival rates as covariate effect. The use of the year of stranding could be of interest to identify a possible shift in life history-traits.

Pink dolphin in colombia © Jaime Rojo.

River dolphins

River and coastal dolphin populations can be monitored both on the basis of individuals found stranded and from capturerecapture programmes. It is the case for the population of Irrawaddy dolphins (Orcaella brevirostris, Owen ) that lives in the Mekong river (Cambodia) and is currently classified as Critically Endangered by the IUCN (Beasley et al. ). It was possible to conduct the two surveys and compare their results. Survival rates were both estimated by a carcass recovery programme and a capture-recapture programme. The authors concluded that the comparison was interesting and provided more information on the bias that could impact the survival estimate for the two monitoring (Beasley et al. ). Coastal dolphins species can be followed through capture-recapture studies such as the Guiana dolphins (Sotalia guianensis, van Beneden ) (Cantor et al. ). It could be of interest to use age-at-death data to compare between the estimates made through CMR monitorings and carcass recoveries for similar species.

Mammal and/or megafauna species

The current challenges in terms of the biodiversity crisis linked to direct anthropogenic impact or indirectly through climate change, require a global cooperative effort to acquire biological data (Urban et al. ). Vital rates are an important source of information for understanding and predicting the effects of anthropogenic impacts and climate change (Paniw et al. ). Demographic databases are particularly interesting given the issues outlined above and allow both sharing and a global effort to produce demographic indicators (Salguero-Gómez et al.

). These databases also provide good sources of information for generating life history knowledge and are important today in a context of digitisation of ecology that requires good and reproducible research practices.

The European wild boar (Sus scrofa, Linnaeus

) is studied using several forms of monitoring (hunting bags, CMR monitoring, age or stage matrix models): (Sáaez-Royuela and Telleriia ; Keuling et al.

; Massei et al. ). For this species, it might be interesting to compare age-at-death estimates and individual long-term monitoring. The Arctic fox (Vulpes lagopus, Linnaeus

) is a mammalian species for which little demographic information exists (Tannerfeldt and Angerbjörn ; Meijer, Norén, and Angerbjörn ). The analysis of the teeth of this species allows the determination of variations in their feeding ecology in addition to age (Ungar et al. ). With the approach of this thesis, it would be possible

Arctic fox © Marco Gaiotti.
to link ecological variations and vital rates of threatened Arctic fox populations.

Crocodilians are an order of sauropsids for which little information is available on population trends. They may be sensitive to pressures such as mercury contamination. The few demographic studies focus mainly on the Mississippi alligator (Alligator mississippiensis, Daudin ; D. Taylor and Neal ). Age is approximated by body size (Hutton ; Eaton and Link ). Although it appears that age-based matrix models do not necessarily capture the effect of factors on populations for crocodilians, it may be of interest to provide baseline estimates of survival rates. Some species are important to indigenous human populations. The S'ami people depended on the presence of reindeer (Rangifer tarandus, Linnaeus

), which they partially domesticated. This is no longer the case, but there is still a strong link between this people and the presence of reindeer in the Arctic. Other peoples such as the Inuit still hunt reindeer. This species seems to be impacted by climate change: (Heggberget, Gaare, and Ball ; Moen ). Recent progress has made it possible to determine the age at death of this species (Van den Berg, Loonen, and Çakırlar

). It could be interesting to link the survival of this species to variations in the environment and food resources in order to better understand its dynamics and to favour the conservation of the culture of certain indigenous peoples.

. Conclusion and perspectives

This thesis project has provided a new and more flexible approach to survival modelling based on linear regression. This approach allowed the inclusion of covariates and random effects. It seems appropriate to model the survival of elusive species taking into account covariates and random effects. This project allowed a new assessment of the common dolphin population in the Bay of Biscay to be conducted. The stratified random sampling developed in the demographic study allowed a better representation of the stranded population and avoided potential selection biases that could affect the survival estimate. The survival modelling approach developed in the project allowed the variation in vital rates over time to be modelled. It also highlighted the effect of bycatch or sex. The associated population projections showed that the viability of the population has potentially deteriorated over the time period considered. The viability of the population appears to be strongly threatened by bycatch as an external mortality force. The multilevel regression with post-stratification approach adapted to estimate bycatches allowed for the first time to use an on-board observer dataset despite its known biases. It also provided an estimate of bycatch and associated parameters at a spatial and temporal scale never achieved before for common dolphins in the Bay of Biscay. Ultimately, the management strategy evaluation approach allows the combination of demographic and bycatch estimates to define sustainable removal limits for the Bay of Biscay common dolphin population. This is primarily a feasibility exercise as the study area considered does not take into account the entire management unit currently recognised for the common dolphin. This approach is the first to take into account several types of data specific to this population in order to define management strategies with regard to the conservation objective proposed by OMMEG in European waters. This work is important from both a fundamental and an applied point of view. Elusive species are defined in this project as species that may have one or more of the following characteristics: difficult to monitor individually, sparsely distributed, difficult to estimate anthropogenic mortality and difficult to implement management policies. The demographic approach developed in this project is an interesting contribution to the literature on these species. It addresses some of the issues inherent to cross-sectional monitoring and analyses. It shows that the sampling of individuals is important to minimise bias in the estimation of vital rates. It provides a method for estimating vital rates that allows the effect of different variables on vital rates to be estimated. It could be very interesting to apply this modelling approach to other elusive species such as the harbour porpoises in the North-East Atlantic. This population presents major conservation challenges. The number of strandings seems to be sufficient for stratified random sampling. We also have long time series that would allow us to test for variation in vital rates over time. The covariates to be tested are also multiple. We have data on contamination by pollutants. We also have strandings in several management units, which would allow us to highlight differences in viability between populations.

The work carried out in this project is mainly associated with the Bay of Biscay. However, the current management unit of the common dolphin is much larger and includes a large part of European Atlantic waters. It is difficult to generalise the demographic results and removal thresholds obtained in this project to the whole management unit. It might be interesting to apply the project's approach using further data covering more of the European coastline. But for the time being, the demographic estimates produced can directly feed into the MSFD. It would even be possible and interesting to revise the classification of demographic indicators within the framework of the MSFD. The work carried out could provide an operational indicator that would make it possible to highlight an alteration in viability before it is translated into a depletion that could be detected by abundance surveys.

page

).

-(

).

Institute of Zoology (

). Cetacean-stressors: The independent and interactive effects of multiple stressors on reproduction and development in cetaceans (cited on page ). was historically difficult and required specific models. We identified a simple linear regression model with individual frailty that can fit a bathtub-shaped hazard, take into account covariates, allow goodness-of-fit assessments and give accurate estimates of survivorship in realistic settings. We first conducted a Monte Carlo study and simulated age at death data to assess the accuracy of estimates with respect to sample size. Secondly, we applied this framework on a handful of case studies from published studies on marine mammals, a group with many threatened and data-deficient species. We found that our framework is flexible and accurate to estimate survivorship with a sample size of 300. This approach is promising for obtaining important demographic information on data-poor species.

K E Y W O R D S

age at death, Monte Carlo study, regression, survival analysis, survivorship

| INTRODUCTION

The protection and conservation of wildlife is a salient challenge in the Anthropocene, where human-induced pressures are both far-reaching in their geographic and temporal scope, and acute in their impact on biodiversity [START_REF] Bongaarts | Summary for policymakers of the global assessment report on biodiversity and ecosystem services of the intergovernmental science-policy platform on biodiversity and ecosystem services[END_REF]. Long-lived species are particularly sensitive to these pressures: they can act as sentinel species for whole ecosystems [START_REF] Heithaus | Predicting ecological consequences of marine top predator declines[END_REF]. Their demographic trajectory (e.g., extinction) can also affect the functioning of ecosystems, especially in the case of top predators [START_REF] Beschta | Large predators and trophic cascades in terrestrial ecosystems of the western United States[END_REF][START_REF] Ritchie | Ecosystem restoration with teeth: What role for predators?[END_REF]. Assessing the demographic viability of long-lived species in the face of current and future pressures is required for their proactive conservation; but such an endeavor may be hampered in practice because of logistical challenges in monitoring and collecting relevant data at the relevant spatiotemporal scales.

The classical way to assess a populations viability is to first construct a static life-table where mortality and fecundity rates for a cohort of individuals are tallied at each age/stage of their life cycle [START_REF] Caughley | Mortality patterns in mammals[END_REF]. Knowledge of mortality at each age/stage allows to investigate the associated hazards, estimate vital rates, such as (cumulative) survival; and to project the population trajectory forward in time using matrix models [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF][START_REF] Leslie | On the use of matrices in certain population mathematics[END_REF]. Historically, the age at death data used to construct life-tables were crosssectional (e.g., [START_REF] Gompertz | On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies[END_REF]: a sample of the population is taken at a specific time and the observed age structure is assumed similar to that of a cohort of individuals had we been able to monitor them from birth to death. This cross-sectional design allows for a rapid demographic assessment [START_REF] Boyd | Marine mammal ecology and conservation: A handbook of techniques[END_REF]Margules & Austin, 1990, pp. 126-127.) whose accuracy however hinges on the assumption that the crosssectional design approximates well a longitudinal one.

Longitudinal designs, whereby an individual is monitored from birth to death are, in wildlife ecology, a byproduct of capture-recapture studies. The initial motivation for the latter was the accurate estimation of abundance in open populations when detectability is less than perfect (Cormack, 1964;[START_REF] Jolly | Explicit estimates from capture-recapture data with both death and immigration-stochastic model[END_REF][START_REF] Seber | A note on the multiple-recapture census[END_REF]. Models (e.g., the Cormack-Jolly-Seber model) tailored for these life history data can provide accurate estimates of survival that can be fed into matrix population models (e.g., [START_REF] Fujiwara | Demography of the endangered North Atlantic right whale[END_REF]. Since the 1980s, the emphasis of capture-recapture studies in wildlife ecology shifted from abundance to survival estimation [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies[END_REF] and the inclusion of individual-level covariates, which paved the way for a better understanding of evolution and natural selection in the wild [START_REF] Cam | Contribution of capture-mark-recapture modelling to studies of evolution by natural selection[END_REF]. Concomitant with this change in focus was the less frequent reporting of life-tables in publications, an omission partly mitigated by the rise of open online databases such as COMADRE/COMPADRE [START_REF] Salguero-Gómez | The compadre plant matrix database: An open online repository for plant demography[END_REF][START_REF] Salguero-Gómez | Comadre: A global database of animal demography[END_REF]. From an applied perspective in conservation, the obvious drawback of capture-recapture studies is the time and manpower required to collect data, especially for long-lived species. [START_REF] White | Modeling the population dynamics of lemon sharks[END_REF] recently proposed a new development that aims to avoid birth to death monitor. From both juvenile mortality and fecundity, it is possible to infer adult mortality. However, it requires monitoring the juvenile part of the whole population in order to model the overall population dynamics. The mismatch between the necessary delay in data acquisition and the urgency of mitigation on conservation decisions can be acute, suggesting interest for crosssectional data that have the additional benefit of being applicable on species where the identification of individuals is difficult [START_REF] Williams | Analysis and management of animal populations[END_REF]. One alternative to species identification, for vital rates estimation, is the use of count data. This approach may involves the building of an N-mixture model relying on the individual-state knowledge. [START_REF] Zipkin | Modeling structured population dynamics using data from unmarked individuals[END_REF] proposed a development that allows taking into account data even if the individual stage is unknown. Another way to obtain vital rates with count data, as time-series, is to conduct an inverse modeling approach: this method also presents the advantage to consider individual-level state data [START_REF] González | Inverse estimation of integral projection model parameters using time series of population-level data[END_REF]. Even if these approaches are promising for hard to monitor species, they remain difficult to apply on some species, especially very mobile ones.

Assuming that cross-sectional data on age at death are available and representative, conservationists now face some further choices, such as non-parametric (e.g., Kaplan-Meier survivorship curves;[START_REF] Clark | Survival analysis part I: Basic concepts and first analyses[END_REF][START_REF] Kaplan | Nonparametric estimation from incomplete observations[END_REF] versus parametric modeling. The latter presents the advantage to provide smoothing and to summarize the data in a handful of parameters. This is small sample attractive: non-parametric approaches such as the Kaplan-Meier estimators (also known as Product Limit Estimators) for survivorship is piecewise constant with discontinuities at the time of death of individuals in the sample. While assumption-lean, this estimator may yield from small samples rough survivorship curves shaped like staircases that can be difficult to compare. Even with little sample size (i.e., few age at death data regarding the overall time series length), parametric modeling aims to estimate smooth survivorship and allow the statistical comparison of parameters values [START_REF] Kleinbaum | Survival analysis[END_REF]. It is important to notice that the estimation hinges on the correct specification of the underlying model. More realistically, the challenge is to find a parsimonious model that will nevertheless accommodate the available data without shoehorning the analysis into a convenient, but not necessarily realistic, template. For long-lived animal species, a realistic model should provide the so-called bathtub-shaped curve whereby there is a high juvenile mortality, followed by a lower and rather constant adult mortality and, finally, a late increase due to senescence [START_REF] Choquet | A semi-Markov model to assess reliably survival patterns from birth to death in free-ranging populations[END_REF][START_REF] Siler | A competing-risk model for animal mortality[END_REF]. Although there are models to accommodate this bathtub pattern [START_REF] Heligman | The age pattern of mortality[END_REF][START_REF] Siler | A competing-risk model for animal mortality[END_REF], they can be difficult to fit (but see [START_REF] Saavedra | Strandcet: R package for estimating natural and non-natural mortality-at-age of cetaceans from age-structured strandings[END_REF], their goodness of fit can be hard to measure; and they usually do not allow for the inclusion of individual-level covariates such as sex.

Our aims are to investigate a parametric approach for the construction of life-tables in a data-poor context, with long-lived vertebrate species in mind. We will first present a simple regression modeling framework for the analysis of age at death data (with or without right-censoring) that can accommodate mortality patterns such as constant, increasing/decreasing, unimodal or bathtub-shaped. Importantly, this framework allows the seamless inclusion of individual-level covariates. We then conduct a Monte Carlo simulation study with five biological scenarios to assess our framework, focusing on the accurate estimation of survivorship and mortality. Finally, we illustrate our methodology on a handful of real case studies, with a focus on marine mammals. This group includes many elusive and long-lived species that are threatened (Avila, Kaschner, & Dormann, 2018), data-deficient [START_REF] Parsons | Why IUCN should replace "data deficient" conservation status with a precautionary "assumed threatened" status -a cetacean case study[END_REF] and difficult to study with a longitudinal approach. The most common age at death data for marine mammals are obtained from strandings (i.e., cross-sectional monitoring) which may be associated to selection biases. However, stranding samples are a source of age data from which it is possible to obtain vital rates.

| MATERIALS AND METHODS

| Definitions

Survival analysis deals with the analysis of the timing of death [START_REF] Clark | Survival analysis part I: Basic concepts and first analyses[END_REF][START_REF] Kaplan | Nonparametric estimation from incomplete observations[END_REF][START_REF] Lesaffre | Bayesian biostatistics[END_REF][START_REF] Tanner | Data-based nonparametric estimation of the hazard function with applications to model diagnostics and exploratory analysis[END_REF]. We assume data y ≥ 0 to be time-to-event data, for example longevity, survival time or age at death data. Such data can be collected from stranding of animals (that is marine animals that are washed ashore), or any recovery of dead specimens (e.g., Lepus europaeus, Pallas 1778; Ovis dalli, Nelson 1884) when aging is possible (e.g., from teeth growth layers). Let the index i denotes the i th individual, and N the sample size. The equation y i D θ ð Þ reads as datum y i follows statistical distribution D of parameters θ and with probability density function f(y; θ) and cumulative density function F y; θ ð Þ= Pr y ≤ t ð Þ= Ð t 0 f y; θ ð Þdy. The survival function S(t; θ) gives the probability of being alive at time t, that is Pr(y > t) = 1 -Pr(y ≤ t) = 1 -F(y; θ). At the population level, this quantity correspond to the fraction of the population that is still alive at age t, that is cumulative survival or simply survivorship. The hazard rate, or age-specific mortality rate, is the instantaneous probability of dying at time t + dt given that and individual i has survived until time t.

h t ð Þ = lim dt!0 Pr t ≤ y i < t + dt;y i > t ð Þ dt ð1Þ
With parametric models, the hazard rate can be expressed in terms of the probability density and survivorship functions:

h t; θ ð Þ= f t;θ ð Þ S t; θ ð Þ , with S t;θ ð Þ> 0 ð2Þ
Estimation of the hazard rate function is the goal of parametric survival analysis [START_REF] Lesaffre | Bayesian biostatistics[END_REF].

| Data simulation scenarios

We considered five biological scenarios corresponding to different patterns in survivorship and the underlying hazard (Figure 1, see Text S1):

1. A unimodal distribution of age at death corresponding to a unimodal hazard, whereby mortality risk increases in early ages, peaks and decreases in late life.

2. A mixture of two hazards corresponding to either a unimodal (a) or bimodal (b) distribution of age at death.

3. A bathtub-shaped hazard due to individual frailty, that is individual-specific risk of mortality.

4. A bathtub-shaped hazard with an additional bump in early life due, for example, to an additional source of mortality.

Each scenario is associated to survivorship and hazard functions (Figure 1).

| Statistical analysis of age at death data

We used parametric models ℳ of age at death data y to estimate hazard and survivorship rates (see Data S1 for model code): the latter are especially of interest to conservationists. We assumed that the exact timing of death is available, but our framework can easily accommodate censoring (that is cases when death is known to have occurred before or after measurement). Our framework consists in modeling the logarithmic transform of y i in a regression framework (location-scale model):

logy i = μ + σ × ε i - Z i β ð3Þ
where μ is a location parameter; σ and 1 β are (positive) scale parameters; and Z i are independent standard exponential deviates. The parameter 1 β quantifies individual frailty [START_REF] Kannisto | Frailty and survival[END_REF][START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF] or persistent demographic heterogeneity sensu [START_REF] Cam | The conundrum of heterogeneity in life history studies[END_REF]. The choice of the statistical distribution for the residuals ε i determines the shape of the underlying hazard rate. We considered three different choices, each corresponding to a model (see Text S2 for the associated survivorship and hazard functions).

• ℳ 1 :

ε i N 0, 1 ð Þand 1 β = 0 (β = + ∞)
This model assumes a log-normal distribution for y i , which corresponds to a unimodal hazard curve.

• ℳ 2 :

ε i G 0, 1 ð Þand 1 β = 0 (β = + ∞)
This model assumes a Gumbel distribution for the residuals ε i , which corresponds to a Weibull distribution for y i . It is known as the Accelerated Failure Model. The hazard rate is monotonic: it can be constant, increasing or decreasing depending on the value of σ.

• ℳ 3 : ε i N 0, 1 ð Þand 1 β > 0 This choice leads to assume a normal-Laplace distribution for logy i , which induces a flexible hazard curve depending on the value of β [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF]. In particular, the hazard can be a bathtub-shaped, as expected for example for long-lived species of vertebrates in the wild [START_REF] Choquet | A semi-Markov model to assess reliably survival patterns from birth to death in free-ranging populations[END_REF].

Our framework given by Equation ( 3) is a flexible location-scale model and boils down to a generalized linear mixed model (GLMM; [START_REF] Bolker | Generalized linear mixed models: A practical guide for ecology and evolution[END_REF] with a handful of parameters θ = (μ, σ, β) to accommodate a large diversity of survivorship and hazard curves (see Text S2 for equations). One attractive feature of this framework is the seamless incorporation of p individual-level covariates x ip in Equation (3) [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF]:

logy i = μ + X p j = 1 γ p x ip + σ × ε i - Z i β ð4Þ
An important restriction of our approach is that only time-invariant individual-level covariates can be included. This covers however interesting cases such as sex-differences in survival or differences due to geography (that is comparing different populations of the same species).

| Monte Carlo study

Our aim is to carry a Monte Carlo study (e.g., [START_REF] Morris | Using simulation studies to evaluate statistical methods[END_REF] to investigate whether our modeling framework can provide accurate survivorship estimates from age at death data. We considered five biological scenarios to cover a diversity of realistic mortality patterns. For each scenario, we simulated 100 data sets of sample size 100, 200, 300, 400, 500 and 1000 to provide recommendations on the minimum sample size required for accurate estimation. Data simulation was carried out in R version 3.6.0 (R Core Team, 2019) using base

F I G U R E 1
The five biological scenarios considered in the Monte Carlo simulation study. Scenario 1 corresponds to a low juvenile mortality, a high adult mortality and no senescence. Scenario 2a corresponds to a low juvenile mortality and a higher adult mortality plateau. Scenario 2b corresponds to a high juvenile mortality and a lower adult mortality that decreases in a linear fashion. Scenario 3 corresponds to a high juvenile mortality, a lower adult mortality and senescence (bathtub shaped hazard). Finally, Scenario 4 corresponds to a high juvenile mortality and a high adult mortality, with a transient dip in mortality risk between these two life-stages functions such as rnorm and rexp. Each simulated data set was then analyzed with our framework that considered three parametric models. Crucially, we considered scenarios for which the true model was not among the set (Table 1). In other words, we assessed the performance of our framework under the possibility that none of the candidate models is correctly specified with respect to the data at hand. For example ℳ 2 is a Weibull model, which is widely used in survival analysis [START_REF] Kleinbaum | Survival analysis[END_REF] it is biologically unreasonable for marine megafauna and top predators as a model over their entirely lifetime because it cannot accommodate the expected bathtub-shape hazard. Model fitting was done with software Stan version 2.18 (Carpenter et al., 2017) called from R via the library rstan (Stan Development Team, 2018). Three chains were run with a warm-up of 500 iterations, followed by an additional 1000 iterations. No thinning was performed but the delta parameter of the NUTS algorithm was increased from 0.80 to 0.95 to avoid divergent transitions and the maximum tree depth increased to 15. Parameter convergence was assumed when its R statistics was lower than 1.10. Upon convergence, the three chains were pooled to obtain a sample of ≈1000 values from the posterior distribution.

Model fit was assessed with the widely applicable information criterion (WAIC, Gelman, Hwang, & Vehtari, 2014), computed with R package loo [START_REF] Vehtari | Practical bayesian model evaluation using leave-one-out cross-validation and waic[END_REF]. Survivorship estimates from each model were computed from the posterior distribution of parameters, and visually compared to the true survivorship curve and a non-parametric (Kaplan-Meier) estimate. Because we carried out a simulation study, the true survivorship curve was known and could have been used to compute the root-mean-squared error (RMSE). However, in practice, this is not the case and we chose instead to compute RMSE with respect to the nonparametric Kaplan-Meier survivorship estimates. The latter were thus assumed to represent the best estimates available to researchers, and the aim was to assess whether a parametric model could provide a fit as good as that from a non-parametric approach.

RMSE ℳ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ŜKS t ð Þ-Ŝℳ t ð Þ À Á 2 h i r ð5Þ
where ŜKS t ð Þ is the Kaplan-Meier estimate of survivorship at age t, Ŝℳ t ð Þ is the corresponding estimate from parametric model ℳ, and the expectation is taken over a sequence of values of t.

In order to assess model selection and model check, it is necessary to determine if the models represent well the data. [START_REF] Conn | A guide to bayesian model checking for ecologists[END_REF] recently reviewed some ways to proceed. We choose to do a prior predictive check to test for adequacy between models and time to event data (Text S3 and Figure S1). We also did a posterior predictive check (Text S3 and Figure S2) to see whereas data simulated through the fitted models are similar to that observed from the [START_REF] Barlow | Interpreting spotted dolphin age distributions[END_REF] data set. The comparison is done with both the true Kaplan-Meier survivorship curve and the ones from posterior simulated data sets (Text S3).

Our focus was on accurate estimation of survivorship, and thus we assessed goodness-of-fit by comparing the expected mean survivorship under each model to the observed Kaplan-Meier estimates. This focus was in line with downstream use of such estimates in matrix population models: here the salient statistics we want our model to reproduce [START_REF] Gelman | A bayesian formulation of exploratory data analysis and goodness-of-fit testing[END_REF] is the survivorship function.

Our study design is summarized in Table 1 andFigure

ℳ 1 + - - - - ℳ 2 - - - - - ℳ 3 - - - + -
Note: Likewise, scenarios 2a, 2b and 4 corresponds to data generated from mixture models not included in the set: they serve as tests of the performance of our framework to obtain accurate estimates with mispecified models. Finally, scenarios 1 and 3 act as positive control as they correspond to data simulated under models ℳ 1 and ℳ 3 , respectively. x 2 ) were generated by sampling from a Bernoulli distribution with probability 0.5. These covariates could represent for example sex or two sub-populations in different geographic areas.

| Applications

We analyzed real data sets from published case studies (Table 2) within our framework, and compared the estimated parametric survivorship curves with the nonparametric Kaplan-Meier one. All species in [START_REF] Rodríguez-Caro | Maintained effects of fire on individual growth and survival rates in a spur-thighed tortoise population[END_REF][START_REF] Sanz-Aguilar | Coexisting with fire: The case of the terrestrial tortoise Testudo graeca in mediterranean shrublands[END_REF]. However, we used these data as if they were age at death data and ignored right-censoring.

| RESULTS

Across all scenarios and sample size, parameter convergence (assessed with R ) was not equivalently reached depending on the model and scenario (see Figures S3 andS4a). ℳ 1 always converged very easily. In contrast, ℳ 2 and ℳ 3 were not as good as ℳ 1 to converge with 1000 iterations (500 as warm up). More precisely, convergence for ℳ 3 was difficult for β given this configuration (see

Figures S5 to S10

). An increased in the number of iterations per chains (here from 1,000 to 2000) solved the problem (Figure 3).

| Hazard rate estimation

Estimated hazard curves are shown in Figure 4 for each combination of scenario, model and sample size. Models ℳ 1 and ℳ 3 were the best fitting ones in the first and third scenarios respectively (i.e., positive control, Table 1): estimates were accurate and precision increased with sample size. For scenarios 2a, 2b and 4, all estimates were biased, and precision increased with sample size (Figure 3). In contrast, model ℳ 2 never provided accurate hazard estimates (i.e., negative control). The same results were obtained when covariates were included (not shown).

| Survivorship estimation

Estimated cumulative survival curves are shown in Figure 5 for each combination of scenario, model and sample size. Across all scenarios and sample sizes, estimates from model ℳ 3 were the most accurate (Figure 5, see Text S4 and Figure S11 for RMSE results). Precision increased with sample size. In particular, survivorship rates estimated with ℳ 3 were very close to Kaplan-Meier estimates for sample size ≥300. This sampling size also provided a good confidence interval precision with a maximum width of 0.04% while estimating survivorship (Figure S12). Predictive ability, as measured with WAIC, was the greatest for model ℳ 3 : it was consistently ranked first across each combination of scenario and sample size, even for scenario 1 where model ℳ 1 was the true datagenerating mechanism (see Text S4 and Figure S13).

| Covariate effects

Covariate effect estimation is summarized on Figure 6 as a difference between survivorship rates. Model ℳ 2 could not estimate difference in survivorship. Model ℳ 1 is only effective for both scenario 1 and 2a. ℳ 3 could estimate accurately covariate effects when the difference is expressed as an unimodal pattern (i.e., scenario 1, 2a and 3). The precision of the estimated effect increased with sample size for models ℳ 1 and ℳ 3 . With small sample size, sign errors on the effect of covariate was possible but disappeared with sample size ≥300.

| Case studies

Estimated survivorship curves from published data sets are plotted against the Kaplan-Meier curves in Figure 7 for each model. Model ℳ 3 was the most flexible: it provided the most accurate estimates for each data set. Uncertainty, as measured with 80% credible intervals were narrower with model ℳ 3 , and overlapped most with Kaplan-Meier estimates. Model ℳ 3 consistently had the lowest WAIC. In the handful of case studies where covariates were available (Table 2), a similar pattern arose. Including covariate can be expected to account for more variation in the data, and a lower WAIC. For both ℳ 1 and ℳ 2 , this was indeed the case. For ℳ 3 it was only true with the Murphy et al., 2012 data set (see Text S4 and Table S1). The estimated covariate effect for both [START_REF] Barlow | Interpreting spotted dolphin age distributions[END_REF] and [START_REF] Kesselring | Coming of age: Do female harbour porpoises (Phocoena phocoena) from the north sea and Baltic Sea have sufficient time to reproduce in a human influenced environment?[END_REF] case studies is summarized on Figure 8 (see also Text 5). Models ℳ 1 and ℳ 3 both estimated a survivorship difference between each covariate. As it is the case without covariates (Figure 7), the ℳ 3 curve fits the Kaplan-Meier estimate better. However, there is a discrepancy between both, for each covariate. ℳ 3 is able to spot a difference depending on covariate, but is not able to perfectly fit the Kaplan-Meier estimate.

| DISCUSSION

We assessed the ability of a simple linear mixed model to estimate hazard and survivorship rates with cross-sectional age at death data. We used Monte Carlo simulations to investigate the accuracy and precision of estimates across a diversity of mortality patterns and several sample sizes. We contrasted the performance of three different models and found one model, the model ℳ 3 ; that was consistently better with respect to prediction of survivorship, even when it was misspecified.

| Age at death data and sampling bias

A crucial but implicit assumption of the cross-sectional approach to survival analysis is that the sample is representative of the larger population, especially with respect to age structure [START_REF] Caughley | Mortality patterns in mammals[END_REF]. This assumption is on the data set, not on the modeling. The way of collecting age at death data is therefore determinant to support this assumption. Transerval designs involving, for example, recording of dead carcasses, hunting bags or population census must match the representativity premise. However, it is possible that an implicit process (e.g., bycatch) shapes the observed age frequency (e.g., from stranded animals). In this case, it is necessary to explore some potentials age-dependent selection biases in the population [START_REF] Barlow | Interpreting spotted dolphin age distributions[END_REF]. With stranding data, the stationary age distribution assumption needs to be substantiated with auxiliary data, but ultimately, it is likely to remain a working hypothesis on which any cross-sectional method will lean on. In the case of cetaceans, many species are not amenable to study with a longitudinal design, and strandings remain an important source of information (e.g., Murphy et al., 2009), and demographic information in particular [START_REF] Ferguson | Climate change and ringed seal (Phoca hispida) recruitment in western Hudson bay[END_REF][START_REF] Saavedra | Strandcet: R package for estimating natural and non-natural mortality-at-age of cetaceans from age-structured strandings[END_REF]. When longitudinal studies are not possible, there are very limited options to obtain demographic information. In some cases, a comparative approach may be possible where information on species with similar life histories can be leveraged [START_REF] Caswell | Harbour porpoise and fisheries: An uncertainty analysis of incidental mortality[END_REF][START_REF] Hashimoto | Estimating the rate of increase for the finless porpoise with special attention to predictions for the Inland Sea population in Japan[END_REF]. This choice relies crucially on the assumption of betweenspecies similarity. Even if comparative data are available, the ecological context may be too different to justify this approach in some instances. Strandings may be the only available source of data, and even if they are suspected to suffer from some selection bias, a pragmatic approach to conservation requires to use them [START_REF] Boyd | Marine mammal ecology and conservation: A handbook of techniques[END_REF], keeping in mind the inherent limitations of these data.

Many conservation instruments specifically requires to use the best available science (e.g., the Marine Strategy Framework Directive EC 2008/56 in Europe). The operative expression "the best available science" may be understood broadly as what lies at the intersection of state-ofthe-art methods, good data and accurate knowledge. In practice, there may be a hiatus between the canonical approach that should be, and the pragmatic one that can be pursued at the time conservation actions need to be decided. A conclusion of a species being data-deficient often leads to the doldrums with respect to conservation decisions [START_REF] Parsons | Why IUCN should replace "data deficient" conservation status with a precautionary "assumed threatened" status -a cetacean case study[END_REF]. Likewise, the many uncertainties that can affect any scientific studies can easily lead to inactions [START_REF] Ascher | Scientific information and uncertainty: Challenges for the use of science in policymaking[END_REF]. It is because we are starkly aware of these limitations that we carried out this study to identify a pragmatic approach to estimate survivorship rates from age at death data. Crucially, all the models we used are conditional of the underlying sample being representative of the population it is taken from. Granting this assumption, we identified a simple model from [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF] to obtain accurate estimates.

| Prediction accuracy and models' goodness of fit

Traditionally, estimating survivorship rates of marine mammals with cross-sectional data was done with parametric models such as the Siler or Heligman-Pollard models [START_REF] Heligman | The age pattern of mortality[END_REF][START_REF] Siler | A competing-risk model for animal mortality[END_REF]. These models aim at reproducing a bathtub-shaped hazard curve, but need several parameters to do so. Furthermore, they can be difficult to fit although new tools have been developed to use these models [START_REF] Saavedra | Strandcet: R package for estimating natural and non-natural mortality-at-age of cetaceans from age-structured strandings[END_REF]. A remaining challenge with these models is to assess goodness of fit, to incorporate individual-level covariates and perform model selection. The simple parametric form of [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF] for analyzing the logarithm of age at death data provides a bathtub-shaped hazard with a linear mixed model, the current workhorse of ecologists (Bolker , 2009). This linear mixed modeling framework allows for a seamless incorporation of covariates, and to use standard tools for model selection and goodness-offit assessments. In other words, model ℳ 3 brings back survival analysis within the comfort zone of ecologists. We harnessed the simplicity of [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF] to conduct our Monte Carlo study, and found that the model suggested by [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF], our model ℳ 3 , was very accurate in predicting survivorship (i.e., it consistently had the lowest RMSE), even in cases when it was not the true model behind the data (see Text S6 to have details about priors). This result is important as it suggests to start building model of increasing complexity from ℳ 3 and use tools such as WAIC (Gelman et al., 2014) to balance model complexity with prediction accuracy.

From our simulations, we can recommend a sample size of at least 300 individuals to obtain accurate and precise estimates of survivorship, from which age-specific survival estimates can be derived (see Figure S14 for parameters estimates). This sample size recommendation is practical and realistic (e.g.. [START_REF] Kesselring | Coming of age: Do female harbour porpoises (Phocoena phocoena) from the north sea and Baltic Sea have sufficient time to reproduce in a human influenced environment?[END_REF]Mannocci et al., 2012;Murphy et al., 2009). This recommendation is mostly to obtain precise estimates, but may be relaxed in some cases where only sparse data may be available (data-poor context hereafter). Estimates from a simple linear model with an individual frailty term, were accurate, if imprecise, with a sample size as small as 100. The possibility to use sparse data is critical as it can help conservation of marine mammals, many of which being classified as data-deficient [START_REF] Parsons | Why IUCN should replace "data deficient" conservation status with a precautionary "assumed threatened" status -a cetacean case study[END_REF][START_REF] Schipper | The status of the world's land and marine mammals: Diversity, threat, and knowledge[END_REF]. Some of the cases studies presented in this paper support this statement, although, with so few data, including covariates in the model will be difficult or will require great care (see, e.g., [START_REF] Cox | High mortality rates in a juvenile free-ranging marine predator and links to dive and forage ability[END_REF]). Rodríguez-Caro et al. ( 2019) recently provided an approach to estimate survival in data-poor settings using inverse modeling, also to obtain accurate estimates of demographic rates. With the latter, population matrix models can then be used to assess population dynamics and the fate of populations over time [START_REF] Caswell | Matrix population models: Construction, analysis, and interpretation[END_REF].

| HAZARD AND FRAILTY

A linear mixed model can fit age at death data very well: this ability comes for the individual frailty term, which

F I G U R E 6 Estimated survivorship difference ( Ŝℳ t ð Þjx = 1 À Á -Ŝℳ t ð Þjx = 0 À Á
) under each scenario. The covariate effect could correspond to that of sex (e.g., x = 1 for females and x = 0 for males). Ten estimated difference curves are depicted in each panel and compared with the truth [Color figure can be viewed at wileyonlinelibrary.com] corresponds to an individual random effect in the mixed modeling framework. Individual frailty in statistical models translate the empirical observation that two similar individuals (with respect to observable features of their phenotypes) can nevertheless differ markedly in their longevity [START_REF] Cam | The conundrum of heterogeneity in life history studies[END_REF][START_REF] Kannisto | Frailty and survival[END_REF]. The parametric form (i.e., exponential) for individual frailty gives extra flexibility to the model, and can accommodate a bathtub-shaped hazard curve [START_REF] Reed | A flexible parametric survival model which allows a bathtub-shaped hazard rate function[END_REF]. However, in our simulations, we found that estimating hazard rates was more difficult than estimating survivorship rates (Figures 3 and4). Thus, even though we identified a model (ℳ 3 ) for reliable and accurate estimation of survivorship, the same model was less reliable with respect to hazard. In other words, the individual frailty term in our model ℳ 3 should not be over-interpreted, and is probably best seen as a statistical device for robust estimation. Hazard estimation is a difficult statistical problem [START_REF] Watson | Hazard analysis[END_REF], for which there are better tools available, especially non-parametric ones (see, e.g., [START_REF] Hanson | Surviving fully Bayesian nonparametric regression models[END_REF], than the simple parametric approach we considered in this study. Non-parametric approaches to infer the shape of the hazard curve are data-hungry: Hanson and Jara (2013) using Bayesian non-parametric, which is better described as a model with a massive number of parameters [START_REF] Hoff | Comment on article by Müeller and Mitra[END_REF]. The traditional Kaplan-Meier approach is truly a non-parametric approach but it does not give access to the underlying hazard, and give rough (i.e., non-smooth; Figure 5) survivorship rates with small sample size. In data-poor settings, parametric modeling remains attractive because it has interpretable parameters (e.g., individual frailty) and because these parameters can smooth out noise in data, yielding more precise estimates if the model is at least approximately correct, or more pragmatically, if it is grounded in theory (e.g., bathtubshaped hazard for natural populations) and cannot be rejected from a goodness-of-fit test. It is precisely in this data-poor setting that we envision our parametric modeling approach to be most useful. Because the approach boils down to linear mixed effects modeling, great flexibility in model specification of additional random effects (e.g., year effects, sex-specific frailties) is possible provided there are enough data to offset the increase in complexity.

| BENEFITS AND LIMITS

We believe our approach is very valuable to estimate survivorship from sparse data but may be inadequate for bimodal age at death data distribution. A bimodal distribution of age at death data may be a sign of selection in data collection [START_REF] Barlow | Interpreting spotted dolphin age distributions[END_REF]. Rather than consider the sampling as biased, our approach can accommodate the selection phenomenon through two ways. Firstly, it is possible to consider the selection bias as a covariate in the study if one such covariate is available (e.g., Bycatch index). If no such covariate is available, it may be possible to build a mixture of two models (e.g., ℳ 1 and ℳ 3 ) in order to take into account an additional mortality on some age classes. However, such a development requires to conduct its own simulation study which is beyond the scope of this study. Since our approach is based on linear regression, it is straightforward to expand the model (e.g., random year effects, mixtures) while using familiar and well-established methods for model selection (e.g., WAIC) and assessment of model fit (e.g., posterior predictive checks, R 2 statistics). These features are assets, and suggest that our approach is complementary to existing ones (e.g., [START_REF] Heligman | The age pattern of mortality[END_REF][START_REF] Saavedra | Strandcet: R package for estimating natural and non-natural mortality-at-age of cetaceans from age-structured strandings[END_REF][START_REF] Siler | A competing-risk model for animal mortality[END_REF] to estimate vital rates in some data-poor species.

| CONCLUSION

Survivorship and hazard estimation are the goals of survival analysis but they remain difficult to achieve, even for some long-lived and charismatic vertebrate species such as cetaceans. User-friendly modeling methods are of primary interest to leverage the demographic information available in sparse, but previous, data that field ecologists have collected. We think that our framework can facilitate the difficult statistical problem of survival analysis in data-poor context by providing ecologists with a flexible method to obtain accurate survivorship estimates from

F I G U R E 8 Estimated survivorship difference ( Ŝℳ t ð Þjx = 1 À Á -Ŝℳ t ð Þjx = 0 À Á
) for both case studies [Color figure can be viewed at wileyonlinelibrary.com] age at death data. A simple linear mixed-model can accommodate various mortality patterns without drastically increasing the number of estimated model parameters. All the methods developed to assess fit quality within models can be applied since our framework is based on a mixed linear regression approach. This framework is also very convenient to deal with small sample size. With as few as 100 data points, survivorship may be estimated to conduct exploratory analysis. With 300 data points, survivorship estimates can be precise enough to build life-tables and project populations trajectories taking into account covariates. Covariate inclusion can allow to distinguish different causes of mortality (e.g., pathology, bycatch, collision and so on) and to quantify their respective influence on survival. This is valuable when the studied population is under various pressures as it may allow to identify the most threatening ones, and to design efficient and relevant conservation policies accordingly.

Leslie Matrixes

Both Leslie matrixes and elasticities were computed using the demogR package on R. To compute the matrix, we assumed a female population taking into account the age-specific cumulative survival ( ) (as it is detailed in the package options) and the age-specific fertility rates ( ). The multiplier for fertility equals (since all age-class are of length one) and the infant.class parameter is equal to TRUE since there are three infant age-class.

Life-tables building

Life-tables are useful tools to explore the demographical strategy of a particular species or of a population during its lifetime. We created North-East common dolphin life-tables on a period-based approach (i.e. vertical life-tables). We built life-tables depending on each modelling approach (i.e. simple; with covariate; with year effect). Here we present the basic structure of our life-table approach. For each age , several rates are computed. The first rate to be computed is which is the probability of dying between age to + 1:

= 1 -( +1 ) with as the survivorship rate to age (probability to reach the age ). From it is possible to compute which is the number dying in the interval to + 1: = × with the number of individuals alive in the middle of the age class . The number surviving to at least age , , can be deducted as:

= -. From we can compute the total number of person-years lived by the cohort from age x until all members of the cohort have died (i.e. the sum of from onwards):

= ∞ = .
Finally, it is possible to derive the life expectancy at age : as = . It is possible to add the age-specific fertility rates obtained with pubescents ( ) which are used to build the Leslie-Matrix.

WAIC and LOOIC of models

INTRODUCTION

Incidental catch, or bycatch (i.e. the undesired catch of nontargeted species in fishing gear), represents the most potent and well-identified threat to small cetaceans' populations [START_REF] Reeves | Global priorities for reduction of cetacean bycatch[END_REF]. Because they are long-lived species, cetaceans can-not sustain high rates of depletion [START_REF] Lewison | Understanding impacts of fisheries bycatch on marine megafauna[END_REF], Mannocci et al. 2012). Bycatch has already led populations or species to serious reduction [START_REF] Reeves | Global priorities for reduction of cetacean bycatch[END_REF], Brownell et al. 2019) or even extinction [START_REF] Turvey | First human-caused extinction of a cetacean species?[END_REF], Brownell et al. 2019), the vaquita porpoise Phocoena sinus Norris andMacFarland, 1958 (Norris &MacFarland 1958) being a potentially immi-ABSTRACT: The population of short-beaked common dolphins Delphinus delphis of the Bay of Biscay (northeast Atlantic) has been subjected to potentially dangerous levels of bycatch since the 1990s. As the phenomenon intensifies, it represents a potent threat to the population. Here, we investigated the relationship between bycatch mortality and oceanographic processes. We assumed that oceanographic processes spatiotemporally structure the availability and aggregation of prey, creating areas prone to attract both common dolphins and fish targeted by fisheries. We used 2 datasets from 2012 to 2019: oceanographic data resulting from a circulation model and mortality data inferred from strandings. The latter allows location of mortality areas and quantification of the intensity of mortality events at sea. We fitted a series of spatiotemporal hierarchical Bayesian models using integrated nested Laplace approximations (INLA). Results provided first insights on how bycatch of common dolphins in the Bay of Biscay might be related to key seasonal and dynamic oceanographic features. We showed that from a statistical predictive point of view, the monthly trend of 2019 bycatch mortality could be predicted with few oceanographic covariates. This study highlights how gaining knowledge about environmental influences on interactions between short-beaked common dolphins and fisheries could have great conservation and management value. Identified relationships with oceanographic covariates were complex, as expected given the dynamic aspects of oceanographic processes, dolphins and fisheries distributions. Further research focusing on smaller time scales is needed to elucidate proximal drivers of common dolphin bycatch in the Bay of Biscay.

KEY WORDS: Spatiotemporal modelling • Reverse drift modelling • Prey-predator relationship • Sea surface temperature • Thermal fronts • Eddies

Resale or republication not permitted without written consent of the publisher

Appendix : Explain and quantify the incidental captures of common dolphins

nent example of the latter outcome [START_REF] D'agrosa | Vaquita bycatch in Mexico's artisanal gillnet fisheries: driving a small population to extinction[END_REF][START_REF] Jaramillo-Legorreta | Saving the vaquita: immediate action, not more data[END_REF], 2019).

Short-beaked common dolphins Delphinus delphis Linné, 1758 in the Bay of Biscay (northeast Atlantic) have been subjected to potentially dangerous levels of bycatch since the 1990s (see [START_REF] Murphy | Conservation management of common dolphins: lessons learned from the north-east Atlantic[END_REF] for a review). In France, the magnitude of bycatch is evidenced by high numbers of strandings of bycaught dolphin carcasses on the Atlantic seaboard. Mannocci et al. ( 2012) estimated that with 1000 bycaught individuals per year, the northeast Atlantic population could be reduced by 20% in 30 yr and could become extinct in a hundred. Still, the phenomenon has be come more intense in the past decade. Since 2016, the number of strandings has increased every year (http://seamap.env.duke.edu). Bycatch estimates from strandings that occurred solely in the Bay of Biscay have been above what the population of common dolphins of the whole northeast Atlantic can theoretically sustain (Mannocci et al. 2012, Peltier et al. 2016, ICES 2020). From January to April 2019 alone, strandings suggested bycatch levels as high as 9500 individuals (95%CI: [6890; 14 200]) [START_REF] Dars | Identifying critical areas to reduce bycatch of coastal common bottlenose dolphins Tursiops truncatus in artisanal fisheries of the subtropical western South Atlantic[END_REF].

Stranding data can be complementary to data from European observer programs and can provide relevant information on cetacean bycatch (Peltier et al. 2016[START_REF] Ijsseldijk | Spatiotemporal mortality and demographic trends in a small cetacean: strandings to inform conservation management[END_REF]. In France, they have been instrumental to estimate the magnitude of common dolphin bycatch in the Bay of Biscay (Peltier et al. 2016) and to reveal a strong seasonality, with increased mortality in winter months (January to March) (Peltier et al. 2016, 2019, 2020, Dars et al. 2019). From 2016 onward, strandings of animals showing bycatch evidence have risen in summer (July and August), although to a lesser extent than in winter [START_REF] Dars | Identifying critical areas to reduce bycatch of coastal common bottlenose dolphins Tursiops truncatus in artisanal fisheries of the subtropical western South Atlantic[END_REF]. Stranding data further made it possible to identify spatiotemporally varying mortality areas (as areas where carcasses of by caught dolphins were released from fishing boats) through reverse drift modelling of carcasses trajectories (Peltier & Ridoux 2015, Peltier et al. 2016[START_REF] Course | Monitoring Cetacean Bycatch: An Analysis of Different Methods Aboard Commercial Fishing Vessels[END_REF]. In winter months, most bycatch events likely occur in the southern part of the continental shelf and slope (Peltier et al. 2016[START_REF] Peltier | Can modelling the drift of bycaught dolphin stranded carcasses help identify involved fisheries? An exploratory study[END_REF], resulting in strandings all along the shore from the south of the Finistère region to the border to Spain, whereas in summer most carcasses strand in southern Finistère (Fig. 1).

Finally, the bycatch pattern displays high interannual variability, with different stranding levels and associated bycatch level estimates from year to year (Peltier et al. 2016).

Stranding data further revealed possible association between bycatch events and common dolphin preys. The reverse drift modelling of carcasses allowed the identification of some potentially high-risk fisheries and their caught fish species [START_REF] Peltier | Can modelling the drift of bycaught dolphin stranded carcasses help identify involved fisheries? An exploratory study[END_REF](Peltier et al. , 2021)). The analysis of stomach content of by caught stranded animals suggested that most dolphins were feeding when death occurred. Moreover, some fish species targeted by common dolphins (e.g. mackerel or sardine) are also included in the diet of predatory fish species of fisheries interest (e.g. hake or seabass) (Spitz et al. 2013). Hence, changes in the local distribution and abundance of prey species might be a substantial driver of the co-occurrence of common dolphins, commercial fish species and fisheries in localized areas.

As marine mobile predators, cetaceans have a dynamic distribution integrating ecological processes across all levels of the trophic web [START_REF] Croll | An integrated approch to the foraging ecology of marine birds and mammals[END_REF][START_REF] Barlow | Links in the trophic chain: modeling functional relationships between in situ oceanography, krill, and blue whale distribution under different oceanographic regimes[END_REF]. The proximal relationship between prey and predator could drive the spatiotemporal variability of the bycatch pattern: fish distribution is notoriously variable in both space and time [START_REF] Hyrenbach | Workshop on fisheries emergency measures to minimize bycatch of short-beaked common dolphins in the Bay of Biscay and harbour porpoise in the Baltic Sea (WKEMBYC)[END_REF] and is governed by many factors ranging from dynamic oceanographic conditions to each species' annual cycle. In this study, we aimed at assessing the influence of oceanographic processes on bycatch of common dolphins in fishing gear. We thus focused on the distal relationship between marine predators and oceanographic processes. We assumed oceanographic processes spatiotemporally structure the availability and aggregation of prey, creating areas prone to attract common dolphins and predatory fish targeted by fisheries and consequently increase the bycatch risk. For instance, meso and submesoscale processes such as fronts (where 2 joined water masses differ by their density), up welling or eddies enhance the enrichment, concentration and retention of nutrients [START_REF] Bakun | Fronts and eddies as key structures in the habitat of marine fish larvae: opportunity, adaptive response and competitive advantage[END_REF]. These processes facilitate the development of trophic networks, from phytoplankton to zooplankton, to fish and finally, to apex predators [START_REF] Bakun | Fronts and eddies as key structures in the habitat of marine fish larvae: opportunity, adaptive response and competitive advantage[END_REF][START_REF] Bakun | Fronts and eddies as key structures in the habitat of marine fish larvae: opportunity, adaptive response and competitive advantage[END_REF]. Knowledge of oceanographic conditions is readily available from either remote sensing or modelling. This information is therefore commonly used to make inferences about prey availability and species distribution [START_REF] Forney | Environmental models of cetacean abundance: reducing uncertainty in population trends[END_REF][START_REF] Becker | Comparing California current cetacean-habitat models developed using in situ and remotely sensed sea surface temperature data[END_REF][START_REF] Best | Online cetacean habitat modeling system for the US East Coast and Gulf of Mexico[END_REF][START_REF] Stephenson | Modelling the spatial distribution of cetaceans in New Zealand waters[END_REF]). Oceanographic data have been previously used in the context of bycatch studies, either to identify high risk areas through the association of species distribution modelling and distribution of fishing effort (Žydelis et al. 2011[START_REF] Murray | Estimating the risk of loggerhead turtle Caretta caretta bycatch in the US mid-Atlantic using fishery-independent and -dependent data[END_REF][START_REF] Dars | Identifying critical areas to reduce bycatch of coastal common bottlenose dolphins Tursiops truncatus in artisanal fisheries of the subtropical western South Atlantic[END_REF][START_REF] López | Living on the edge: overlap between a marine predator's habitat use and fisheries in the northeast Atlantic waters (NW Spain)[END_REF] or to identify possible drivers of bycatch events [START_REF] Gardner | Hierarchical modeling of bycatch rates of sea turtles in the western North Atlantic[END_REF][START_REF] Cosandey-Godin | Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic[END_REF][START_REF] Hahlbeck | Oceanographic determinants of ocean sunfish (Mola mola) and bluefin tuna (Thunnus orientalis) bycatch patterns in the California large mesh drift gillnet fishery[END_REF], Scales et al. 2018). [START_REF] Cosandey-Godin | Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic[END_REF] suggested the forecasting of high bycatch risk areas to identify mitigation measures. We investigated relations between mortality of common dolphins in fishing gear and oceanographic processes in the Bay of Biscay. Predicting high bycatch areas was beyond the scope of this paper, although it constitutes a promising venue for further investigations.

This work capitalized on 2 sources of data: the bycatch-mortality index inferred from strandings and fine-scaled essential oceanographic variables (EOVs) [START_REF] García-Barón | Essential ocean variables and high value biodiversity areas: targets for the conservation of marine megafauna[END_REF][START_REF] Tew-Kai | Dynamic coastal-shelf seascapes to support marine policies using operational coastal oceanography: the French example[END_REF]. We selected physical variables significantly contributing to the characterization of the ocean realm (Tew- [START_REF] Tew-Kai | Dynamic coastal-shelf seascapes to support marine policies using operational coastal oceanography: the French example[END_REF]. We used hierarchical Bayesian models to account for spatiotemporal processes: bycatch mortality areas, oceanographic processes, and common dolphin and fisherie distributions are dynamic in both space and time. The area of study was limited to waters of the Bay of Biscay, with a small part of the Celtic Sea (Fig. 1). After model fitting, we used outof-sample, 1-step-ahead prediction to predict bycatch mortality in 2019 from oceanographic processes.

MATERIALS AND METHODS

Dolphin mortality index from stranding data

Mortality index

For most bycatch events, no data or location at sea is available. However, some bycatch events are detectable from the stranding of common dolphins presenting bycatch evidence (Kuiken 1994). To determine the origin of carcasses, a reverse drift trajectory can be computed with the drift prediction model MOTHY (Modèle Océanique de Transport d'Hydrocarbure, developed by Météo France; [START_REF] Daniel | Drift modeling of cargo containers[END_REF], integrating drift conditions and characteristics of dolphins' carcasses (see Peltier & Ridoux 2015, Peltier et al. 2012). Reverse drift trajectories of bycaught animals were used to compute a mortality index (MI) to identify probable mortality areas and to quantify the intensity of mortality events at sea in space and time (Fig. 2).

From 2012 to 2019 stranding data, only carcasses presenting bycatch evidence or which stranded during an intense mortality event were selected. Intense mortality events are when more than 30 dolphins strand on 200 km of coastline within 10 d (Peltier et al. 2016). We further selected carcasses with estimated mortality windows up to 15 d (2384 out of 4544, i.e. 52%). For each carcass, a reverse drift trajectory was calculated with MOTHY, with one point every 10 h. Drift tracks were limited from 0 to 5 or from 5 to 15 d depending on the mortality window estimated from the decomposition state of the stranded carcasses. The computation of the MI on a daily basis, using a regular grid of 0.1° × 0.1° resolution (i.e. approximately 26 nmiles 2 or 89 km 2 ) for the Bay of Biscay, assumed an equal probability of bycatch oc curring at each drift point. For a day d, a grid cell c and a carcass i, MI was defined as follows:

(

The more a carcass stagnates in a grid cell, the higher MI reflects a higher probability that death occurred in that grid cell. For a given cell on a given day, there could be reverse drift points associated with several carcasses. The total MI for d and c was then:

(2)

The greater the number of carcasses that have drifted through a grid cell, the higher its MI. Finally, daily MIs were summed for each grid cell over each month from 2012 to 2019 resulting in the following spatial monthly MI, for month m and grid cell c (Fig. 2):

(3)

The total sum of MI over each month is equal to the number of stranded dolphins included in the dataset. In total, 1789 carcasses stranded all along the shore of the Bay of Biscay (Fig. 1) were used for the present study, with substantial seasonal and inter-annual variations (Fig. 3). Monthly numbers are provided in Supplement 1; www. int-res. com/ articles/ suppl/ m679 p195 _ supp/.

Stranding probability

A wide range of factors lead a carcass to strand, including buoyancy (the carcass must be floating) and drift conditions (wind being one of the main drivers). Consequently, not all carcasses of bycaught common dolphins do strand (Peltier et al. 2016). Drift conditions can sometimes keep carcasses at sea rather than push them to the coast. Buoyancy of carcasses is probably influenced by several parameters (i.e. depth at which death occurred, buoyancy at time of death; [START_REF] Moore | Dead cetacean? Beach, bloat, float, sink[END_REF]) so that they may either sink or float when they are released from fishing boats. The calculated MI thus contained 'false' zeros, associated to grid cells where conditions would not have led to strandings if carcasses happened to drift through them. Conversely, 'true' zeros were associated to grid cells where conditions could have led drifting carcasses to strand. To account for this, we systematically included stranding probability as a covariate in all models. Stranding probabilities were estimated from drifts of simulated dolphin carcasses with MOTHY [START_REF] Peltier | The stranding anomaly as population indicator: the case of harbour porpoise Phocoena phocoena in north-western Europe[END_REF](Peltier et al. , 2014(Peltier et al. , 2016)). It was calculated over 10 d periods and averaged over each month. A conditional autoregressive (CAR) model was then applied to the result to leverage spatial dependence and smooth estimated probabilities. The spatial resolution of estimated probabilities was larger than that of MI (Fig. 4): each grid cell of the MI grid was matched to its closer neighbour in the stranding probability grid (using R package sf of Pebesma 2018). True zeros should be associated with high stranding probabilities whereas false zeros should be associated with low stranding probabilities.

Oceanographic data

Monthly EOVs and derived variables were computed from the 3-dimensional HYCOM numerical model of the French Service Hydrographique et Océanographique de la Marine (Shom). Based on the original HYCOM model [START_REF] Bleck | An oceanic general circulation model framed in hybrid isopycnic-Cartesian coordinates[END_REF], it was set with a number of numerical developments to optimize the model for coastal zones [START_REF] Pichon | Internal tides modelling in the Bay of Biscay. Comparisons with observations[END_REF][START_REF] Morel | Time splitting and linear stability of the slow part of the barotropic component[END_REF][START_REF] Lahaye | A numerical scheme for modeling tidal wetting and drying[END_REF] and has already been successfully used over the Bay of Biscay area (e.g. [START_REF] Pichon | Internal tide interactions in the Bay of Biscay: observations and modelling[END_REF]. Hindcasts were calculated hourly from 1 January 2012 to 31 December 2019 to compute monthly EOVs and derived variables. EOVs are outputs of the model (temperature, salinity, cur-

MI MI MI , , , , c m d m d c d m i d c i ∑ ∑ ∑ = = ∈ ∈

MI

No. of drift points of in grid cell Total no. of drift points of , ,

i c i d c i = MI MI , , , d c i d c i ∑ = 198 1 2 3 4 5 6 7
Month Total IM per month 8 9 Fig. 3. Evolution of the total mortality index (MI) per month and per year. The monthly sum of spatial MI over the study area is equal to the number of stranded dolphin carcasses for which a reverse drift was calculated, thus giving a mortality signal Author copy rent), and derived variables have been calculated from these EOVs. One EOV and 2 derived variables, hereafter referred as oceanographic variables (OVs), were integrated in this study: (1) the sea surface temperature (sst), (2) the mean sea surface temperature gradient (mean_sst_grad), quantifying the intensity and wideness of thermal fronts, and (3) eddy kinetic energy (eke), quantifying the turbulent component of the residual (non-tidal) surface current. sst was chosen for its strong seasonal component [START_REF] Lambert | Habitat modelling predictions highlight seasonal relevance of marine protected areas for marine megafauna[END_REF]) but also as it is commonly in tegrated in habitat modelling of multiple marine taxa [START_REF] Doniol-Valcroze | Influence of thermal fronts on habitat selection by four rorqual whale species in the Gulf of St. Lawrence[END_REF][START_REF] Mellin | Geographic range determinants of two commercially important marine molluscs[END_REF][START_REF] Hughes | Changes in the spatial distribution of spawning activity by north-east Atlantic mackerel in warming seas: 1977-2010[END_REF][START_REF] Castro | Oceanographic determinants of the abundance of common dolphins (Delphinus delphis) in the south of Portugal[END_REF]). In addition, thermal fronts and eddies have been identified as key structures for the conservation of marine predators [START_REF] Scales | On the front line: frontal zones as priority at-sea conservation areas for mobile marine vertebrates[END_REF][START_REF] Scales | Fisheries bycatch risk to marine megafauna is intensified in Lagrangian coherent structures[END_REF]. Such interfaces might trigger bottom-up processes. As nutrient mixing and retention in the photic zone can boost primary productivity, interfaces like fronts and eddies might attract apex predators through a cascading effect from the lower trophic levels to the highest ones [START_REF] Bakun | Fronts and eddies as key structures in the habitat of marine fish larvae: opportunity, adaptive response and competitive advantage[END_REF][START_REF] Bakun | Fronts and eddies as key structures in the habitat of marine fish larvae: opportunity, adaptive response and competitive advantage[END_REF][START_REF] Scales | On the front line: frontal zones as priority at-sea conservation areas for mobile marine vertebrates[END_REF]). This association was observed for several marine predators [START_REF] Nel | Exploitation of mesoscale oceanographic features by grey-headed albatross Thalassarche chrysostoma in the southern Indian Ocean[END_REF][START_REF] Doniol-Valcroze | Influence of thermal fronts on habitat selection by four rorqual whale species in the Gulf of St. Lawrence[END_REF][START_REF] Bailey | Effect of oceanographic features on fine-scale foraging movements of bottlenose dolphins[END_REF][START_REF] Miller | Basking sharks and oceanographic fronts: quantifying associations in the north-east Atlantic[END_REF][START_REF] Snyder | Crossing the line: tunas actively exploit submesoscale fronts to enhance foraging success[END_REF] along with further associations with bycatch events [START_REF] Hahlbeck | Oceanographic determinants of ocean sunfish (Mola mola) and bluefin tuna (Thunnus orientalis) bycatch patterns in the California large mesh drift gillnet fishery[END_REF], Scales et al. 2018). Moreover, the Bay of Biscay encompasses a shelf area with complex dy namic mesoscale processes following a seasonal cycle [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]), and we meant to integrate some of these features in our analysis through the dynamic variables eke and mean_ sst_ grad.

As the spatial resolution of the oceanographic data was the nautical mile, it was upscaled to match that of the MI. All data can be visualized through maps and histograms at http:// pelabox. univlr. fr: 3838/ pelagis/ DdSeaByc/ (Tab: '1. Data').

Spatiotemporal modelling framework

Our goal was to elucidate if oceanographic conditions were associated with bycatch of common dolphins in fishing gear with statistical modelling. We considered a general set up MI ~ OVs where MI is the response variable whose variations are to be correlated with OVs, with linear effects for covariates.

Spatial and temporal considerations

Common dolphin bycatch -and stranding of animals showing bycatch evidence -is a seasonal phenomenon in the Bay of Biscay [START_REF] Dars | Identifying critical areas to reduce bycatch of coastal common bottlenose dolphins Tursiops truncatus in artisanal fisheries of the subtropical western South Atlantic[END_REF], ICES 2020). The mortality signal, defined here as the number of stranded common dolphin showing bycatch evidence, is higher in January to March (winter) and in July and August (summer) but is lower in between (Fig. 3). Oceanographic conditions also follow a seasonal cycle with different dynamics in winter and summer and transition regimes in between [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Our exploratory analysis (not shown here) suggested that the relationship between the re sponse variable and oceanographic covariates varies across months. Hence, we chose to fit one model per month to get a climatology over the study period. Each monthly model gathered data for the corresponding month from 2012 to 2019, resulting in 12 models (from January to December). A similar approach was used in Foravanti et al. (2021). This climatological approach had the advantage of reducing computational cost.

The relation between OVs and MI in one given month can vary from one year to the next, for in -

Author copy

Mar Ecol Prog Ser 679: 195-212, 2021 stance because of a timelag in cyclic phenomena [START_REF] Huret | Survey timing vs. ecosystem scheduling: degree-days to underpin observed interannual variability in marine ecosystems[END_REF] or because of singular climatic events. The dynamic of mesoscale processes is complex in the Bay of Biscay [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Singular phenomena could be especially attractive for preys of dolphins and predatory fish targeted by fisheries depending on their extent, their duration and their time of occurrence. The effect of OVs on bycatch risk (and thus on MI) must be varying in time, as suggested by the intensification in strandings from 2016 [START_REF] Dars | Identifying critical areas to reduce bycatch of coastal common bottlenose dolphins Tursiops truncatus in artisanal fisheries of the subtropical western South Atlantic[END_REF]. This led us to consider annually varying co efficients for OVs (i.e. random slopes). We further added an annually varying intercept (random intercept) to account for different annual levels of by catch. Using random slopes amounts to considering an interaction with year: time-series of the effect of OVs on MI for each month and each year were an output from models over the study period. Generalized linear models (GLM) and generalized additive models (GAM) usually assume all relevant covariates have been included in the model (Zuur et al. 2007a,b). This assumption is unrealistic in most cases, and omitted variables may result in a residual de pendence that, if not accounted for, may result in misleading inferences [START_REF] Legendre | The consequences of spatial structure for the design and analysis of ecological field surveys[END_REF][START_REF] Valcu | Spatial autocorrelation: an overlooked concept in behavioral ecology[END_REF]. Omitted variables bias may manifest in residual autocorrelation, either temporal, spatial, or both. Because there was one model per month, the time lag between each 'measurement' was 11 mo, and we assumed no temporal autocorrelation. Yet, spatial autocorrelation, if not taken into account, could suggest spurious correlations with OVs. Consequently, all models included explicitly a spatial component in addition to the (implicitly spatial) OVs.

Hierarchical Bayesian framework and

Gaussian random field

We used a hierarchical Bayesian framework as they are adapted to the modelling of complex spatial or spatiotemporal phenomena [START_REF] Cosandey-Godin | Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic[END_REF][START_REF] Martínez-Minaya | Species distribution modeling: a statistical review with focus in spatio-temporal issues[END_REF]. Including a spatial random effect, e.g. with a Gaussian random field (GRF), is straightforward [START_REF] Blangiardo | Spatial and spatio-temporal models with R-INLA[END_REF][START_REF] Cosandey-Godin | Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic[END_REF]. The GRF, indexed on each grid cell, is a stochastic process that will gather all the spatially implicit information in the data that are not accounted for by other model components, in our case oceanographic covariates [START_REF] Cosandey-Godin | Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic[END_REF]. To model this random field, we chose a CAR process [START_REF] Besag | A Bayesian image restoration with two applications in spatial statistics[END_REF][START_REF] Besag | On conditional and intrinsic autoregressions[END_REF] based on the notion of nearest neighbours (the so-called first law of geography: 'Everything is related to everything else, but near things are more related than distant things'; Tobler 1970, p. 236). For the vector of grid cells (η 1 , η 2 , ..., η n , here n = 3487) it can be written as follows:

(4)
where n i is the number of neighbours for grid cell i and τ is a precision parameter. i ~ j means that the 2 grid cells i and j are neighbours.

Model set up

MI was first transformed to log(MI + 1) and then modelled by a Gaussian distribution with an identity link function. The spatial CAR process was considered constant over years, so that there would be one spatial field per month. The stranding probability (SP) was always included as a covariate with a linear effect. For oceanographic variables effects and for the intercept, we included the possibility for the coefficients to vary from one year to the next by specifying a Gaussian random (independent and identically distributed) effect on years (random slopes). For an overview of our monthly model, we had for the response variable MI in year t and grid cell c, with oceanographic covariates X:

(5)

where N defines the normal distribution of location parameter π c,t and scale parameter σ, Id is the identity link function, γ the coefficient associated with SP, β 0,t the intercept for year t, β j,t the annual coefficient of covariate X j,c,t at cell c in year t and η c the spatial effect associated to grid cell c, which is assumed to result from a CAR model. Finally,  c,t are unstructured errors, i.e. residuals. Parameters estimated in the linear predictor, i.e. γ, β 0,t , β j,t , η c and  c,t constitute the latent field. They are assumed to follow Gaussian distributions [START_REF] Rue | Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations[END_REF]). These parameters depend on a vector of hyperparameters θ that are not necessarily Gaussian [START_REF] Rue | Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations[END_REF]. In this manner, the monthly model defined in (Eq. 5) is a latent Gaussian model (LGM) [START_REF] Rue | Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations[END_REF]. It presents a spatial component (η c ) along with annual components (β 0,t and β j,t ).

For all in 1:12 dropping the subscript below for clarity log MI

X m c t c t c t c t t j j t j c t c c t ∑ ( ) 1 ~, Id SP , , , , 0 , , , , , m N
+ π σ π = γ × +β + β × +η + ε ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ( ) η η ≠ τ × τ | , , ~1 , 1 i j N n n i j i i 200 ( ) ( ) ( )
Author copy Inference for such complex spatial models is computationally demanding and requires approximation to be fully scalable with a large amount of data. For a large dataset with 3487 grid cells observed 8 times (i.e. years) per analysis, i.e. 27 896 observations per month, classical methods like Markov Chain Monte Carlo may be prohibitively slow [START_REF] Blangiardo | Spatial and spatio-temporal models with R-INLA[END_REF]). We used INLA, which offers a very efficient alternative giving fast and reasonable estimations [START_REF] Rue | Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations[END_REF]. Especially adapted to LGMs, this algorithm uses a combination of analytical approximations and numerical integration for the inference of posterior marginals and likelihood, which greatly reduce computational demand while being extremely accurate [START_REF] Rue | Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations[END_REF][START_REF] Moraga | Geospatial health data: modeling and visualization with R-INLA and Shiny[END_REF]. Inference with INLA was carried out with the R-INLA package in R statistical software v4.0.2 (R Core Team 2020).

Before model fitting, all oceanographic variables were standardized (mean-centered and scaled to unity), and eke and mean_sst_grad were logtransformed to account for skewness and strict positivity. Weakly informative priors were chosen so that estimations would mostly be driven by the data (Gabry et al. 2019). Fitted and predicted values that were negative after back transformation were set to zero.

Model selection

All possible combinations of the different effects and covariates were tested, with the SP systematically included as a confounding factor (list of tested models is provided in Supplement 2). The first combinations were the simplest models with only linear effects for oceanographic covariates (i.e. in Eq. 5, β j replaces β j,t ) and no spatial field (8 possibilities with the null model). Second, coefficients were allowed to vary between years for oceanographic covariates (7 possibilities) with random slopes. Finally, the CAR process was added, with and without oceanographic covariates, considering strict linear effects or annual random slopes (15 possibilities). The 30th model was the more complex, defined in Eq. ( 5), where j ∈ [1, 3] and X j are thus sst, eke or mean_sst_grad. Model selection was conducted using the Wanatabe Akaike information criterion (WAIC) [START_REF] Watanabe | Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory[END_REF]. WAIC gives an approximation of the out-of-sample predictive accuracy using the in-sample data (Gelman et al. 2014).

Model checking

The coefficient of determination R 2 for each monthly model was computed. R 2 is the proportion of variance in the response variable accounted for by explanatory variables [START_REF] Johnson | Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models[END_REF]. We approportioned the total variance to different model components, i.e. spatial or annual, and assessed their respective contributions. For random slopes models, some variance components depend on observations [START_REF] Johnson | Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models[END_REF]. For instance, the variance of β sst,t , coefficient of variable sst on year t, depends on values of sst during year t. We used the matrix formula (11) of [START_REF] Johnson | Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models[END_REF]; see Supplement 3 for further details.

Finally, we assessed the predictive power of models, first by conducting out-of-sample cross-validation (CV), then by testing different prediction scenarios. While the dataset was complete from 2012 to 2018, for 2019 only values of oceanographic variables were given to get predictions of MI for this year. As the intercept and covariates coefficients were estimated annually, there was an index associated to each year. For the CV procedure, we considered 2019 as a new year (predicted from the posterior distribution) and attributed to this year a new and unique index. In addition, we tested different repetition scenarios (RS): instead of attributing a new index to year 2019 and consider it as a new sample, we considered 2019 as a repetition of a previous year by attributing to 2019 its index. This way, all model components fitted during the 'repeated' year were used (covariates' coefficients, effect of the stranding probability, spatial CAR process) for the prediction of MI in 2019, still considering 2019's oceanographic conditions. We tested all 7 possibilities, i.e. we considered 2019 as a repetition of 2012, then of 2013, 2014, 2015, 2016, 2017 and 2018. Then, we confronted the predicted total mortality for 2019 to 2019's total mortality from the dataset (never seen by the models). We also computed a root mean squared error (RMSE) per RS and for the CV, based on the monthly differences between predicted and observed (from strandings) total monthly mortality. With these repetition scenarios, we aimed at addressing whether oceanographic conditions of past years affected bycatch mortality risk in the same way and could predict the observed mortality in 2019.

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Mar Ecol Prog Ser 679: 195-212, 2021

RESULTS

Model selection

For most months, the most complex model detailed in formula (Eq. 5) had the smallest WAIC values (table with WAIC of all tested models provided in Supplement 4). It included a CAR process, yearvarying intercepts and year-varying slopes for the 3 covariates sst, eke and mean_sst_grad. In May, June and November the model with the smallest WAIC values excluded eke and for September sst was excluded. However, the difference in WAIC between these models and the most complex one selected for other months was small (relative difference of 0.01, 0.03, 0.1, 0.06% in May, June, September and November, respectively). Hence, we chose to keep the most complex model with all 3 covariates for all months to facilitate comparison of results over the 12 months.

Oceanographic covariates effects on the MI

Slope estimates of the effect of sst, eke and mean_sst_grad on spatiotemporal MIs were precise with tight credible intervals (see http:// pelabox. univlr. fr:3838/pelagis/DdSeaByc/; Tab: '2. Results 2.1.'. There were no estimates for year 2019 as this year was not used in model fitting but in out-of-sample, 1-step-ahead cross-validation and predictions. Overall, effects were more intense in the winter season, from January to March (Fig. 5). They were close to zero for the rest of the year, except in July and August for 2016, 2017 and 2018 for eke and mean_sst_grad. The winter variation pattern of the effect of eke and mean_sst_grad was different every year. Their effect on MI was sometimes positive and sometimes negative, depending on years. In July and August, however, the effect of these 2 covariates was consistent from 2016 to 2018. It was negative for eke and positive for mean_sst_grad. The winter variation pattern of the effect of sst was consistent be tween years, with interannual differences in its intensity. It switched sign from one month to the next, for most years. In summer, the effect of sst was close to zero, except in 2017, when it was slightly negative. For all 3 covariates, 2017 emerged as a singular year.

The intercept was estimated as a mean level of MI with OVs being set at their mean values (given that they were standardized, that is mean-centered). It was greater in winter, with mostly positive values in January and February but negative values in March, except in 2017, when it was positive for all 3 months (Fig. 5). Its pattern of variation was more or less consistent over the study period, with differences in its intensity. In summer, the intercept was slightly positive only in 2016 and 2017. Author copy

Variance partitioning

The computation of R 2 allowed us to differentiate the contribution of the different model components to integrate MI variations (Fig. 6). The annual component included annual coefficients of covariates (random slopes) along with annual intercepts. The spatial component was the CAR process whose contribution in variance showed little between-month variation. The residual component included what was not accounted for by either of the 2 precedent components. In total, models better accounted for MI variations from January to April, the annual component being the most contributive component on this period. From January to March, 21 to 25% of the variance were accounted for by annual and spatial components. A rise in the annual component contribution in July and in the spatial component contribution in August resulted in a rise in the overall variance accounted for these 2 months. Afterward, most of the variations of MI were integrated in residuals. Note that, using formula (11) of [START_REF] Johnson | Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models[END_REF], co variances between components were included: some negative covariances terms explain that contributions of the 3 components naïvely sum to more than 100% for some months.

CV and predictions of 2019 mortality with repetition scenarios

For CV, 1-step-ahead prediction was performed. Each monthly model was calibrated on 2012-2018 data, so MI data from 2019 was not used in model fitting. 2019 oceanographic data were thus used for prediction, taking between-year variations into account. The total monthly MI of the dataset (observed) for 2019 fell within the confidence interval of total monthly predicted values, for both the winter and the summer mortality periods (Fig. 7). Total mean predicted values were below observed ones. In addition, 2019 was assumed as a repetition of each year between 2012 and 2018 (RS), to highlight if past oceanographic influences could predict ob served mortality in 2019. Among the 7 RS, observed values of total monthly mortality of 2019 during the winter mortality period was within (or close to) confidence intervals of predicted values for 2 RS, when 2019 was considered as a repetition of 2013 (RS 2019(RS = 2013) ) and as a repetition of 2017 (RS 2019 = 2017) (Fig. 7). For RS of other years, predicted values and confidence intervals were lower than observed monthly mortality, from January to March. The summer mortality season was not reproduced by the models for RS of years before 2016. Overall, 2017 RS showed the lowest RMSE and gave the best prediction of 2019 monthly mortality levels (Table 1). RMSE for CV was below RMSE of RS of years 2012 and 2015 and close to that of 2018.

Additional results such as maps of fitted or predicted values and maps of the estimated spatial effects are available at http://pelabox.univ-lr.fr:3838/ pelagis/ DdSeaByc/ (Tab: '2. Results') for all 12 months. A temporal series of total monthly mortality of fitted values and ob served values are provided in Fig. A1 in the Appendix.

DISCUSSION

We assumed bycatch risk could be correlated to the proximal relationship between the dynamic distribution of common dolphins, commercial predatory fish species targeted by fisheries and that of their common prey species, the latter being partially driven by oceanographic processes. Spatiotemporal information on at-sea bycatch mortality, fisheries, dolphins and fish prey species distribution is not easily accessible to study this proximal relationship. Thus, we used proxies, a mortality index inferred from strandings and oceanographic descriptors, to study the dis-

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Mar Ecol Prog Ser 679: 195-212, 2021 tal relationship between common dolphin bycatch and oceanographic processes. We mobilized 8 yr of stranding data (from 2012 to 2019) and explored spatial and temporal links between basal and apex nodes of the trophic network, from oceanographic processes to short-beaked common dolphins (and fisheries). We fitted a series of 12 spatiotemporal monthly models that included 3 oceanographic variables as covariates potentially contributing to the spatiotemporal structuring of bycatch risk of common dolphins in the Bay of Biscay. Results provided first insights into how the intensity of common dolphin bycatch in the Bay of Biscay might be modulated by key seasonal and dynamic oceanographic features.

Two seasons of mortality, two different underlying processes

Strandings highlight 2 seasons of high bycatch mortality in the Bay of Biscay: a main one from January to March and another (only observed from 2016 onward) in July and August (Fig. 3). Results reflected different patterns for these 2 mortality seasons, suggesting differences between the underlying phenomena leading to bycatch mortality. Outside these seasons, the explanatory power of oceanographic processes was very limited, and the effect of OVs on the mortality index was close to zero. Thus, it seems that OVs seasonally affect bycatch risk and should not be considered as season-invariant drivers of by catch mortality.

Winter high mortality season, from January to March

The annual component, including OVs, accounted for most variations in bycatch risk during the winter mortality season (Fig. 6), suggesting a substantial influence of the oceanographic processes. Moreover, January, February and March were the months when OVs had the strongest effects on bycatch risk (Fig. 5). Depending on OVs, their effects (i.e. monthly estimated linear coefficients) showed different inter-annual patterns. The month-to-month effect of sea surface temperature was the only one to be consistent from one year to the next. Sea surface temperature partially drives the distribution and seasonal migrations of multiple marine taxa, from small pelagic fish [START_REF] Lanz | Small pelagic fish catches in the Gulf of California associated with sea surface temperature and chlorophyll[END_REF][START_REF] Hughes | Changes in the spatial distribution of spawning activity by north-east Atlantic mackerel in warming seas: 1977-2010[END_REF] to bigger fish [START_REF] Hahlbeck | Oceanographic determinants of ocean sunfish (Mola mola) and bluefin tuna (Thunnus orientalis) bycatch patterns in the California large mesh drift gillnet fishery[END_REF]) and high-level predators [START_REF] Weltz | Precision in bycatch estimates: the case of Tuna Purse-Seine Fisheries in the Indian Ocean[END_REF]). In the Bay of Biscay, the influence of sea surface temperature affects the distribution of identified prey species of common dolphins and commercial predatory fish species (Spitz et al. 2013), such as anchovies and sardines [START_REF] Planque | Modelling potential spawning habitat of sardine (Sardina pilchardus) and anchovy (Engraulis encrasicolus) in the Bay of Biscay[END_REF][START_REF] Petitgas | Modelling the variability in fish spatial distributions over time with empirical orthogonal functions: anchovy in the Bay of Biscay[END_REF][START_REF] Politikos | A coupled movement and bioenergetics model to explore the spawning migration of anchovy in the Bay of Biscay[END_REF]. Therefore, a correlation between mortality and sea surface temperature can be expected if the underlying mechanisms are linked to preys of both common dolphins and commercial fish species targeted by fisheries. The effects of the mean temperature gradient and eddy kinetic energy were of the same magnitude as that of sea surface temperature but presented high inter-annual variations. Mean temperature gradient quantifies the intensity and wideness of thermal fronts, whereas eddy kinetic energy is associated with turbulent surface features such as eddies. Because they were averaged over each month, they should refer to relatively persistent structures. In winter, the Bay of Biscay environment is characterized by a seasonal cross-shore (west to east) surface temperature gradient with lowest temperature close to shore and intense frontal activity parallel to the coast (north to south) [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. These frontal structures are freshwater fronts, correlated with the mixing of oceanic waters and cold freshwater inputs from river plumes [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Their location and timeline concur with observed patterns regarding bycatch mortality: during winter months, strandings were from all the Atlantic coast (http:// seamap. env. duke. edu), and high mortality areas were previously identified on the continental shelf (Peltier et al. 2021). Moreover, most of highest MI values were located offshore of the coast from the south of the Finistère region to the south French border (not shown, see http://pelabox.univ-lr.fr:3838/pelagis/DdSeaByc/; Tab: '1. Data'), especially in 2017 when strandings were the more numerous over the 2012 to 2018 period. Together with the effects of the associated OVs on the MI, results suggested these seasonal fronts may be targeted by both fisheries and common dolphins as areas where fish aggregate, thereby putting the latter at risk of bycatch by the former.

Summer high mortality season in July and August

July and August were the only 2 other months when OVs accounted for a non-negligable fraction of variations in bycatch mortality (Fig. 6). The amount of total explained variance accounted for by OVs (included in the annual component) was more limited compared to the winter mortality season and was below the contribution of the CAR process. This could suggest that, during this summer season, other drivers (currently omitted in the model) might drive mortality risk. However, this was expected as strandings are at least one order of magnitude lower in summer and only became substantial from 2016 on ward. Nevertheless, different patterns emerged re garding the effects of oceanographic covariates. Conversely to the winter season, the sea surface temperature coefficient was close to zero for all years, and the effects of mean temperature gradient and eddy kinetic energy on the MI were, respectively, quite consistent for all 3 years when a summer mortality peak was observed (2016,2017,2018). For these 3 years and in July and August, the effect of the mean temperature gradient was slightly positive, meaning that thermal fronts were associated with higher MI and that of eddy kinetic energy was slightly negative, meaning that bycatch mortality probably occurred in low turbulence waters. In July and August, the mesoscale dynamic activity of the Bay of Biscay is rather different than in winter. In winter, the frontal activity in the Bay of Biscay is dominated by freshwater fronts resulting from input of cold freshwater from rivers whereas in summer, there are mainly fronts due to tidal flow [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Winter freshwater fronts display more spatiotemporal variability (on a given year and interannually) as they are directly correlated to river flows [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Summer tidal fronts are conversely quite consistent from one year to the next because they are correlated to a repetitive process (i.e. tides) [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. During summer, the main frontal activity is a seasonal tidal front, called the Ushant Front, and it is located in front of the French Finistère region [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Its activity peaks in July and August [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Turbulent components of surface currents are relatively low around this structure (not shown, see http://pelabox.univ-lr.fr:3838/ pelagis/ DdSeaByc/; Tab: '1. Data'), except at its most northern part. Again, the location of this typical frontal structure concurred with the location of hot spots of fitted MI as well as with the location of bycaught common dolphin strandings, mainly occurring on the coast of the south Finistère region during the summer mortality season (http:// seamap.env.duke.edu).

Other drivers of seasonal differences

The distribution of common dolphins in the Bay of Biscay differs between seasons (Laran et al. 2017). In winter, they are more numerous on the shelf, whereas in summer they are mainly distributed on the shelf break. This probably results in spatial differences in the local abundance of common dolphin between seasons, directly affecting bycatch risk. Seasonal differences in bycatch-induced mortality may further be linked to annual biological cycles of common dolphins and fish. The calving and mating season of short-beaked common dolphins of the northeast Atlantic is from April to September, with a likely peak of activity in July and August [START_REF] Murphy | Mating strategy in the male common dolphin (Delphinus delphis): what gonadal analysis tells us[END_REF](Murphy et al. , 2009)). Feeding strategies and movement patterns of dolphins differ during those periods, resulting in a different response to oceanographic processes. This could also be true with respect to dolphins' preys, many of which present an annual biological cycle. As for the fisheries, they might likewise change their searching strategies, or their fishing gear, depending on the season and target species. The fishing effort and fishing targeted areas also likely differ between seasons (ICES 2020). A combination of those parameters might affect the difference between the 2 seasons regarding patterns of oceanographic variables effects on bycatch mortality.

Although the mechanisms were different between winter and summer mortality seasons, an influence of oceanographic processes and especially seasonal thermal front structures was highlighted for the 2 seasons of mortality. The association between marine predator and thermal fronts was shown and studied elsewhere [START_REF] Doniol-Valcroze | Influence of thermal fronts on habitat selection by four rorqual whale species in the Gulf of St. Lawrence[END_REF][START_REF] Gannier | SST fronts and the summer sperm whale distribution in the northwest Mediterranean Sea[END_REF][START_REF] Cox | Oceanographic drivers of marine mammal and seabird habitat-use across shelf-seas: a guide to key features and recommendations for future research and conservation management[END_REF], Scales et al. 2018), and thermal fronts were identified as priority conservation areas for marine megafauna [START_REF] Scales | On the front line: frontal zones as priority at-sea conservation areas for mobile marine vertebrates[END_REF]). If correlations are complex and not straightforward, this study still provides first hints of an association between short-beaked common dolphins and this type of mesoscale frontal structure in the Bay of Biscay, likely linked to bycatch mortality.

Between-month variations

For all 3 oceanographic covariates, in winter, interpretation is further complicated as their effects on the mortality index often switched sign from one month to the next. From a statistical point of view, there could be 2 reasons for this. First, because we conducted different models for each month, there is no dependence between consecutive months. This could be improved by putting additional model structure such as a first order random walk for example. However, this would also substantially increase the computational burden, multiplying the number of observations (3487 spatial cells 'observed' once per month and per year, so 334 752 observations for the complete dataset). Second, it is likely that the effect of oceanographic processes on the mortality of dolphins due to bycatch is not linear or additive. Non-linear relationships with environmental covariates are routinely evidenced in marine species distribution modelling [START_REF] Forney | Environmental models of cetacean abundance: reducing uncertainty in population trends[END_REF][START_REF] Stoner | Spatially explicit analysis of estuarine habitat for juvenile winter flounder: combining generalized additive models and geographic information systems[END_REF][START_REF] Kai | Influence of mesoscale eddies on spatial structuring of top predators' communities in the Mozambique Channel[END_REF] and fisheries [START_REF] Maury | Hierarchical interpretation of nonlinear relationships linking yellowfin tuna (Thunnus albacares) distribution to the environment in the Atlantic Ocean[END_REF][START_REF] Walsh | Generalized additive model and regression tree analyses of blue shark (Prionace glauca) catch rates by the Hawaii-based commercial longline fishery[END_REF], Zagaglia et al. 2004) or even bycatch studies (Žydelis et al. 2011[START_REF] Dars | Identifying critical areas to reduce bycatch of coastal common bottlenose dolphins Tursiops truncatus in artisanal fisheries of the subtropical western South Atlantic[END_REF][START_REF] Hahlbeck | Oceanographic determinants of ocean sunfish (Mola mola) and bluefin tuna (Thunnus orientalis) bycatch patterns in the California large mesh drift gillnet fishery[END_REF]. Given the extra complexity of our response variable, it is quite conceivable that the influence of oceanographic features on short-beaked common dolphins bycatch is not linear. Combined with the fact that oceanographic processes might have different effects depending on their location and timing of appearance (that might vary from month to month), it could explain changes in signs of mean monthly coefficients of the oceanographic covariates. Again, relaxing the linear assumption would be possible with, for example, functional regression, but is beyond the scope of the present study.

There could also be ecological reasons for these between month variations. The winter oceanographic dynamics of the Bay of Biscay is characterized by high-frequency processes with rapid changes, sometimes within a month. Changes in the sign of OVs effects on the MI could be due to dynamic processes affecting common dolphins and fisheries interactions. The possibility that the effect of OVs could change from one month to the next motivated the use of random slopes. Taking thermal fronts as an example: these can aggregate preys of both common dolphins and predatory fish and do not necessarily increase bycatch risk. If fronts are quite wide and persistent, they could be wide enough so that they could attract fisheries on, say, their shore side and common dolphin on their seaside: spatial overlap be tween common dolphins and fisheries would be limited. However, if thermal fronts are very spatially localized, then spatial overlap between common dolphin and fisheries could be substantial, and bycatch risk would be higher. Yet, the frontal dynamics in the Bay of Biscay shows seasonal and interannual variability [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. The width and intensity of thermal fronts can therefore vary from one month to the next (and for a given month, from one year to the next), which may also explain fickle statistical relationships.

The bycatch mortality of 2019:

a predictable event?

Bycatch mortality in 2019 was the highest ever observed on the French Atlantic coast [START_REF] Peltier | Etat des connaissances sur les captures accidentelles de dauphins communs dans le golfe de Gascogne -Synthèse 2019. Rapport scientifique dans le cadre de la convention avec le MTES[END_REF], exceeding 2017 levels that were also unprecedented at that time. Still, crossvalidation results suggested it was statistically predictable as the observed monthly mortality tally from the dataset was within the confidence interval of the prediction (Fig. 7). Furthermore, results of the repetition scenarios showed how 2019's mortality was possibly due to circumstances already met in the past, either related to oceanographic conditions, or to other component included in models through the spatial process. The MI in 2019 was more accurately predicted when 2019 was considered as a repetition of 2013 and 2017 (Table 1, Fig. 7). For RS 2019 = 2017, the observed monthly mortality of 2019 was close to predicted values from oceanographic covariates (in 2019) and past relationships with OVs, both for winter and summer mortality seasons. As mentioned above, bycatch mortality in 2017 was also unprecedented at that time as an all-high record. 2017 indeed stands out as a singular year regarding the effect of all 3 covariates on bycatch risk (Fig. 5). This suggests maybe singular oceanographic phenomena in 2017 that were especially conductive of bycatch events, phenomena which recurred in 2019. A more surprising result was the remarkable good prediction made for winter months, when 2019 was considered as a repetition of 2013 (RS 2019(RS = 2013)). Observed mortality in 2013 from the dataset was 156 individuals (as fresher stranded carcasses showing bycatch evidence or stranded during an intense mortality event, see Section 2), whereas it was 317 in 2017 and 413 in 2019, through January, February and March. However, the observed mortality in January 2013 was close to that of January 2017 (Fig. 3) and was the second highest value for this month in the study period. In addition, mean coefficient estimates of eke and sst were part of the highest from 2012 to 2018. In January, the mean coefficient estimate of mean_sst_grad in 2013 was close to the mean coefficient estimate in 2017. In January, the mean effect of the mean temperature gradient was also the same in 2013 and in 2017. Despite the lower levels of observed mortality in 2013, it seems that winter oceanographic conditions in 2013, 2017 and 2019 favored common dolphin and fishery interactions, with an intensifying trend over the years. Given the complexity of our results regarding oceanographic covariate effects on bycatch risk (with inter-annual and between-month variations), the ad ditional value of repetition scenarios was to highlight whether similar associations occurred in different years.

Overall, these results showed that, from a statistical point of view, with few oceanographic covariates, the unprecedented bycatch mortality of 2019 was at the very least unsurprising as the pattern observed in 2019 could be accurately predicted from 2012-2018 data. This highlights how gaining knowledge about environmental influences on interactions between short-beaked common dolphins and fisheries could have great conservation and management value [START_REF] Scales | Fisheries bycatch risk to marine megafauna is intensified in Lagrangian coherent structures[END_REF]. In this study, we adopted a modelling approach to determine whether bycatch mortality can be explained by oceanographic processes. However, with evidenced oceanographic influences on bycatch incidence, a similar analysis could be designed to forecast potential hotspots of mortality. The modelling approach would have to be different, as explanatory and predictive modelling are distinct and are associated with different 'modelling paths' [START_REF] Shmueli | To explain or to predict?[END_REF]. For example, other sources of environmental data, available in a smaller timeframe, might have to be considered for near real-time prediction. Remote sensing could be an option, as it is already used in the support of fisheries activities [START_REF] Santos | Fisheries oceanography using satellite and airborne remote sensing methods: a review[END_REF]. The forecasting of high mortality zones might make it possible to define dynamic time-area closures for fisheries [START_REF] Cosandey-Godin | Applying Bayesian spatiotemporal models to fisheries bycatch in the Canadian Arctic[END_REF], which represent a promising tool to reduce bycatch pressure on the common dolphin population while limiting the economic impact [START_REF] Dunn | Spatio-temporal management of fisheries to reduce by-catch and increase fishing selectivity[END_REF][START_REF] Maxwell | Dynamic ocean management: defining and conceptualizing real-time management of the ocean[END_REF], Hazen et al. 2018). In this regard, further studies modelling the dynamic distribution of areas of co-occurrence of both short-beaked common dolphins and fisheries in the Bay of Biscay would be highly valuable for the conservation of this population.

Limitations and prospects for improvement

Several limitations can be imposed regarding the present work and results. First of all, strandings give only minimal estimations of bycatch-caused mortality. Some carcasses may never strand, and some others strand in advanced states of decomposition that do not allow their examination nor the identification of the cause of death of the animals. Moreover, due to the reverse drift modelling, the MI is associated with an uncertainty that is difficult to measure completely. Peltier et al. (2012) quantified the average distance between observed stranding locations of animals that were tagged when released from fishing boats and stranding locations predicted with MOTHY (27.1 ± 207 Fig. 7. Prediction for 2019. Monthly total mean mortality index from the dataset ('Observed MI') and from fitted values ('Predicted MI') for 2019 resulting from the 12 spatiotemporal models inferred with integrated nested Laplace approximations (INLA), with (1) a cross-validation (CV) procedure, i.e. the mortality index was predicted from previous years data and from values of oceanographic covariates for 2019 (2) repetition scenarios (RS), analogous to the latter except 2019 was considered as a repetition of a previous year of the study period. 24.5 km), which is accurate enough for practical purposes. Yet, there is no estimate of uncertainty associated with drift points as no information is available on where dolphins died. Developing the reverse modelling approach, on which the MI depends, was precisely motivated by infering high-probability atsea mortality areas (Peltier & Ridoux 2015[START_REF] Peltier | Can modelling the drift of bycaught dolphin stranded carcasses help identify involved fisheries? An exploratory study[END_REF].

The spatial scale for the present analysis (approximately 89 km -2 ) was coarse compared to that of OVs. This coarse resolution required upscaling for OVs, which could have smoothed out too many high frequency processes such as fronts and eddies. It is thus possible that the relationship between the MI and OVs was weakened, impacting the results concerning oceanographic coefficients and explanatory power. The association between the MI and oceanographic variables is likely scale dependent as are associations between oceanographic covariates and marine top predators' distribution and foraging [START_REF] Logerwell | The distribution of sea birds relative to their fish prey off Vancouver Island: opposing results at large and small spatial scales[END_REF][START_REF] Fauchald | Scale-dependent predator-prey interactions: the hierarchical spatial distribution of seabirds and prey[END_REF][START_REF] Pinaud | Scale-dependent habitat use in a long-ranging central place predator[END_REF][START_REF] Pirotta | Scale-dependent foraging ecology of a marine top predator modelled using passive acoustic data[END_REF]. The link of interest here could be detectable at a specific spatial scale and not at another, and for high frequency processes the scale could be relatively small. Getting mortality data at a finer scale (and defining the spatial uncertainty associated with the reverse drift modelling) would then improve our understanding of the physical and ecological processes at stake.

Lastly, the models we developed for this study accounted for a low proportion of variance of the MI (25% at the highest, in February). This limits the range for interpretation, especially for months when this proportion was below 10%. This was expected: we aimed at quantifying links that are, at best, indirect between OVs and a marine top predator. Many trophic levels might be involved in the relationships of interest here, from phytoplankton to zooplankton and fish, each of these levels being influenced by several environmental and physiological factors. Moreover, the response variable we used is only a proxy of realized bycatch risk. Nonetheless, models performed well enough to reproduce annual patterns of total monthly MI from 2012 to 2018 (Appendix). Models performed better for the winter season with increased mortality and to a lesser extent for the summer mortality season, which were of particular interest regarding the seasonal pattern of bycatch. The CV results finally confirmed the acceptable performance of our models, suggesting an association between chosen oceanographic variables and bycatchinduced mortality of common dolphins in the Bay of Biscay.

This association should not be over-interpreted: relationships highlighted here are not straightforward, and they have been highlighted only for a few months, and not for all years. The limited explanatory power of the oceanographic covariates even for months when bycatch mortality was relatively high suggests that bycatch risk in the Bay of Biscay may be influenced by other non-environmental factors, e.g. characteristics of fisheries like the type of fishing gear used, the overall fishing effort or the timeline of the fishing operation (day or night). There were no significant changes in the pattern of the relationship between oceanographic processes and bycatch risk that could help explain the rise in bycatch observed from 2016 onward. This further suggests an influence of additional factors not considered in the present study. This rise of bycatch could be associated with a change in the distribution of common dolphins in the Bay of Biscay or a change in fishing practices. Bycatch risk is likely affected by a mix of different factors, oceanographic processes being one among others.

CONCLUSIONS

This study demonstrated the relevance of stranding data and modelled OVs from coastal operational oceanography to better understand this intense bycatch phenomenon which causes a serious threat to the short-beaked common dolphin population of the northeast Atlantic (Peltier et al. 2016[START_REF] Murphy | Conservation management of common dolphins: lessons learned from the north-east Atlantic[END_REF]. The need to explicitly integrate spatial and temporal dimensions for such complex interactions was justified: our results effectively highlighted the spatiotemporal influences of structuring oceanographic processes on the risk of common dolphin bycatch in the Bay of Biscay. However, the underlying mechanisms remain unclear, and the relationship between environmental characteristics and dolphin bycatch must not be overinterpreted, as our model only indirectly links OVs with dolphin bycatch. Because of the dynamic aspects of both structuring oceanographic processes and dolphin and fishery distributions, further research focusing on smaller time scales is needed. For instance, extreme stranding events are observed on a weekly timeframe [START_REF] Peltier | Can modelling the drift of bycaught dolphin stranded carcasses help identify involved fisheries? An exploratory study[END_REF], suggesting short periods of intense interactions between dolphins and fisheries, so a similar analysis on a weekly timeframe and focusing on these events might be more informative. Furthermore, we showed that the pattern of seasonal bycatch events can be directly inferred from past data on strandings and oceanographic conditions. This highlights the value of establishing functional relationships to eventually identify adapted and acceptable management measures to ensure both the longterm conservation of the population of short-beaked common dolphins of the Bay of Biscay and fisheries activities.

INTRODUCTION

Bycatch, the non-intentional capture or killing of non-target species in commercial or recreational fisheries, is a world wide threat to protected, endangered or threatened species (PETS) of marine megafauna [START_REF] Gray | Bycatches of endangered, threatened and protected species in marine fisheries[END_REF], including seabirds [START_REF] Martin | Threats to seabirds: a global assessment[END_REF], elasmobranchs [START_REF] Pacoureau | Half a century of global decline in oceanic sharks and rays[END_REF] and cetaceans (Avila et al., 2018). Bycatch in fishing gears, such as gillnets, is currently driving some small cetacean species to extinction (Brownell et al., 2019;Jaramillo-Legorreta et al., 2019). The European Commission recently issued infringement procedures against several Members States for failing to correctly transpose some provisions of European environmental law (the Habitats Directive, Council Directive 92/43/EEC), in particular the obligations related to the establishment of a coherent monitoring scheme of cetacean bycatch 1 . The Data Collection Framework (DCF) provides a common framework in the European Union (EU) to collect, manage, and share data within the fisheries sector (Anonymous, 2019a). The Framework indicates that the Commission shall establish a Multi-Annual Union Programme (EU-MAP) for the collection and management of fisheries data which should be inclusive of data that allows the assessment of fisheries' impact on marine ecosystems. With respect to PETS (including cetaceans), the collection of high quality data usually requires a dedicated sampling scheme and methodology, and is generally different from those applied under the DCF [START_REF] Stransky | Scientific, Technical and Economic Committee for Fisheries (STECF) -Revision of the EU-MAP and Work Plan Template (STECF-19-12)[END_REF]: "EU MAP remains not well-suited for the dedicated monitoring of rare and protected bycatch in high-risk fisheries since its main focus is the statistically-sound random sampling of all commercial fisheries (Ulrich and Doerner, 2021, p. 126)." In practice, the introduction of any programme on PETS bycatch under the DCF may be met with caution because of its perceived potential to disrupt data collection for fisheries management [START_REF] Stransky | Scientific, Technical and Economic Committee for Fisheries (STECF) -Revision of the EU-MAP and Work Plan Template (STECF-19-12)[END_REF]. This perception implicitly relegates PETS bycatch as a side issue for fishery management rather than an integral part of it. It may explain the usually poor quality of bycatch data on PETS (ICES, 2020a).

Recent EU legislation (Regulation 2019/1241), referred to as the Technical Measures Regulation (TMR), requires Members States to collect scientific data on cetacean bycatch for the following métiers: pelagic trawls (single and pair), bottomset gillnets and entangling nets; and high-opening trawls (Anonymous, 2019b). Unlike its predecessor (Council Regulation EC No. 812/2004), this Regulation does not require the establishment of dedicated observer schemes for cetacean bycatch data collection [START_REF] Dolman | Implications of new technical measures regulation for cetacean bycatch in European waters[END_REF]. Furthermore, only vessels of an overall length of 15 m or more are to be monitored, but these represent a small fraction of the European fleet (less than 10% in 2019) 2 . This vessel length criterion introduces bias in the bycatch monitoring data as the sample of vessels larger than 15 m is almost certainly dissimilar to smaller 1 https://ec.europa.eu/info/news/july-infringements-package-commissionmoves-against-member-states-not-respecting-eu-energy-rules-2019-jul-26_en 2 https://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=fish_fleet_alt& lang=en vessels. Even within the sample of vessels that are monitored, pragmatic considerations can complicate sampling. For example, in the United States, observer sampling trips are allocated first by region, port, and month, then randomly to vessels of particular categories within those monthly and spatial strata (ICES, 2009). Random allocation of observers to vessels follows sound statistical methodology and increases the likelihood of collecting unbiased data (Babcock and Pikitch, 2003). In France, observer days are allocated by port and by month for each fishery, but the exact vessel allocation is then negotiated and left at the discretion of skippers (ICES, 2009). Allocation is no longer random as skippers may only accept observers when cetacean bycatch risk is low (Benoît and Allard, 2009). Nonrandom allocation means potential bias in the collected data for monitoring bycatch as the sub-sample of skippers accepting an observer may be very different from skippers refusing to do so (Babcock and Pikitch, 2003).

One pragmatic solution bypassing observers is to mandate skippers to self-declare the non-intentional capture or killing of any PETS, as already required under the DCF (Anonymous, 2019a). In France, a national law from 2011 mandate fisheries to declare (without fear of prosecution) the bycatch of any cetacean species, but this law remained largely unknown to French fishermen until late 2019 [START_REF] Cloâtre | Projet LICADO -Limitation des Captures Accidentelles de Dauphins Communs dans le Golfe de Gascogne[END_REF]. In general, self-reported PETS bycatch data are sub-optimal as they may be heavily biased, non-representative (ICES, 2009) and typically provide poor information on which to base management decisions (National Marine Fisheries Service, 2004). Once again, the set of skippers who choose to declare bycatch may differ markedly from those who do not: for example the former take the extra time required to fill logbooks and thus provide accurate data while the latter do not. If this behavior is correlated to other attributes, e.g., a more acute awareness of threats to PETS resulting in practices that tend to minimize impact on PETS, data collected from skippers reporting bycatch would not be representative. There may also be an element of skippers genuinely forgetting to log PETS bycatch in the bustle of the fishing operation but this is random and unlikely to introduce bias. In addition, ground-truthing, for example with remote-electronic monitoring (REM; [START_REF] Course | What's in the Net? Using Camera Technology to Monitor, and Support Mitigation of[END_REF], would be required in order to ensure the quality and accuracy of self-reported data before their statistical analyses.

Another hurdle, of the statistical kind, with cetacean bycatch is the low frequency of these events. Assuming that implementing a representative sampling program were feasible, if bycatch is a rare event [START_REF] Komoroske | Addressing fisheries bycatch in a changing world[END_REF], then few events would be observed for realistic sampling effort (Babcock and Pikitch, 2003;ICES, 2009). This paucity of observed event means a large uncertainty in statistical estimates: with a bycatch rate of the order of 0.01 event per fishing operation, a sample size of 1,100 observed operations would be required to obtain, in the best case scenario (no bias, statistical independence, etc.), the US recommended coefficient of variation of 30% (National Marine Fisheries Service, 2005[START_REF] Parsons | Why IUCN should replace "data deficient" conservation status with a precautionary "assumed threatened" status -a cetacean case study[END_REF]ICES, 2009;[START_REF] Carretta | Recommendations for Pooling Annual Bycatch Estimates when Events are Rare[END_REF]. The amount of observer coverage needed to reach this precision depends on fishery size and trip duration (Babcock and Pikitch, 2003). In practice, the sampling error depends on the overall design of the survey, of which the sample size is only one factor: for example a larger sample size could be needed if there are large "skipper-effects" as the same vessels would contribute fishing operations, and these would not be statistically independent. With a small sample size, uncertainty may be so large as to prevent using estimates altogether, even if one were to assume no bias in the data (Babcock and Pikitch, 2003). Given this challenge and the lack of uptake of dedicated monitoring programmes of cetacean bycatch in Europe over the last decade or more [START_REF] Stransky | Scientific, Technical and Economic Committee for Fisheries (STECF) -Revision of the EU-MAP and Work Plan Template (STECF-19-12)[END_REF], it would appear prudent to seek methods of analysis that can handle the few and nonrepresentative data available to robustly estimate bycatch rates.

The problem of having non-representative samples to carry out statistical analyses is ancient [START_REF] Hansen | The problem of nonresponse in sample surveys[END_REF] and widespread: it pops up in many applied disciplines, including election forecasting [START_REF] Wang | Forecasting elections with non-representative polls[END_REF][START_REF] De Jonge | Predicting state presidential election results using national tracking polls and multilevel regression with poststratification (MRP)[END_REF], political sciences [START_REF] Lax | How should we estimate public opinion in the states?[END_REF][START_REF] Zahorski | Multilevel Regression With Post-Stratification for the National Level VIBER/Street Poll on the 2020 Presidential Election in Belarus[END_REF], social sciences [START_REF] Halsny | Nonresponse bias in inequality measurement: cross-country analysis using luxembourg income study surveys[END_REF], addiction studies [START_REF] Rhem | The elusiveness of representativeness in general population surveys for alcohol[END_REF] or epidemiology [START_REF] Zhang | Multilevel regression and poststratification for smallarea estimation of population health outcomes: a case study of chronic obstructive pulmonary disease prevalence using the behavioural risk factor surveillance system[END_REF][START_REF] Downes | Multilevel regression and poststratification: a modeling approach to estimatingpopulation quantities from highly selected survey samples[END_REF]. In these disciplines, there are also intrinsic limits on improving the representativeness of sampling. For example, in polling, nonresponse rates can be above 90% [START_REF] Forsberg | Polls and the US presidential election: real or fake?[END_REF]. In other cases, some populations of interest may be hard to reach [START_REF] Rhem | The elusiveness of representativeness in general population surveys for alcohol[END_REF], or answers may not be honest (St. John et al., 2014). Challenges lie in the accurate estimation of quantities of scientific interest (e.g., the true magnitude of bycatch in a fishery; Babcock and Pikitch, 2003) with the construction of statistical weights that can calibrate a non-representative survey sample to the population targets. Such weights are implicit with simple random sampling where each unit in a population has the same, non-nil, probability of being included in the sample. When inclusion probabilities differ between units, weights inversely proportional to the former can be used to adjust the sample. However, constructing survey weights is in general more elaborate than using inverse probabilities of selection in the sample [START_REF] Gelman | Struggles with survey weighting and regression modelling[END_REF]. Model-based approaches, and multilevel regression modeling with post-stratification in particular, has become an attractive alternative to weighting to adjust nonrepresentative samples [START_REF] Gelman | Struggles with survey weighting and regression modelling[END_REF].

Multilevel regression modeling allows researchers to summarize how predictions of an outcome of scientific interest vary across statistical units defined by a set of attributes or covariates (Gelman et al., 2021, p. 4): for example bycatch events are a binary outcome at the fishing operation level (a unit) associated with attributes, such as date-time, location, gears and vessels (e.g., [START_REF] Palka | Bycatch Estimates of Coastal Bottlenose Dolphin (Tursiops truncatus) in US Mid-Atlantic Gillnet Fisheries for 1996 to 2000[END_REF]. Post-stratification is a standard technique to generalize inferences from a sample to the population by adjusting for known discrepancies between the former and the latter. Post-stratification is a form of adjustment whereby statistical units are sorted out according to an auxiliary variable (hereafter a stratum) after completion of data collection; stratum-level effects (i.e., effects within each stratum or cell) are then estimated, and finally averaged with weights proportional to stratum size to obtain the population-level estimate. Poststratification differs from blocking as the latter is done before data collection to ensure balance and representativeness at the design stage. Post-stratification is a post hoc statistical adjustment done at the analysis stage: it can remove bias, but at the price of an increased variance of estimates. [START_REF] Lennert | Estimates of incidental marine Mammal Bycatch in California Gillnet Fisheries for July Through December[END_REF] provided an early example of model-based estimates of bycatch with post-stratification.

In small samples post-stratification can degrade estimate precision, especially if the number of strata is large as each stratum will typically include very few data, or even not a single datum (the so-called "small-area" problem). In practice, adequate post-stratification may require handling hundreds of cells (the crossing of several attributes; e.g., week by statistical area by gears). Some predictions for each cell may be too noisy, especially if there are sparse or no data for that particular combination of attributes. Multilevel regression can offer a solution as it borrows strength from similar units to improve and stabilize (i.e., regularize) these predictions [START_REF] Cam | Each site has its own survival probability, but information is borrowed across sites to tell us about survival in each site": random effects models as means of borrowing strength in survival studies of wild vertebrates[END_REF]. In other words, multilevel regression allows an efficient use of a sparse sample to estimate the outcome of interest within each cell, even if these cells are very numerous (e.g., several hundreds). The key insight of combining multilevel regression modeling with poststratification is thus: even if observations are not a representative sample of the population of interest, it may be possible to construct a regression model to first predict unobserved cases, and then post-stratify to average the fitted regression model's predictions over the population of interest (Gelman et al., 2021, p. 313). Good predictions may be obtained with regularization by means of multilevel models with structured priors (including so called "random-effects" models). The latter can increase precision by inducing shrinkage of parameter estimates across similar post-stratification cells, where similarity is encoded in the model specification (e.g., by using random effects that assume exchangeability). The amount of shrinkage, or partial-pooling across cells, is model-based and thus data-driven. However, in order to be able to leverage the information in the data, some model structure on the parameters of interest is necessary hence the need for structured priors. Relying on a model rather than just empirical means of the response variable addresses the bias-variance problem intrinsic to having a large number of cells in post-stratification, and leverages the large toolbox of regression-based models.

Technically, when data arise as signal plus noise, overfitting occurs when a regression model captures too much of the noise compared to the signal; that is in using an ill-conditioned (unstable) model that will provide an excellent in-sample fit but make poor out-of-sample predictions [START_REF] Authier | Variable selection and accurate predictions in habitat modelling: a shrinkage approach[END_REF][START_REF] George | Comment: regularization via Bayesian penalty mixing[END_REF]. Overfitting may result when using richly parametrized models without using adequate estimation methods such as regularization to stabilize parameter estimates and buffer them against noise (Gelman et al., 2021, p. 459-460). Weakly-informative priors in a Bayesian framework regularize the estimation of the large number of parameters that may be present in a multilevel model. Multilevel modeling takes into account complex data structures with structured prior models for batches of parameters; the simplest example are so-called "random effects" whereby a common (Gaussian) distribution centered on zero and with an unknown variance to be estimated for data is assumed for a group of parameters; for example years or sites [START_REF] Cam | Each site has its own survival probability, but information is borrowed across sites to tell us about survival in each site": random effects models as means of borrowing strength in survival studies of wild vertebrates[END_REF]. This common distribution for the parameters is a prior model, and this model for parameters means that the latter are not independently estimated but in tandem according to the postulated prior model. For example, [START_REF] Sims | Modeling spatial patterns in fisheries bycatch: improving bycatch maps to aid fisheries management[END_REF] used a model-based approach to obtain spatially smoothed estimates of bycatch in a gillnet fishery. Spatial-smoothing (also known as "small-area estimation"; [START_REF] Fay | Estimates of income for small place: an application of james-stein procedures to census data[END_REF] was used to stabilize estimated bycatch rates by using a Conditional Autoregressive prior model that leverages information from spatial neighbors to improve the prediction at a specific location. Prior models add some soft constraints to the overall model and these constraints are very useful in data sparse settings to mitigate variance and bias in predictions. In other words, these prior models represent additional assumptions about the data, assumptions, which if approximately correct, add information in the analyses and increase the precision and stability of predictions at the cost of a usually small estimation bias. Introducing bias to reduce variance is a common statistical technique known as shrinkage or regularization [START_REF] George | Comment: regularization via Bayesian penalty mixing[END_REF].

Regularized multilevel regression with post-stratification is thus the combination of several important ideas to obtain accurate predictions (Gao et al., 2019). First, post-stratification is a survey technique to re-align the sample with the population and addresses the problem of non-representative samples. In practice, post-stratification requires to sub-divide the population of interest into many cells corresponding to the combination of important attributes. Multilevel regression can be used to accommodate all these cells in a single model, but the problem has now moved to how to obtain useful estimates for all these cells, which can number in the several hundreds. Regularization solves this estimation problem: it introduces model-driven bias in statistical estimates in order to stabilize them. These new developments in the statistical analysis of non-representative samples may help in obtaining a better quantification of bycatch rates and numbers. Our aim is to assess with simulations, the potential of regularized multilevel regression with poststratification for analyzing already collected bycatch data, with the full knowledge that these data are non-representative and biased in several respects. These biases in sampling are manifold (see above): bias may be due to regulation exempting certain vessels (e.g., no monitoring for vessels smaller than 15 m); to non-dedicated observers or because sampling is driven for other purposes than bycatch monitoring of PETS (commercial discards, stock assessment); or in the case of dedicated schemes, to over-sampling a few "cooperative" skippers or focusing sampling in métiers with the highest or lowest bycatch risk. Our focus will be narrower, honing in on specific sampling scenarios whereby observer coverage is correlated to bycatch risk. In other words, we will assess the potential of regularized multilevel regression with post-stratification to estimate accurately bycatch numbers with samples preferentially collected either during lowor high-bycatch risk periods. Our investigation is largely framed from our knowledge on small cetacean bycatch in European waters, such as short-beaked common dolphin (Delphinus delphis, lower observer coverage when bycatch risk is higher) in the Bay of Biscay (Peltier et al., 2021) or harbor porpoises (Phocoena phocoena, higher observer coverage when bycatch risk is higher) in the Celtic Seas [START_REF] Tregenza | Harbour porpoise (Phocoena phocoena L.) by-catch in set gillnets in the Celtic Sea[END_REF]. In the remainder, we first introduce methods and notations to detail the proposed model to perform multilevel regression with post-stratification with bycatch data, using dolphins as an example. Next, we explain our data simulation scenarios and how we emulate non-representative sampling. We then compare the results (i.e., estimates of bycatch) from the proposed modeling approach with those from the method currently used by the working group on bycatch of protected species from the International Council for the Exploration of the Sea (ICES WGBYC) before concluding on some recommendations for future investigations.

MATERIALS AND METHODS

We carried out Monte Carlo simulations to assess the ability of regularized multilevel regression with post-stratification to estimate bycatch risk and bycatch numbers from representative and non-representative samples. ICES WGBYC collate data through an annual call from dedicated and DCF surveys collecting data on the bycatch of PETS through onboard observers or REM. These surveys may be qualified as "designbased" in the sense that, ideally, a representative coverage of fisheries would be sought in order to scale up the observed sample to the whole population using ratio-estimators. There are many caveats around the use of these ratio-estimators as EU MAP is not well-suited for monitoring PETS bycatch (Ulrich and Doerner, 2021). Given these shortcomings in the collection of bycatch data under EU MAP, the data available to ICES WGBYC are unlikely to be representative of fisheries of interest but nevertheless, ratioestimators are used as part of a Bycatch Risk Approach (BRA) to identify relative risk of bycatch across species and metiers [START_REF] Ices | Report from the Working Group on Bycatch of Protected Species, (WGBYC)[END_REF]. Cetacean bycatch observer programmes may aim at achieving a pre-specified precision for bycatch rates (with a coefficient of variations less than 30%; National Marine Fisheries Service, 2005Service, , 2016;;ICES, 2009;[START_REF] Carretta | Recommendations for Pooling Annual Bycatch Estimates when Events are Rare[END_REF]. Achieving this is very difficult in practice, and a given coverage of effort deployed by the total fleet is, instead, aimed at: for example 10% (5%) for pair-trawlers (level-3 métier PTM) larger (smaller) than 15 m in France. Data from onboard observer programmes are then used to estimate total bycatch using ratio estimators [START_REF] Lennert | Estimates of incidental marine Mammal Bycatch in California Gillnet Fisheries for July Through December[END_REF][START_REF] Julian | Estimates of marine mammal, turtle, and seabird mortality for two California Gillnet Fisheries: 1990-1995[END_REF]Amandè et al., 2012) and the bootstrap or a classical approach (Clopper-Pearson) for uncertainty quantification (ICES, 2018, p. 57). We used an approach similar to that of WGBYC (hereafter referred to as a "design-based" approach) as a benchmark to compare against results from regularized multilevel regression with poststratification. We honed in on the accurate estimation of the number of bycatch events for a complete fleet. We assume that information on the total effort deployed by a fleet operating in a spatial domain are available and measured without error. This assumption is necessary to scale estimates from the sample to the population. We also assumed that there are no falsenegatives in the sample, that is no bycatch event went unrecorded by onboard observers (assuming thereby a dedicated observer programme). These two assumptions are customary with ratio estimators, whether design-or model-based, and do not deviate from current norms. We assume however that these population data on total effort can be disaggregated at a finer temporal scale in order to post-stratify on calendar weeks. This assumption of accurate measurement of effort at the week-level is crucial for post-stratification.

Notations

The logit transform maps a quantity p ∈]0, 1[ to the real line: logit(p) = log p 1-p . Its inverse is denoted by logit -1 (x) = 1 1+e -x (sometimes called the "expit" transform). Let y ijkl denote the ith fishing operation of vessel j in week k of year l, with y ijkl = 1 if a bycatch event occurs and 0 otherwise:

y ijkl ∼ Bernoulli p jkl = logit -1 µ + β kl + α j (1)
where p jkl is the product of the probability of a bycatch event occurring and the probability of dolphin presence. This unconditional probability p jkl , or "bycatch risk" hereafter, is not indexed by i: although there may be several fishing operations of vessel j in week k of year l, the risk is assumed constant over these. Bycatch risk is a function of several parameters (on a logit scale): µ is the intercept (overall risk), α j ∼ N(0, σ vessel ) are (unstructured, normal random effects) vessel-effects accounting for heterogeneity (e.g., "fishing style" of skippers); and β kl are time effects, modeled with a Gaussian Process. A Gaussian process is written as GP(m, c) where m and c are the mean and covariance functions respectively (Gelman et al., 2014, p. 501).

The Gaussian Process prior on the vector of week effects in year l, β l , defines this vector as a random function for which the values at any week 1, . . . , k, . . . , w are drawn form a w-multivariate normal distribution:

β 1l , . . . , β wl ∼ N m(1), . . . , m(w) , (2)
with mean m and covariance . The function c specifies the covariance between any 2 weeks k and k ′ , with an w × w covariance matrix with element k, k ′ = c(k, k ′ ). A Matérn covariance function of order 3 2 and range parameter fixed to 3 2 was assumed:

c k, k ′ = σ 2 year × 1 + 2 √ 3×d(k-k ′ ) 3 × exp -2 √ 3×d(k-k ′ ) 2
, where d(kk ′ ) is the temporal distance (in weeks) between weeks k and k ′ . The distance function was the absolute difference between calendar weeks within the same year:

d(k-k ′ ) = |k-k ′ |.
The choice of the Matérn covariance function translate an assumption of smoothness in the temporal profile of bycatch risk: bycatch risk is assumed to change gradually across weeks, with no abrupt increase or decrease. The range parameter is fixed and not estimated from data. This choice represents an additional assumption whereby the temporal correlation is 0.05 after 4 weeks corresponding to temporal independence after a month. This choice is to some extent arbitrary and represents an additional assumption. In theory, the range parameter could also be estimated from data but we assumed a data sparse setting with limited information (more so with Bernoulli data) to estimate this parameter.

The mean function m of the Gaussian process was modeled (on a logit scale) with a first order random walk, which was evaluated at specific values k ∈ [1,...,w] corresponding to week number within a year:

     m(1), . . . , m(w) = (ε 1 , . . . , ε w ) ε k = 0 k = 1 ε k+1 ∼ N (ε k , σ week ) k > 1 (3)
The order of the random walk prior was assumed fixed at 1 and not estimated from data. This prior choice smooths the first order differences between adjacent elements of ε and represents an additional assumption, mainly to limit the number of parameters to estimate from the typically sparse data on bycatch. A random walk was chosen as an effective way to reveal the shape of the average risk profile without specifying a family of parametric curves. The model in Equation ( 1) is a decomposition of bycatch risk into a time-varying component (at the week-scale, Equation 3; and with an interaction with year, Equation 2) and timeinvariant component which can be interpreted as fishing-style effects whereby some skippers may have consistent practices that increase or decrease bycatch risk. Importantly, bycatch risk is modeled here with no attempt to model dolphin presence directly as relevant data to do so may be missing in the general case. Bycatch risk is thus to be estimated for each week of a year, and each of these weeks represent de facto a stratum. In any applied case, additional factors, such as statistical area, may need to be included in Equation ( 1) for improved realism. For simplicity, we did not consider space in simulations, and solely focused on time.

Data Simulation

To test the ability of model 1 to estimate bycatch risk, data were simulated (Figure 1).

1. Bycatch probability conditional on dolphin presence was constant and set to 0.3, that is roughly one fishing operation out of 3 generates a bycatch event when dolphins are present (corresponding to a high risk fishery, e.g., the trawl fishery in the Bay of Biscay). 2. Dolphin presence is seasonal (loosely inspired from the observed pattern of common dolphin in the Bay of Biscay where abundance is higher closer to the coasts in winter; Laran et al., 2017): it peaks at the beginning and end of the year, but quickly drops to 0 for roughly 2 thirds of a year. 3. A fishery of 20 vessels is operating all year round, with an overall activity rate of 80% each week (that is, for any week, 20 × 80 100 = 16 vessels are fishing). Each fishing day (5 days per week), on average 2.3 fishing operations are carried out. The expected total number of fishing operations for a year is 5 × 52 × 2.3 × 16 ≈ 10,000. These values were loosely taken from an exploratory analysis of onboard observer data collected on PTM flying the French flag. During each of these operations, a bycatch event may occur depending on dolphin presence at the time and on a skipper-specific risk factor (drawn randomly from a normal distribution with scale parameter set to log(2) FIGURE 1 | Inputs for data simulation. Top: bycatch probability if dolphins are present during a fishing operation. Middle: dolphin presence during a year. Bottom: Probability for a skipper to accept an observer onboard. Left: sampling is unbiased; Middle column: sampling is biased downwards (under-sampling). Right: sampling is biased upwards (over-sampling). Each line corresponds to one of the 100 data simulations that were carried out. The y-axis is on a square-root scale to better visualize small values.

4. Observers are accepted onboard vessels either with a constant probability of 0.05 corresponding to a coverage of 5% of all fishing operations (unbiased sampling scenario) or with a probability that covaries with dolphin presence (biased sampling scenarios). In the latter case, realized coverage is a random variable. With under-sampling, the bulk of the observer data is collected when bycatch risk (the product of dolphin presence and bycatch probability) is nil (Figure 1). With over-sampling, the bulk of the observer data is collected when bycatch risk is high but no data are collected when the risk is nil (Figure 1). 5. In a year, the number of fishing operations is ≈ 10,000, and the number of bycatch events ≈ 300, which yields a rate of ≈ 3%. This rate is not large, but is not extremely rare either.

Bycatch events were simulated for each fishing operations during a day when an observer was present from a Bernoulli distribution according to the product of bycatch probability given dolphin presence and dolphin presence probability for that day. If no observer was present, no data were recorded. The datagenerating mechanism used a parametric function for dolphin presence probability and was different from the statistical model used to analyzed the data (see https://gitlab.univ-lr.fr/mauthier/ regularized_bycatch). For each sampling scenario, 100 datasets were generated for 1, 5, 10, or 15 years. All data simulations were carried out in R v.4.0.1 (R Core Team, 2020). When simulating only 1 year of data, Equation ( 2) is not necessary as there is no between-year variation to estimate: the model can be simplified with the omission of β l . Our Monte Carlo study had a comprehensive factorial design crossing (a) sampling regime (either unbiased or not) and (b) sample size as controlled with the number of years for which the observer programme was assumed to have been in operation.

Estimation

Estimation of the parameters of model 1 from simulated data was carried out in a Bayesian framework using programming language Stan (Carpenter et al., 2017) called from R v.4.0.1 (R Core Team, 2020) with library Rstan (Stan Development Team, 2020). Stan uses Hamiltonian dynamics in Markov chain Monte Carlo (MCMC) to sample values from the joint posterior distribution (Carpenter et al., 2017). Weakly-informative priors were used for regularization:

         µ ∼ N(0, 3 2 )
prop ∼ D(1, 1, 1)

σ total ∼ GG( 1 2 , 1 2 , log 2 10 )
where D() denotes the Dirichlet distribution for modeling proportions (such that 3 i=1 prop i = 1) and GG() the Gamma-Gamma distribution for scale parameters (Griffin and Brown, 2017;Pérez et al., 2017). With this simplex parametrization, chosen to improve mixing and ease estimation with Monte Carlo methods [START_REF] He | Re-considering the variance parametrization in multiple precision models[END_REF], the several variance components of the model were:

     σ 2 vessel = σ 2 total × prop 1 σ 2 week = σ 2 total × prop 2 σ 2 year = σ 2 total ×
prop 3 These priors are weakly-informative (Gabry et al., 2019): the prior for the intercept covers the whole interval between 0 and 1 on the probability scale but is informative on the logit scale. The prior for the scale (square-root of the variance) is heavy tailed and has a median set to log 10 2 (Griffin and Brown, 2017;Pérez et al., 2017), which translate an assumption about the plausible range of variations in bycatch risk spanning a priori two full order of magnitude from one tenth to a ten-fold increase compared to the mean bycatch rate. Thirty random realizations from our choice of priors are depicted on Figure 2: the whole interval between 0 and 1 is covered, and between-week variations can be large or small.

For each simulated dataset, four chains were initialized from diffuse random starting points (Carpenter et al., 2017, p. 20) and run for a total of 1,000 iterations, discarding the first 500 as warm-up. Default settings for the No-U-Turn Sampler (NUTS) were changed to 0.99 for adapt delta and 15 for max treedepth (Hoffman and Gelman, 2014). NUTS uses Hamiltonian Dynamics in MCMC and typically requires shorter runs than other MCMC algorithms both to reach convergence and to obtain an equivalent Effective Sample Size from the posterior (Hoffman and Gelman, 2014;[START_REF] Monnahan | Faster estimation of bayesian models in ecology using Hamiltonian Monte Carlo[END_REF]. Parameter convergence was assessed using the R statistics [START_REF] Vehtari | Rank-Normalization, Folding, and Localization: an Improved R for Assessing Convergence of McMC[END_REF] and assumed if R < 1.025. Upon diagnosing convergence of all parameters, a combined sample of 4×500 = 2, 000 MCMC values were obtained to approximate the joint posterior distribution.

Let μ[m] , β[m] kl , σ [m]
vessel denote the mth MCMC sample for parameters µ, β kl and σ vessel . Bycatch risk pj * kl for a randomly chosen vessel j * operating in week k of year l was computed from the mth MCMC draw from the joint posterior distribution as:

p[m] j * kl = logit -1 μ[m] + β[m] kl + α[m] j * (4)
where α[m]

j * ∼ N(0, σ [m] vessel )
. This predicted bycatch risk incorporates between-vessel variability, that is it takes into account the fishing style of skippers. The predicted risk (on a logit scale) for a random chosen skipper is α[m] j * and was drawn from the posterior predictive distribution: not all skippers may be observed in the sample, and but the subset of skippers that accept an observer can be used to estimate a between-skipper variance in bycatch risk. In practice, the number of fishing operations carried out in the course of a week in a year by individual skippers is unknown, although the aggregated number of fishing operations may be known. If totals by skippers were available, and all skippers had been sampled, it would be more efficient to use skipper-specific estimated risk, but we did not assume that this would necessarily be the case.

The total number of bycatch events, T bycatch was estimated as the average over the 2,000 MCMC draws from the posterior:

Tmodel-based bycatch = 1 2000 2000 m=1 nyear l=1 n week k=1 p[m] j * kl × N kl (5)
where N kl is the total number of fishing operations that took place is week k of year l. The total number of strata for post-stratification was n year × n week , with a maximum of 15 × 52 = 780 cells. Highest Posterior Density credible intervals at the 80% level were computed with function HPDinterval from package coda [START_REF] Plummer | CODA: convergence diagnosis and output analysis for MCMC[END_REF] for uncertainty evaluation. Equation ( 5) is an instance of a ratioestimator with post-stratification, except that it uses modelbased estimates of bycatch risk. This model-based approach regularizes estimates with partial pooling [START_REF] Gelman | Philosophy and the practice of Bayesian statistics[END_REF]: the variance of estimates is greatly reduced by introducing some bias with structured priors (Gao et al., 2019). Our results were benchmarked against an approach similar to that of ICES WGBYC whereby total number of bycatch events was estimated1 as:

Tdesign-based bycatch = nyear l=1 pl × n week k=1 N kl (6)
where pl is the average bycatch risk estimated as the mean from the observed sample in year l. Confidence intervals at the 95% level were computed using either the bootstrap or the Clopper-Pearson approach as customary in ICES WGBYC. Both were considered as the Clopper-Pearson approach is known for being more conservative: it produces confidence intervals that above the nominal level (i.e., wider than necessary) but generates non-nil confidence intervals even if no bycatch has been observed [START_REF] Northridge | Annual Report on the Implementation of Council Regulation (EC) No 812/2004 During 2018[END_REF]. In practice, ICES WGBYC often pooled several years to stabilize the estimate of p (e.g., ICES 2018, p. 57-58;[START_REF] Carretta | Recommendations for Pooling Annual Bycatch Estimates when Events are Rare[END_REF]: Equation ( 6) translate an ideal case that is rarely met in practice.

FIGURE 2 | Prior predictive checks sensu (Gabry et al., 2019). Bycatch risk (pijkl in Equation 1) is depicted: 30 random realizations from the priors are depicted.

ICES WGBYC usually works on bycatch rates (in number of PETS per unit effort), not bycatch risk. We focused on risk for simplicity, but scaling bycatch risk to a rate is straightforward by multiplying with the average number of PETS bycaught in a bycatch event. Dolphin presence was seasonal in the datagenerating mechanism for simulations: pitching a method that can explicitly accommodate such seasonality against one that does not may be viewed as knocking down a strawman. However, current estimates of PETS bycatch in Europe are stratified by flag, ICES statistical areas, and métiers but not by season (e.g., Table 2 p. 17 in ICES 2019; Northridge et al., 2019, p. 27).

The comparison remains relevant and topical as it matches current practices.

RESULTS

Convergence across all simulations and scenarios was assumed to be reached, with all R < 1.025, for all parameters. For each simulation, chains were combined in a single sample of 2,000 values to approximate the joint posterior distribution of the model defined by Equations ( 1), (2), and (3).

Design-vs. Model-Based Approach

Comparing the design-and model-based approach was done with simulating 1 year of data. When data sampling was unbiased, both the design-and model-based approach were able to recover the true number of bycatch events (Figure 3; Table 1).

Estimates of bycatch events were statistically unbiased but their precision low with a (frequentist 95%) confidence or (Bayesian 80%) credible interval (CI) as large as 100% of the point estimate (Table 1), as could be expected with only 15 bycatch events were recorded on average by onboard observers (Table 1). With undersampling, design-based estimates were negatively biased (that is, they were under-estimates) whereas model-based estimates were still unbiased on average (Figure 3; Table 1). With oversampling, design-based estimates were positively biased (that is, they were over-estimates) but so were model-based estimates, although bias was 5 times smaller (Figure 3; Table 1). In all cases, coverage was 100% but largely as a result of low precision:

precision was very low with CI spanning some 200% of the point estimate for the unbiased and under-sampling scenarios. This low precision was the result of having to work with as few as 5 observed bycatch events on average (Table 1). Precision improved with over-sampling, but was still as high as 50% of the point (over-)estimate. The model-based approach was wellcalibrated in both the unbiased and under-sampling scenarios (Figure 4): model-based estimates were on average equal to the truth whereas this was only the case with design-based estimates when sampling was unbiased. In addition, the model-based approach was able to recover the temporal profile of bycatch risk (Figure 5) in these two scenarios, but with an increased accuracy and precision if sampling was unbiased. In the over-sampling scenario, both the design-and model-based approaches were not well -calibrated (Figure 4) and the model-based approach overestimated bycatch risk when no data were collected (Figures 1,5).

Model-Based Approach With Several Years of Data

With several years of data, the model-based approach was able to yield nearly unbiased estimates: the bias was smaller than 3 bycatch events when sampling was unbiased, but as large as 10 (on average) with biased sampling and 3 years of data. The precision of estimates improved with several years of data, as expected with larger sample size. Precision of model-based estimates with over-sampling were already acceptable with 3 years of data: an 80% credible interval width of 50% corresponds to a coefficient of variation of 50 2.5 ≈ 20% assuming a normal distribution for the posterior. The model-based approach allowed to obtain estimates at the weekly scale (Figure 6): these estimates were approximately unbiased in the unbiased and over-sampling scenarios, but were biased for the under-sampling scenario. In that latter case, the bias was correlated with the temporal pattern used to simulate dolphin presence (Figure 1): it was the largest when dolphin presence was at its highest but positive at the beginning of a year and negative at the end of the same year. Both biases were greatly attenuated with increased sample size.

DISCUSSION

Using Monte-Carlo simulations, we investigated the statistical properties of a model-based approach, regularized multilevel regression with post-stratification, to estimate the total number of bycatch events in a fishery operating year-round. Simulations were broadly informed from the case of common dolphins and pair-trawlers in the Bay of Biscay and from harbor porpoises and set-gillnets in Celtic Seas. A salient feature of simulations was biased sampling with observers being preferentially accepted onboard when bycatch risk was either high or low. Data simulations in that latter case, which is the most realistic one in the Bay of Biscay (Peltier et al., 2016), resulted in as few as 5 observed bycatch events per year on average (Tables 1,2). This aligns with the ubiquitous description of small cetacean bycatch being a rarely observed event. It was nevertheless possible to fit a regularized multilevel regression model on these data. Importantly, estimates from this model-based approach were statistically less biased than the design-based estimates when sampling was biased. Model-based estimates were, however, imprecise but this is largely to be expected (Amandè et al., 2012), especially with as few as 5 observed bycatch events per year. The design-based approach was also imprecise, even in the unbiased data sampling scenario of 5% coverage of the fleet, which is not reached in practice (Anonymous, 2016;ICES, 2020b). The design-based approach was very sensitive to how data were collected: this approach severely under-or overestimated bycatch when sampling was biased, whereas the modelbased approach was still well-calibrated with under-sampling, but not with over-sampling (Figure 4). Biases in onboard observer data are pervasive and widely acknowledged (Babcock and Pikitch, 2003;Benoît and Allard, 2009;Peltier et al., 2016). Enforcing coverage as required to achieve a pre-specified precision in estimates can be challenging in practice. For example, in 2016, France only achieved a coverage rate less than 2% for most métiers and concluded on the impossibility of scaling-up observed bycatch rates to the whole fleet (Anonymous, 2016, p. 24). There were, however, 9 bycatch events of common dolphins in pair-trawlers targeting European hake (Merluccius merluccius). From these numbers, bycatch was described a "rare" event (Anonymous 2016, p. 23). Such a conclusion would be warranted if sampling were representative, in which case the design-based estimate could be used, even though its precision would still be very low. On the other hand, with under-sampling, this conclusion is misleading as our simulations further illustrated: although only 5 bycatch events were observed on average (Figure 4), the true number of bycatch events was on average 60 times larger (Figure 4). In our simulations, the true bycatch rate was on average ≈ 3% over a year, which is not rare, but not frequent either. Moreover, interviews with French skippers deploying trawls or gillnets in the Bay of Biscay revealed that more than 80% of respondents declared to having experienced at least one small cetacean bycatch event in a year [START_REF] Cloâtre | Projet LICADO -Limitation des Captures Accidentelles de Dauphins Communs dans le Golfe de Gascogne[END_REF]. Such a large proportion contradicts the idea of common dolphin bycatch being a rare event in the Bay of Biscay, but rather suggest severe biases in onboard observer data that result in the rare reporting of bycatch events, rather than a rarity of events per se. The common dolphin in the Bay of Biscay illustrates how under-sampling may distort the perception of bycatch as a very rare event when it can, in fact, be widespread. This is a catch-22 situation whereby cetacean bycatch is described as a rare event because it is rarely reported, and this perceived rarity may serve to argue against ambitious dedicated monitoring programmes out of cost-effective considerations, thereby preventing to dispel the initial misconception.

Finding an optimal sampling plan for fisheries with rare bycatch events is a long standing problem (ICES, 2009). Several strategies have been attempted: for example in the United States, one strategy is "pulsed sampling" whereby a particular fishery or métier is very heavily sampled for a short period of time in order to maximize the chance for observers to record any bycatch that might occur (ICES, 2009). This pulsed sampling strategy corresponds to our over-sampling scenario wherein monitoring effort is positively correlated with bycatch FIGURE 4 | Regression lines of point estimates against the true number of bycatch events, showing the calibration of the design-and model-based approach. The x-axis shows the true number of bycatch events across 100 simulations, spanning between 150 and 400 events. The red dotted line shows the identity line, i.e., no bias. Left: data sampling was unbiased and all methods yielded statistically unbiased estimates. Middle: Under-sampling scenario: only the model-based approach was well-calibrated. Right: Over-sampling scenario: both the design-and model-based approaches were not calibrated to the truth. risk. Under this scenario, the absence of any sampling at all when bycatch risk was low was detrimental to the accurate estimation of bycatch events with our model. Model-based estimates were, however, less biased than design-based estimates. Arguably, this comparison is somewhat artificial as a correct comparison would use all the available information and uses estimators that are season-specific to account for under-sampling when bycatch risk is low if such a period is known to the investigator. Notwithstanding this shortcoming, model-based estimates represented an improvement and allowed to infer the bycatch risk profile accurately, especially with several years of data.

We showed with our Monte-Carlo simulations that regularized multilevel regression with post-stratification can nevertheless be used to analyze bycatch data despite concerns about non-representative sampling. Model-based approaches [START_REF] Palka | Bycatch Estimates of Coastal Bottlenose Dolphin (Tursiops truncatus) in US Mid-Atlantic Gillnet Fisheries for 1996 to 2000[END_REF], with post-stratification [START_REF] Lennert | Estimates of incidental marine Mammal Bycatch in California Gillnet Fisheries for July Through December[END_REF], or machine learning [START_REF] Carretta | Regression Tree and Ratio Estimates of Marine Mammal, Sea Turtle, and Seabird Bycatch in the California Drift Fishery: 1990-2015[END_REF], or multilevel regression [START_REF] Sims | Modeling spatial patterns in fisheries bycatch: improving bycatch maps to aid fisheries management[END_REF][START_REF] Martin | Bayesian inference and assessment for rare-event bycatch in marine fisheries: a drift gillnett fishery case study[END_REF] have previously been used to estimate bycatch rates. Traditional, design-based, ratio estimates are biased if sampling is biased; imprecise if observer coverage is low (as is the usual case in the North East Atlantic; see for example Figure 14, p. 114 in ICES, 2020b); and volatile if bycatch events are only observed occasionally [START_REF] Carretta | Regression Tree and Ratio Estimates of Marine Mammal, Sea Turtle, and Seabird Bycatch in the California Drift Fishery: 1990-2015[END_REF]. The traditional remedy to stabilize estimates and improve precision is to bypass yearspecific estimation and pool several years together [START_REF] Carretta | Recommendations for Pooling Annual Bycatch Estimates when Events are Rare[END_REF][START_REF] Ices | Report from the Working Group on Bycatch of Protected Species, (WGBYC)[END_REF]. This pragmatic solution improves precision but does not address the problem of biased sampling. It also introduces estimation bias for any year-specific estimates by pooling completely several years in order to stabilize the variance of estimates (ICES, 2009, p. 36): any between-year differences are thus ignored in order to obtain a better precision of estimates. It is a reasonable approach in practice, but one that can be improved. Model-based approaches offer a trade-off between no-pooling (keeping all years separate) and completepooling with a third option: partial pooling or regularization [START_REF] Gelman | Philosophy and the practice of Bayesian statistics[END_REF]. Regularization is a general term for statistical procedures that give more stable estimates. Our model-based approach achieves regularization by leveraging, via a structured prior model (Equations 2 and 3, see section 2), the within-year information at the weekly scale. The result were more stable and accurate annual bycatch estimates at the cost of some modeling assumptions and weakly-informative priors. Importantly, weekly estimates could also be obtained with our model-based approach.

Our model-based approach is semi-parametric as it uses a random walk prior to learn from the data the weekly pattern in bycatch risk. This prior is also ensuring some smoothness in the temporal risk profile as it translates an assumption on the correlation between 2 consecutive weeks. This random walk model remains simple as the order is fixed to 1. We further expanded this model to allow for between-years variation in the weekly risk profile with a Gaussian Process prior [START_REF] Neal | Regression and classification using Gaussian process priors[END_REF][START_REF] Goldin | Fast and flexible Bayesian species distribution modelling using gaussian processes[END_REF]. Importantly, these two prior choices (a random walk and a Gaussian Process prior) add structure to the model and help in leveraging the information present in the sparse data typical of onboard observer programmes. Even when with over-sampling, these choices were not detrimental as model-based estimates were statistically unbiased and precise with 3 years of data (Table 2). The explicit consideration of time effects is key to mitigate bias in sampling. In our simulations, dolphin presence was caricaturally seasonal, and observers could be preferentially allowed on fishing vessels when dolphins were less or more likely to be present (Figure 1). Our model was still able to provide statistically unbiased estimates of bycatch in those scenarios, although these estimates were very imprecise with under-sampling. However, they were not more imprecise than the traditional (but biased) design-based estimates (Table 1) if 80% credible interval were used. In addition to being unbiased, these estimates could also reveal with accuracy the temporal risk profile (Figure 5). It is important to keep in mind here that our model is different from the data-generating model used in simulating data: our results were not simply an instance of using a true model, which is impossible in practice as a model is by definition a simplification used to capture the salient features of a phenomenon. Our model had some shortcomings: for example, bias increased with 3 years of data compared to 1 year for the under-sampling scenario (contrast Tables 1,2). This increased bias (toward the prior model) was the result of partial pooling but came with a gain in precision as evidenced in the width of credible intervals. The bias progressively wore off with more years of data, illustrating thereby the attractiveness of partial pooling and structured priors to regularize estimates [START_REF] Gelman | Philosophy and the practice of Bayesian statistics[END_REF]Gao et al., 2019). The gain in reducing bias in estimates and increasing their precision was most evident with over-sampling (Tables 1,2).

Our model could also provide weekly bycatch estimates which were largely unbiased except in the under-sampling scenario where a positive and negative bias remained at the beginning and end of a year respectively, even with 15 years of data (Figure 6). With under-sampling, few observed bycatch events can be collected by design because observers are very unlikely to be accepted on board by skippers. Weekly estimates were too high at the beginning of a year but too low at the end, but this somewhat canceled out at the year-level. There was still a slight overestimation bias resulting from our choice of a non-symmetric pattern for dolphin presence and a symmetric pattern for biased coverage: observing bycatch events at the end of a year was comparatively more difficult than at the beginning of a year because overlap between a non-nil coverage and dolphin presence was smaller at the end of year (Figure 1). These shortcomings illustrate that a model-based approach should be tailored to the context of the study, and we designed our simulations largely from our knowledge on the common dolphin in the Bay of Biscay. However, the framework of regularized multilevel regression with post-stratification is very flexible and we believe our proposed model has large potential for generality as it simply translates a decomposition of bycatch risk into a smooth time-varying and (unstructured) time-invariant effects. The model can easily be made more complex, data permitting, to accommodate spatial effects with, for example, a Besag-type prior [START_REF] Sims | Modeling spatial patterns in fisheries bycatch: improving bycatch maps to aid fisheries management[END_REF]Morris et al., 2019).

Several important assumptions are structurally built into our model: in particular, a first order random was assumed for the mean function of the Gaussian Process prior, with no attempt to estimate from data the correlation parameter (e.g., using an AR(1) prior instead). The choice of a first order random walk was not aiming at uncovering the true datamechanism: our aim were to reveal a temporal pattern in bycatch risk from sparse data using a flexible, yet parsimonious approach. This was particularly true in the under-sampling scenario where few bycatch events could be observed in any given year of simulated data. In the other scenarios, other choices than the first order random walk could be considered as more data are collected. We also assumed that the range parameter of the covariance function in the Gaussian Process prior for week effects was known and such that bycatch risk was temporally uncorrelated after 4 weeks. Fixing the range parameter is usually not recommended but was motivated by consideration of the data-to-parameter ratio, and computation convenience. Bycatch data are binary and can be sparse: these two features underscore how little information may be available. In this context, limiting the number of parameters to estimate can be justified on pragmatic consideration. The model we are proposing is parameter-rich, but some structure are assumed on these parameters in the form of the prior used. These priors represent choices from the analyst and may be reconsidered and tested, data permitting. There was some evidence that bycatch risk was under-smoothed in the over-sampling scenario which resulted in an over-estimation of bycatch risk (Figure 5, rightmost panel). Model expansion is seamless with Stan (Gabry et al., 2019), and the above mentioned parameters could be estimated, rather than fixed, with adequate data. Despite somewhat arbitrary prior and modeling choices, our model provided more accurate estimates of bycatch numbers and bycatch risk in under-and oversampling scenarios. This satisfactory predictive ability points to another important limitation.

Our model is phenomenological, i.e., it is agnostic of the causes behind the temporal variations in bycatch risk. Bycatch risk is the product of dolphin presence and bycatch probability given presence (the latter was constant in our simulations). The model only estimates this product of two probabilities and thus cannot disentangle them without other sources of data. This limitation seems inconsequential in our simulations for the aim of accurate estimation of the total number of bycatch events as interest lies in the effects of causes (how much bycatch?) rather than in the causes of effects (why bycatch occurred?). A straightforward model expansion (as pointed out by a reviewer) would be the consideration of p vessel-level covariates (z 1j , . . . , z pj ) in Equation ( 1):

α j ∼ N p b=1 ξ b × z jb , σ vessel (7)
Candidate covariates such as vessel length or gear-attributes (e.g., mesh size) could be incorporated in the analysis to improve the exchangeability assumption on vessel-effects. An obvious covariate to consider for detecting self-selection of skippers into observer programme participation is to include whether a skipper has ever accepted an observer, or the number of times it did so in the past: a negative regression coefficient could be interpreted as voluntary skippers having an intrinsically lower risk of bycatch.

Including skipper-level covariates could reduce the betweenskipper variance σ 2 vessel , and improve ultimately precision of bycatch estimates. Consideration of other distributions than the normal (e.g., a skew-normal, or a Student-t distribution with a fixed degree of freedom) would be straightforward with Stan but is probably worthwhile only with large enough amount of data for all practical purposes [START_REF] Mcculloch | Misspecifying the shape of a random effects distribution: why getting it wrong may not matter[END_REF].

An important assumption underlying accurate estimation is that the information on the total effort must also be accurate and available at the scale of weeks for post-stratification. This assumption is crucial to scale-up estimates from the (potentially biased) sample to the population, but it does not necessarily hold with fisheries effort as the latter is more often estimated rather than measured directly [START_REF] Julian | Estimates of marine mammal, turtle, and seabird mortality for two California Gillnet Fisheries: 1990-1995[END_REF][START_REF] Ices | Report from the Working Group on Bycatch of Protected Species, (WGBYC)[END_REF]ICES, , 2020b). Here we assumed that the total number of fishing operations (e.g., number of tows for trawls; Tremblay-Boyer and Berkenbusch, 2020) are available as auxiliary information for post-stratification. This assumption about the availability of disaggregated data stems from the explicit consideration of time as an important predictor of variations in bycatch risk. This assumption is necessary for using post-stratification to align the sample with the population targets but may be difficult to meet in practice. Currently, ICES WGBYC uses in its BRA a coarse, but admittedly comparable proxy across fisheries and countries to quantify fishing effort, namely days at sea (ICES, 2019). A day at sea is any continuous period of 24 h (or part thereof) during which a vessel is present within an area and absent from port (Anonymous, 2019a). Importantly, this definition is not at the level of a fishing operation, and effort thus quantified is already aggregated at a level above that at which bycatch data are collected. This coarsening of fisheries effort data is fundamentally a measurement problem, and one that modeling should not be expected to remedy easily. BRA uses an estimate of total fishing effort for the fisheries of concern in a specific region, together with some estimate of likely or possible bycatch rates that might apply for the species of concern, in order to evaluate whether or not the total bycatch in that area might be a conservation issue. A regularized multilevel regression model could be used to obtain estimates of bycatch rates to be used in BRA. Post-stratification could also be attempted using the coarse days at sea proxy for effort, and thus our framework could be adapted to match the requirements of ICES WGBYC.

Assuming that our framework were to be adopted to produce bycatch estimates, how would both fisheries and Non-Governmental Organizations (NGOs) react given the salience of bycatch as a policy issue in Europe? Such a prospective question inevitably entails some speculations (as with all "whatif " questions), but may nevertheless bring some insights as highlighted by a reviewer. Within Europe, the conservation reference currently available for assessing bycatch is that established under the Agreement on the Conservation of Small Cetaceans of the Baltic, North East Atlantic, Irish and North Seas. The agreement has the conservation objective to minimize anthropogenic removals of harbor porpoises (and other smallsized cetaceans), and to restore and/or maintain population depletion to/at 80% or more of the carrying capacity in each assessment unit [START_REF] Ascobans | Annex O -report of the IWC-ASCOBANS Working Group on Harbour Porpoises[END_REF]ICES, 2020c). Methods for setting conservation reference points were agreed in March 2021 at the meeting of the Biodiversity Committee of the Olso-Paris Regional Sea Convention. This committee adopted the use of the Removals Limit Algorithm for harbor porpoises in the North Sea assessment unit and a modified Potential Biological Removal (Wade, 1998) for common dolphins in the North-East Atlantic (Genu et al.) 3 . Accurate bycatch estimates will be needed for assessment against these reference points. However, fisheries may challenge the accuracy of estimates precisely because they will result from a new statistical model. While a healthy skepticism is warranted, and model improvements are certainly possible, it must be kept in mind that our model only addresses the issue of having a correlation between observer coverage and bycatch risk, and does so with some assumptions. There would remain many biases to be addressed in bycatch data (Babcock and Pikitch, 2003), and many of them would be best addressed with a proper random allocation of professional observers to vessels (that is better design and better measurement). A purely model-based solution can be brittle [START_REF] Sarewitz | How science makes environmental controversies worse[END_REF], and may lead to displacement of the problem of bycatch assessment to a never-ending problem of model improvement that would delay any corrective measures or decision (Rayner, 2012). Model-based estimates offer a pragmatic approach to the analysis of already collected data, but should not deflect from improving survey design where possible. Assuming that model-based estimates would be endorsed by a fishery industry, NGOs could challenge in court any reference point that is not zero for PETS, since by definition, it ought to be zero. The Habitats Directive requires strict protection and prohibits "all forms of deliberate capture or killing" (emphasis added) of all species listed on its Annex IV which includes all cetacean species. The Court of Justice of the European Union has consistently ruled that the adjective "deliberate" is to be understood in the sense of "conscious acceptance of consequences" [START_REF] Trouwborst | Domestic cats (Felis catus) and European nature conservation law-applying the EU birds and habitats directives to a significant but neglected threat to wildlife[END_REF]: in other words, using knowingly a gear that may potentially catch a protected species contravenes the Habitat Directives. What will eventually play out remains to be seen, but strongly hinges on how polarized the bycatch issue is. As scientists, our duty remains to provide the best available evidence on bycatch and to outline all management actions and their consequences in light of this evidence [START_REF] Pielke | The Honest Broker -Making Sense of Science in Policy and Politics, 1st Edn[END_REF]. Our model is unlikely to change bycatch management in France in the near term: both fisheries and NGOs are at loggerheads, vying for public and official support. They are building constituencies and advertising unyielding positions in diverse medias: we content that a legal confrontation at a national or supra-national level is extremely likely and probably being prepared. We nevertheless think our model, by making use of data already collected within the DCF framework and by encouraging further, ideally dedicated, monitoring; can be part of a messy solution to the wicked problem [START_REF] Frame | Wicked' , 'Messy' , and 'Clumsy': long-term frameworks for sustainability[END_REF] of dolphin bycatch in the medium to long term, once the gavel hits and the dust settles.

CONCLUSION

We investigated with simulations the ability of multilevel regularized regression with post-stratification to estimate cetacean bycatch for observer programmes when coverage is correlated to bycatch risk. Our aims were to provide a first investigation on model-based estimates obtained from samples preferentially collected either during low-or high-bycatch risk periods. The unbiased sampling case is unrealistic (Babcock and Pikitch, 2003): biased sampling, either under-sampling or oversampling (ICES, 2009), may be the general case. We considered both of these cases, under quite extreme scenarios whereby data collection was highly correlated with bycatch risk, resulting in either very few observed events with under-sampling, and a large number of observed events with over-sampling. In both cases, multilevel regularized regression with post-stratification was able to produce nearly unbiased bycatch estimates with as few as 5 observed events data. With only 1 year of data, precision was low, especially with under-sampling, and there was some estimation bias with over-sampling one. These results stemmed from the extreme scenarios we considered but illustrate nevertheless that a model cannot be expected to solve all the deficiencies of data collection and measurement. Good measurement is key for accurate estimation and our results actually re-emphasize the importance of design. However, they also show that a good data collection design and an adequate modeling framework are synergistic and allow to extract a lot of information for sparse data. Assuming a normal distribution for the bycatch estimates (which is not necessary as the posterior is available, but the following are back-of-the-envelope calculations to be used for deriving heuristics), a 80% Bayesian CI width divided by 2.5 gives an idea of the associated coefficient of variation: the model-based approach can yield a coefficient of variation of 50% with as few as 15 observed events if sampling is unbiased. With under-sampling, one would need 10 years of data (under our data simulation schemes) to obtain the same precision. This re-iterates the need to (i) have dedicated observer schemes, (ii) ensure adequate observer coverage and (iii) use a model-based approach tailored to extract as much information as possible from sparse data, as the first two points are very difficult to live up to in practice.

The key assumptions behind regularized multilevel regression with post-stratification in our simulations are that bycatch risk changes smoothly through time and that accurate data on the number of fishing operations at the same temporal scale are available (e.g., number of tows for trawls; Tremblay-Boyer and Berkenbusch, 2020). When both assumptions can be reasonably entertained, we showed how a model-based approach using recent methodological developments is attractive, irrespective of how data were collected. A further asset of the explicit consideration of a temporal scale is that it may help in pinpointing more precisely windows of heightened risk in order to target adequate mitigation measures (e.g., spatio-temporal closures). The framework of multilevel modeling is very flexible and can accommodate spatial effects, etc., data permitting. Regularization will, in general, be needed to mitigate data sparsity and leverage partial pooling in order to obtain stable estimates of bycatch. Given the satisfactory performance of regularized multilevel regression with post-stratification in our simulations, we recommend further investigations using this technique to estimate bycatch rate and numbers from both representative or non-representative samples. The modeling choices we made (e.g., a first order random walk for the mean function, or fixing the range parameter in the covariance function of the Gaussian Process prior) are not prescriptive, and other choices of prior models for parameters should be investigated. Investigations should be tailored to the context, and modeling choices motivated by the latter: given the complexity of PETS bycatch, a one-size-fits-all solution is unlikely. A re-analysis of > 15 years of observer data on common dolphin bycatch in pair trawlers flying the French flag is currently underway (Rouby et al.) 4 in order to obtain better bycatch estimates that could be further used to estimate conservation reference points in order to better manage this fishery in the long run [START_REF] Cooke | Improvement of fishery-management advice through simulation testing of harvest algorithms[END_REF][START_REF] Punt | Can we manage marine mammal bycatch effectively in low-data environments?[END_REF].

Estimating Bycatch From Non-representative Samples (II): A Case Study of Pair Trawlers and Common Dolphins in the Bay of Biscay

Etienne Rouby 1,2,3 , Laurent Dubroca 4 , Thomas Cloâtre 5 , Sebastien Demanèche 5 , Mathieu Genu 1,3 , Kelly Macleod 6 † , Hélène Peltier 3,7 , Vincent Ridoux 1,2,3 and Matthieu Authier Marine megafauna plays an important functional role in marine ecosystems as top predators but are threatened by a wide range of anthropogenic activities. Bycatch, the incidental capture of non-targeted species in commercial and recreational fisheries, is of particular concern for small cetacean species, such as dolphins and porpoises. In the North-East Atlantic, common dolphin (Delphinus delphis, Linné 1758) bycatch has been increasing and associated with large numbers of animals stranding during winter on the French Atlantic seashore since at least 2017. However, uncertainties around the true magnitude of common dolphin bycatch and the fisheries involved have led to delays in the implementation of mitigation measures. Current data collection on dolphin bycatch in France is with non-dedicated observers deployed on vessels for the purpose of national fisheries sampling programmes. These data cannot be assumed representative of the whole fisheries' bycatch events. This feature makes it difficult to use classic ratio estimators since they require a truly randomised sample of the fishery by dedicated observers. We applied a newly developed approach, regularised multilevel regression with post-stratification, to estimate total bycatch from unrepresentative samples and total fishing effort. The latter is needed for post-stratification and the former is analysed in a Bayesian framework with multilevel regression to regularise and better predict bycatch risk. We estimated the number of bycaught dolphins for each week and 10 International Council for the Exploration of the Sea (ICES) divisions from 2004 to 2020 by estimating jointly bycatch risk, haul duration, and the number of hauls per days at sea (DaS). Bycatch risk in pair trawlers flying the French flag was the highest in winter 2017 and 2019 and was associated with the longest haul durations. ICES divisions 8.a and 8.b (shelf part of the Bay of Biscay) were estimated to have the highest common dolphin bycatch. Our results were consistent with independent estimates of common dolphin bycatch from strandings. Our method show cases how non-representative observer data can nevertheless be analysed to estimate fishing duration, bycatch risk and, ultimately, the

INTRODUCTION

Over the last 50 years, the conservation status of cetaceans has been deteriorating (Brownell et al., 2019). Over 80 species of cetaceans occur worldwide and bycatch, the non-intentional capture or killing of non-target species in commercial or recreational fisheries [START_REF] Hall | On bycatches[END_REF][START_REF] Davies | Defining and estimating global marine fisheries bycatch[END_REF], remains a threat, especially to small-sized species [START_REF] Scarff | The international management of whales, dolphins, and porpoises: an interdisciplinary assessment[END_REF][START_REF] Read | Bycatch of marine mammals in us and global fisheries[END_REF]Avila et al., 2018;[START_REF] Anderson | Cetacean bycatch in Indian Ocean tuna gillnet fisheries[END_REF]. Success stories in small cetacean conservation are the exception rather than the rule (e.g., [START_REF] Bessesen | Vaquita -Science, Politics, and Crime in the Sea of Cortez[END_REF]. Both [START_REF] Rogan | Empty promises: the European union is failing to protect dolphins and porpoises from fisheries by-catch[END_REF] and [START_REF] Bearzi | Shifting baselines of cetacean conservation in Europe[END_REF] opined of institutional failures to conserve cetaceans in European Waters in spite of current legislation (for example, the Habitats Directive, the Marine Strategy Framework Directive) or regional agreements such as the Agreement on the Conservation of Small Cetaceans of the Baltic and North Seas (ASCOBANS, see Table 1 for acronyms; ICES, 2020c). Over 20 species of small cetaceans have been registered in the North-East Atlantic, with roughly half of which occurring regularly [START_REF] Course | Monitoring Cetacean Bycatch: An Analysis of Different Methods Aboard Commercial Fishing Vessels[END_REF]. Because of their slow life histories and their limited potential rates of increase, small cetaceans are particularly at risk of decline when anthropogenic activities induce additional mortality on populations [START_REF] Read | The looming crisis: interactions between marine mammals and fisheries[END_REF]. Anthropogenic activities and their cumulative impacts can take a heavy toll on populations. Common species may disappear, such as short-beaked common dolphins (Delphinus delphis, hereafter called common dolphins) in the Adriatic Sea [START_REF] Bearzi | Shifting baselines of cetacean conservation in Europe[END_REF], or are under many threats, e.g., in the Bay of Biscay [START_REF] García-Baron | which are the main threats affecting the marine megafauna in the Bay of Biscay? Continent[END_REF][START_REF] Murphy | Conservation management of common dolphins: lessons learned from the North-East Atlantic[END_REF].

In 2013, the common dolphin's conservation status in the European Marine Atlantic, as assessed under Article 17 of the Habitats Directive, was "Unfavourable-Inadequate" because of fishery bycatch [START_REF] Murphy | Conservation management of common dolphins: lessons learned from the North-East Atlantic[END_REF]. Common dolphin bycatch in the Bay of Biscay, in particular, has attracted a lot of media coverage since 2017 in international outlets 1 and motivated (with bycatch of Harbour porpoise Phocoena phocoena in the Baltic Sea) a special request of Non-Governmental Organisations to the European Commission in 2019. The International Council for the Exploration of the Sea (ICES) advised in 2020, for the common dolphin in the Bay of Biscay, a combination of temporal closures of all métiers (i.e the combination of gear, target species, and fishing area) of concern and application of pingers on pair trawlers to mitigate bycatch outside of the period of closure (ICES, 2020b). Temporal closures, restricted to winter months in , 2021). This decision against the advice of ICES was motivated by a lack of knowledge on common dolphins, including its abundance at the level of the whole North-East Atlantic (the currently recognised management unit: [START_REF] Murphy | The short-beaked common dolphin (Delphinus delphis) in the North-East Atlantic: distribution, ecology, management and conservation status[END_REF] and the extent of bycatch. The issue of managing uncomfortable knowledge through interpretation of scientific uncertainty can be raised [START_REF] Schweder | Distortion of uncertainty in science: Antarctic fin whales in the 1950s[END_REF]Rayner, 2012); yet it should not eclipse that there are genuine difficulties in estimating accurately the true magnitude and the extent of bycatch of small cetaceans [START_REF] Moore | Estimating bycatch mortality for marine mammals: concepts and best practices[END_REF]. Several types of fishing gear are known to cause cetacean bycatch: drift nets, set gill, trammel nets, both pair and single midwater trawls, and some demersal trawls [START_REF] Rogan | Megafauna bycatch in drift nets for albacore tuna (Thunnus alalunga) in the ne atlantic[END_REF][START_REF] Fernández-Contreras | Incidental bycatch of short-beaked common dolphins (Delphinus delphis) by pairtrawlers off Northwestern Spain[END_REF]Peltier et al., 2016). Accurate quantification of bycatch rates by fishing gears or métiers remains a challenging endeavour (Babcock et al., 2003;ICES, 2019). Traditionally, bycatch data are collected by onboard observers monitoring fishing operations and recording the unwanted catch of non-commercial species [START_REF] Course | Monitoring Cetacean Bycatch: An Analysis of Different Methods Aboard Commercial Fishing Vessels[END_REF]. Ratio estimators, based on the number of observed hauls with bycatch over the total number of monitored hauls, are used [START_REF] Alverson | A Global Assessment of Fisheries Bycatch and Discards[END_REF]page 18) but are plagued by large uncertainties due to low coverage and the usual small number of hauls with small cetacean bycatch (Babcock et al., 2003;[START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF][START_REF] Course | Monitoring Cetacean Bycatch: An Analysis of Different Methods Aboard Commercial Fishing Vessels[END_REF]. It may also happen that some bycatch events may not be reported by nondedicated observers since they may drive observations for other purposes than report bycatch (e.g., commercial discards or stock assessments). A critical assumption behind the use of such ratio-estimators is that of a representative sample: this assumption is difficult to sustain unless monitoring is dedicated to marine mammals, and allocation of observers to fishing vessels is truly randomised (that is, not at the discretion of skippers). Even if we are willing to assume representative sampling, if coverage is low, the main challenge remains to extrapolate from sample to the whole fisheries. In France, monitoring of cetacean bycatch in fisheries is non-dedicated [START_REF] Cornou | Captures et rejets des métiers de pêche français[END_REF], and the collected data are described as non-representative of the bycatch events, preventing the use of ratio-estimators (Anonymous, 2016;page 24).

This non-dedicated nature and the sparseness of the bycatch data complicates the use of state-of-the-art spatio-temporal models such as Vector-Autoregressive Spatio-Temporal (VAST) [START_REF] Thorson | Guidance for decisions using the vector autoregressive spatio-temporal (VAST) package in stock, ecosystem, habitat and climate assessments[END_REF]. This framework accommodates densitydependence, spatial and temporal scales to estimate biomass or abundance or presence of a species [START_REF] Thorson | The importance of spatial models for estimating the strength of density dependence[END_REF]. Spatio-temporal models are also used to model the co-occurrence of commercial and bycaught species, allowing the estimate of bycatch risk with time-varying spatial effects [START_REF] Ward | Using spatiotemporal species distribution models to identify temporally evolving hotspots of species co-occurrence[END_REF]. These types of model-based approaches methodologies allow modelling spatial and temporal auto-correlation through the use of Gaussian process priors. It is difficult to transfer a priori the same model-based structure to analyse small cetacean bycatch. Models such as VAST capitalise on the availability of catch data that are collected as part of fisheries monitoring. In contrast, bycatch monitoring is not as developed or efficiently enforced in many fisheries in Europe (ICES, 2019(ICES, , 2020a;;[START_REF] Stransky | Scientific, Technical and Economic Committee for Fisheries (STECF) -Revision of the EU-MAP and Work Plan Template (STECF-19-12)[END_REF], and bycatch data are typical of low quality and unrepresentative [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. In Europe, fisheries monitoring is carried out under the Data Collection Framework (DCF) but "remains not well-suited for the dedicated monitoring of rare and protected bycatch in high-risk fisheries since its main focus is the statistically-sound random sampling of all commercial fisheries" (Ulrich and Doerner, 2021). Because of these data quality issues, [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF] conducted a simulation study to gauge the potential of investigating recent methods for the analysis of nonrepresentative samples (for a recent example of a model-based approach to estimate bycatch, refer to Luck et al., 2020) in the context of small cetacean bycatch: they concluded the potential of regularised multilevel regression with post-stratification to infer more accurately bycatch rates (although uncertainties remained large). The approach of [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF] also makes use of Gaussian process priors but does not necessarily assume that a large dataset has been collected.

We analysed historical bycatch monitoring data collected by onboard observers (from 2004 to 2020) on PTM, a métier historically associated with high levels of dolphin bycatch in the Bay of Biscay (ICES, 2019;[START_REF] Murphy | Conservation management of common dolphins: lessons learned from the North-East Atlantic[END_REF]. Leveraging recent modelling developments (see companion article; [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF], we jointly estimated bycatch risk, haul duration, and number of hauls per days at sea (DaS) from an updated and revised observer dataset on common dolphin bycatch. The modelling procedure accounts for the sparseness of the bycatch incident dataset and the low observer coverage through constraints. This type of constraint (which can be viewed as some sort of penalisation) is also called regularisation. We used structured priors, such as Gaussian processes, to achieve regularisation and leverage the within-year information at the weekly scale (inducing correlation between some weeks). Structured priors allow inducing some spatial-or temporal-dependency between so called random-effects whereas unstructured priors do not induce such dependency (but both assume exchangeability). Importantly, we used this model-based approach to disaggregate bycatch risk at the level of calendar weeks in order to document within-year variations. Estimates were summed over a whole year to investigate between-year variations in the number of bycaught dolphins. We compared these model-based estimates with strandings, both within-and between-years. Finally, we concluded with recommendations on conservation and mitigation.

MATERIALS AND METHODS

Materials

Study Division

The study area (Figure 1) encompasses 10 ICES divisions within area 27: it includes the Bay of Biscay, the English Channel, and part of the Celtic seas. These zones are associated with submesoscale and mesoscale oceanographic processes, such as eddies and upwelling, that enhance ecosystem productivity and result in high availability of fishes, including commercial species (e.g., European seabass Dicentrarchus labrax, Sardine Sardina pilchardus or Anchovy Engraulis encrasicolus). Each division can roughly be classified as oceanic or neritic: divisions 7.d, 7.e, 7.f, 7.g, 7.h, 8.a, 8.b, and 8.c are related to neritic ecosystems while divisions 7.j, and 8.d are related to oceanic ecosystems.

Data Sources

Two main sources of data were used. The first dataset, called ObsMer3 ("Observation des captures en Mer"), is collected as part of an onboard observer program set up within the Data Collection Framework of the Common Fisheries Policy. The ObsMer program is carried out by Ifremer ("Institut Français de Recherche pour l'Exploitation de la Mer"), under the supervision of the Directorate of Fisheries and Aquaculture ("Direction des pêches maritimes et de l'aquaculture, " DPMA). ObsMer observers' primary duty is to register the length and weight composition of catches. Still, they have to report any bycatch event if they witness such events. ObsMer data on PTM cover 4, 484 hauls between 2004 and 2021, of which 82 were associated with a bycatch event of at least 1 and up to 50 common dolphins. ObsMer provides, among other information, the geographic position, timing, and duration of hauls. Although ObsMer is aiming at a coverage of 10 and 5% of fishing effort for (level-3 métier) PTM for vessels of more than 15 m and less than 15 m, respectively, these figures are rarely, if ever, reached in practice: accepting onboard observers remains entirely at the discretion of skippers. The effort is quite low overall, ranging from 0 to 11% of Days at Sea (DaS) (Table 2). A DaS is any continuous period of 24 h (or part thereof) during which a vessel is present within an area and absent from the port (Anonymous, 2019). The number of observed hauls with at least one bycatch record is very small because the yearly percentage of observed hauls with a bycatch event never exceeded 4.5% and was 0 in nearly half of the surveyed years. ObsMer data on pair-trawlers are an unrepresentative sample of hauls, largely because allowing an observer remains largely at the discretion of skippers (Babcock et al., 2003;Benoît and Allard, 2009).

The second dataset provides monthly estimates of total fishing effort in each division. This dataset is generated from the algorithm SACROIS developed by Ifremer and integrates data from Vessel Monitoring System, log-books, and landing statistics (for boats longer than 18 m from January 1, 2004, and longer than 15 m from January 1, 2005;Système d'Information Halieutique, 2017). SACROIS aims at (1) correcting errors that could exists in the integrated dataset due to recording or collecting errors and

(2) reconstitute métiers during the fishing trip as they are not recorded in logbooks or fish market data [START_REF] Cornou | Captures et rejets des métiers de pêche français[END_REF]. The SACROIS dataset provides the best available estimates of total effort, in DaS, between 2004 and 2020 (Table 2). There are also refusals from skippers due to administrative and security reasons. Skippers must file an application for authorisation to embark observers and even if they decide to file, the authorisation may be declined due to security reasons (e.g., not enough room or rails not high enough).

These two datasets are complementary for our purposes: ObsMer provides micro-level data on marine mammal bycatch at the resolution of hauls. From these data, bycatch risk may be estimated (Luck et al., 2020). Fishing trips effort data, on the other hand, are macro-level: they provide spatialised effort data at the scale of a whole fishing fleet. These population-level data on effort allows the post-stratification of bycatch risk estimated from observer data to obtain the number of bycaught dolphins [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. Descriptive statistics of both datasets are displayed in Table 2. Used in tandem, both datasets allow using regularised multilevel regression with post-stratification to estimate cetacean bycatch from non-representative samples [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF].

Estimates were finally compared to strandings along the French Atlantic seaboard. The French Stranding Network, founded in the 1970s, is dedicated to the monitoring of marine mammal strandings along the shores of France (mainland and overseas). Around 400 trained volunteers are currently taking an active part in the network. These volunteers make the complete coverage of French coastlines possible. Standardised training of volunteers by permanent Observatoire Pelagis staff, which takes place two times a year, ensure the homogeneity, comparability, and standardisation of data collection procedures in the field. Observatoire Pelagis is mandated by the French Ministry of Ecology to train and deliver authorisation to handle carcasses of marine mammals (which are all protected species under national law). It also collates the data and analyse it to inform on the status of marine mammal populations. Stranding data for the period 2004-2020 were used. Only common dolphins found with lesions diagnostic of bycatch in fishing gear were considered (Kuiken, 1994) as well as those stranded during multiple stranding events, or "unusual mortality events" related to lesions diagnostic of bycatch. Multiple stranding events were defined as high numbers of strandings occurring in a restricted area with a common cause of death. The threshold was defined at 30 cetaceans over 10 consecutive days recorded along a maximal distance of 200 km in the Bay of Biscay, and 10 individuals per 10 days per 200 km of coastline along the coast of the western Channel (Peltier et al., 2014). Reverse drift modelling uses a deterministic drift model developed by Meteo France (Peltier et al., 2012) to reconstruct the trajectory of every stranded common dolphin from its stranding location to its likely area of death at sea. The number of dead stranded animals in each cell is then corrected by the cellspecific probability of being stranded (Peltier et al., 2016). These probabilities were estimated by numerical experiment in which the drift of carcasses in the study area was simulated in order to assess with which frequency they would reach a coastline (Peltier and Ridoux, 2015).

To account for strict positivity, fishing duration is modelled either with a Gamma or a log-normal likelihood:

d ijkt ∼ G β, β djkt (2a) d ijkt ∼ LN djkt , σ (2b)
The number of hauls per DaS is modelled assuming a zerotruncated Poisson likelihood: jkt are rates. The linear predictors α jk are vectors of week-level parameters related to ICES division j and year k (dropping the superscript for convenience):

n jkt ∼ P + DaS jkt × λ jkt (
         α jk ∼ GP (δ k , division ) δ k ∼ GP ǫ, year ǫ t = µ t = 1 ǫ t+1 ∼ N (ε t , σ week ) t > 1 (4)
Parameter µ is the intercept. The vector ǫ aggregates the mean weekly effects (on the linear predictor scale) which are modelled with a first-order random walk to ensure some smoothness in between-week variations [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. The vector δ k are year-specific deviations from the mean weekly pattern ǫ. 2 and range parameter fixed to ρ = 3 2 was assumed: t, t

′ = 1 + 2 √ 3×d(t-t ′ ) 3 × exp -2 √ 3×d(t-t ′ ) 2
where d(tt ′ ) = |tt ′ | is the temporal distance (in weeks) between weeks t and t ′ . The choice of the range parameter induces a temporal correlation of 0.05 after 4 weeks (that is, temporal independence after a month; [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. The correlation matrix is assumed known and is depicted in interaction between week, year, and division. The joint model defined in Equations ( 1), (2a), and (3) includes a time-varying component at the week-scale with interaction with year and division.

Simpler models without such interactions, and with only additive effects, were also fitted to the data. The simplest model included only additive random (unstructured) effects (dropping the superscript for convenience):

               α jkt = ǫ t + δ * k + α * j α * j ∼ N (0, σ division ) ∀ j δ * k ∼ N 0, σ year ∀ k ǫ t = µ t = 1 ǫ t+1 ∼ N (ε t , σ week ) t > 1 (6)
Models are multilevel, accommodating week-, year-, and division-level variations. They also use structured priors such as Gaussian processes or random walks to regularise estimation (Gao et al., 2019). More information on these models, and on applying (regularised) multilevel regression with poststratification in the context of estimating bycatch, are detailed by [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. Estimation was carried out in a Bayesian framework using programming language Stan (Carpenter et al., 2017) called from R v.4.0.1 (R Core Team, 2020) with library Rstan (Stan Development Team, 2020). Stan uses Hamiltonian dynamics in Markov chain Monte Carlo (MCMC) to sample values from the joint posterior distribution (Carpenter et al., 2017). Four chains were initialised from diffuse random starting points and run for a total of 2, 000 iterations, discarding the first 1, 000 as a warm-up. Default settings for the No-U-Turn Sampler (NUTS) were changed to 0.99 for adapt delta and 15 for max treedepth (Hoffman and Gelman, 2014). Priors are reported in Table 3. We fitted a total of 6 models of differing complexity (Table 4): we compared models assuming either gamma or a log-normal likelihood for haul duration, and models assuming additive effects vs. interactive effects of the week, year, and divisions. Model fitting was carried out on the supercomputer facilities of the "Mésocentre de calcul de Poitou Charentes (Université de Poitiers/ISAE-ENSMA/La Rochelle Université)." Codes are available at https://gitlab.univlr.fr/mauthier/cdptmbycatch. For confidentiality reasons, the actual dataset cannot be shared: a synthetic dataset, generated by predicting from the posterior distribution, is provided instead.

Estimating the Total Number of Hauls and Bycatch Events

The number of unobserved hauls N jkt that happened in ICES statistical division j in week t of year k can be estimated from the number of observed DaS in ObsMer (DaS ObsMer The total number of bycatch events in ICES statistical division j in week t of year k is estimated as the sum of events observed in ObsMer (Bycatch ObsMer

jkt

) and the number of unobserved hauls multiplied by bycatch risk ( p jkt ):

Bycatch jkt = Bycatch ObsMer jkt + Njkt × p jkt (8)
Similarly, for each year, the number of common dolphins bycaught in pair-trawlers can be estimated using the observed number of bycaught dolphins in ObsMer, the estimated number of unobserved hauls (Equation 7), bycatch risk, and either the median number of dolphins involved in a bycatch event (Table 2, or the grand median of m = 2 for years with no observed bycatch event). We used the median to attenuate the influence of some bycatch events involving up to 50 dolphins (Table 2). These estimates are thereafter referred to as model-based estimates.

Comparing Model-Based Estimates With Strandings

The sample provided by ObsMer, a non-dedicated observer scheme of marine mammal bycatch, may not be representative of all bycatch. In addition, it provides very sparse data, with less than 100 observed events over 17 years (Table 4) when strandings have reached several hundred per week in recent years (ICES, 2020d) (for all causes of death). Despite this, the weekly pattern of bycatch risk provided by ObsMer roughly matches that of strandings, with an increase in winter (Figure 3). Despite this rough match, the ObsMer data also suggest a heightened risk in summer, especially in the 2000s, whereas strandings suggest such an increased risk in very recent years (Peltier et al., 2021). The number of stranded common dolphins with evidence of bycatch can be used to estimate the total bycatch mortality with reverse drift modelling (Peltier et al., 2016). These strandingbased estimates are now used in international working groups (ICES, 2020d). Reverse drift modelling corrects for at-sea drifting conditions, but cannot inform on which fishing gears were responsible for bycatch. Hence, strandings-based estimates are total estimates of bycatch and can be compared to model and observation based estimates of bycatch by French pairtrawlers. These model-based estimates use data independent from strandings, but they should not exceed stranding-based estimates. Second, whether model-estimates correlate with strandings-based ones is of interest to shed light on the increased mortality witnessed in the Bay of Biscay (Peltier et al., 2021). For each year, we checked the magnitude of model-based estimates against stranding-based ones and computed Pearson's correlation coefficient between the two time-series at the month level. To account for drift, these correlations were computed with and without a lag of 2 weeks when aggregating model-based estimates at the month level.

RESULTS

We built and compared six models (Table 4). Convergence was reached for all parameters with all R < 1.05. Model M 6 had the lowest WAIC and was selected as the best model for Models are ordered in increasing order of W ÂIC (the smaller, the better the fit). se stands for "standard error". 4) and the dotted black line the yearly average (parameters ǫt in Equation 4).

further inferences. Model M 6 included an interaction between week, year, and ICES division (Equations 4 and 5). All codes to fit models are available at https://gitlab.univ-lr.fr/mauthier/ cdptmbycatch.

Bycatch Risk, Haul Duration, and Haul Number Per DaS

Haul duration, hauls per DaS, and bycatch risk per haul (Equations 7 and 8) were jointly estimated. Their temporal variations are displayed in Figure 3 for each week between 2004 and 2020. Haul duration was the highest in week 1 with a posterior median estimate of 5.8 h that decreased to 4.0 h in week 16, before dropping to 2 h in week 24. Haul duration increased up to 3 h in week 32 and plateaued until the end of the year. Remarkable years were 2017, 2019, and 2020 with the longest haul durations estimated from week 1 to 10. From week 10 onwards, years before 2012 displayed some variations in haul duration. In particular, duration was consistently smaller in 2004. In 2016, an increase in haul duration was estimated in week 48 (5 vs. 3 h on average across years).

Bycatch risk was maximum in week 1 (around 0.1) and decreased to almost 0 from week 8 onwards. 2017, 2019, and 2020 were the years with the highest estimated bycatch risk in the first 8 weeks. In particular, the risk was as high as 0.20 in 2017 for the first four consecutive years of the year. Two years prior to 2012 were associated with an increased risk between weeks 30 and 36. The year 2016 showed a rise in bycatch risk in week 48. Bycatch risk and haul duration were positively correlated with weeks in years associated with the highest risk and also having the longest haul duration. Numbers of hauls per DaS were negatively correlated with weeks with longer haul duration. There was little variation across years in numbers of hauls per DaS, but substantial within year variations.

Spatial variations in bycatch risk and haul duration are available as supplementary information. There were noteworthy differences between divisions regarding bycatch risk (see supplementary information). The overall signal was similar to the one observed in Figure 3 with the highest risk values estimated between weeks 1 and 8. Risk in 2017 and 2019 was higher by a factor of 5 in week 1 compared to other years. After week 8, this difference disappeared. With respect to divisions, division 8.a. was the one with the highest bycatch risk, with an estimate as high as 0.50 in winter 2017 and 2019.

Number of Bycaught Dolphins

The estimated total number of bycaught dolphins for each year is reported in Table 5. The study area was further divided into three strata: a neretic stratum in ICES subarea 7 (divisions 7.defgh) and another in subarea 8 (divisions 8.abc); and an oceanic stratum spanning subareas 7 and 8 (divisions 7.j and 8.d). Estimates were the lowest in the oceanic stratum of the study area and the largest in the neretic stratum spanning ICES subarea 8. The largest bycatch estimate was in 2017, with a posterior median of > 600 common dolphins bycaught in PTM operating in the neretic stratum spanning ICES subarea 8. There were large betweenyear variations in estimates, ranging from less than a hundred (in 2018) to more than one thousand (in 2017). Uncertainties around model-based estimates were also large.

Comparison and Correlations With Strandings

Strandings data were used to estimate common dolphins mortality due to fisheries following method described in Peltier et al. (2016) for each month from 1990 to 2020. Strandingbased estimates aggregate mortality due to all fisheries and do not distinguish between gears or métiers. Nevertheless, we correlated stranding-based estimates with our model-based estimates of mortality from PTM flying the French flag both between years (Figure 4) and within each year (Figure 4). For yearly estimates, correlations were computed on raw and standardised (mean centered and unit variance) values (Figure 4). Model-based estimates of bycatch by PTM were always below stranding-based estimates (which do not allow to disaggregate by métiers) save for 2010 (Figure 4). In 2010, model-based and stranding based estimates were 465 and 343, respectively, with a large overlap in credibility interval. At the year level, the Pearson correlation between stranding-based and model-based estimates was 0.25. Yearly variations between the two time series were more in phase from 2015 onwards (Figure 4). At the within year (between month) level, correlations between the two time-series were always positive. These within year correlations generally increased by 47% (median) when model-based estimates were aggregated by month with a lag of 2 weeks to account for drift (Figure 4). The temporal trend in within year correlation was negative over the study period.

DISCUSSION

From a non-representative sample of bycatch events of common dolphins collected over more than 15 years, we estimated bycatch risk and number of dolphins bycaught in PTM. Leveraging recent methodological developments in the analysis of nonrepresentative samples (Gao et al., 2019;[START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF], we built a joint model of bycatch risk, haul duration, and haul number per DaS to investigate changes within and between years in common dolphin bycatch. The years 2017 and 2019 were associated with the highest bycatch risk and the longest haul duration in winter.

Within-Year Variations in Bycatch Risk

We uncovered the within-year pattern in bycatch risk of common dolphins. Bycatch risk is the highest in winter, during the first weeks of a calendar year. This pattern is largely congruent with the pattern seen in strandings of common dolphins in the Bay of Biscay [START_REF] Gilbert | Spatiotemporal models highlight influence of oceanographic conditions on common dolphin bycatch risk in the Bay of Biscay[END_REF]. Both stranding and observer data, which are independent, identified 2017 and 2019 as years with the highest risk of bycatch [START_REF] Gilbert | Spatiotemporal models highlight influence of oceanographic conditions on common dolphin bycatch risk in the Bay of Biscay[END_REF]Peltier et al., 2021). A limitation of stranding data is how the location of bycatch events must be inferred with reverse drift modelling (Peltier and Ridoux, 2015). The ObsMer data in contrast included geolocalised bycatch events, with a spatial resolution at the level of ICES divisions kept for analysis. Despite this coarse resolution, we could identify divisions 8.a and 8.b as the ones with the highest risk of bycatch by PTM.

The ICES Working Group on Bycatch (WGBYC) estimate bycatch of protected species, including common dolphins, in the North East Atlantic. Using data collected by onboard observers collected between 2005 and 2017, bycatch rates for ICES divisions on the continental shelf of the Bay of Biscay were estimated with ratio estimators (ICES, 2019). These estimates are not produced at the week level, but ICES (2019) also identified divisions 8.a and 8.b as the ones with the highest of bycatch in midwater trawls for common dolphins over the period 2005-2017 (p. 61). ICES (2019) estimated yearly rates ranging between 0.285 and 0.372 dolphins per DaS and warned against extrapolation given the low observer coverage (p. 61). Our model-based approach overcomes this limitation [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF] and was able to identify, within each year, that weeks 3 to 5 were the ones with the highest bycatch numbers for both divisions 8.a and 8.b. These results were concomitant with the seasonal stranding pattern observed each year on the French seashore (that is, winter strandings; [START_REF] Gilbert | Spatiotemporal models highlight influence of oceanographic conditions on common dolphin bycatch risk in the Bay of Biscay[END_REF]: around 80% of all common dolphin strandings on the French Atlantic seashore is observed between the end of January and the beginning of April.

A key feature of our model-based approach is how it leverages correlations between bycatch risk, haul duration, and number of hauls per DaS (Figure 3). Some of the correlations are expected, such as the negative correlation between haul duration and the number of hauls per DaS. However, average haul duration is not constant within a year, with the variations reflecting the change in the commercial fish species targeted by PTM at different time of the year. These variations at the week-level were quite substantial and were taken into account when estimating bycatch in our model. There was a positive correlation between haul duration and a bycatch risk, with at least a two-fold increase in the later when haul duration exceeds 5 h (Figure 3). This was particularly evident in weeks 1 to 5 in 2017 and 2019 and week 48 in 2016. The latter was due to a single fishing trip with 5 hauls that lasted > 10 h, each of which resulted in a bycatch event. We recommend, in light of the within-year pattern in haul duration (Figure 3), to investigate management actions and mitigation measures on limiting haul duration in winter to assess whether bycatch may also be reduced.

Another possible mitigation measure is to manage common dolphin interactions with PTM with spatio-temporal closures (and acoustic repulsive devices such as pingers) during the first week of a year, when bycatch is the highest. Such measures were explored by WKEMBYC (ICES, 2020d) to reduce bycatch mortality across several scenarios. The performance of each scenario was assessed with the Potential Biological Removal (Wade, 1998), bycatch reduction rate, and fishing effort reduction rate. WKEMBYC (ICES, 2020d) defined an efficiency score by the ratio between the latter two rates. This efficiency score is a trade-off between the expected bycatch reduction and the cost for the fishing industry (without direct economic consideration). WKEMBYC (ICES, 2020d) identified one scenario (scenario L) wherein 2 months closure from mid January to mid March for all fishing métier (and the use of pingers for "Bottom pair trawlers" (PTB) and PTM the rest of the year) was efficient. This scenario appears as a good compromise between bycatch reduction and a reduced cost for the industry. Another efficient scenario (scenario N) involves a 3-month closure from January to March and another 1 month from mid July to mid August for all métier (and the use of pingers for PTB and PTM the rest of year). This scenario can achieve the highest level of bycatch reduction but incurs a high cost to the industry. However, scenarios considered by WKEMBYC are emergency measures meant to reduce punctually common dolphin bycatch. Systematic spatio-temporal closures, which are usually not favoured by the fisheries, were not considered and remained to be explored. In contrast, mitigation measures relying on the large scale deployment of acoustic repulsive devices and the development of new such devices are underway (e.g., in the CetAMBICion project4 ).

Between-Year Variations in Bycatch Risk

There were large between-year variations in model-based estimates of common dolphin bycatch in the study area. To some extent, these variations were explained by other factors than bycatch risk. For example, the (posterior median) estimate is >600 dolphins in 2017 down to <100 in 2018. The total effort in DaS in the Bay of Biscay (divisions 8.a and 8.b) in the first 10 weeks of 2017, when bycatch risk was highest, is two times the value of total effort in 2018. The median number of dolphins involved in a bycatch event in 2017 was also two times the number in 2018 (2 and 1, respectively, Table 2). All else being equal, the estimate for 2017 is expected to be at least four times that of 2018. A further improvement of the model-based approach is to jointly model the number of dolphins involved in a bycatch event. This improvement will require accomodating a large overdispersion, but there were however less than 100 such events in the dataset and we chose to use the median. This is a cautionary choice since the median is less sensitive to the few events for more than 10 dolphins. The uncertainty in the median number of dolphins involved in a bycatch event is currently ignored: incorporating it in future development will further widen credibility intervals (which are already large; [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. Thus, the model-based estimates are conservative estimates of bycatch by PTM.

Bycatch risk was also very variable between years: the large between-year variations may be due to ecological factors. Bycatch risk results from both fisheries activity within a particular division at a particular time and dolphin presence. The highest bycatch risk values were estimated for the 8 or 10 first weeks of each year within each division of the study area (Figure 3). [START_REF] Astarloa | The role of climate, oceanography, and prey in driving decadal spatiotemporal patterns of a highly mobile top predator[END_REF] found evidence of an increased abundance of common dolphins in the Bay of Biscay in recent years but weak correlations with biological and oceanographic variables, such as chlorophyll a concentration or sea surface temperature. ICES divisions 8.a and 8.b cover the continental shelf parts of the Bay of Biscay (Figure 1). These neritic divisions are witness to sub-mesoscale oceanographic processes and nutrient offloads from the Gironde estuary. Gilbert et al., 2021 correlated eddies and frontal structures with common dolphin mortality areas at sea in the Bay of Biscay (although these authors also concluded that oceanographic accounted for a small fraction of the overall variance in stranding numbers). In winter, the Bay of Biscay environment is characterised by a seasonal cross-shore (West to East) surface temperature gradient with the lowest temperature close to shore and intense frontal activity parallel to the coast (North to South) [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. These frontal structures are freshwater fronts, correlated to the mixing of oceanic waters and cold freshwater inputs from river plumes [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. These seasonal fronts may be targeted by both fisheries and common dolphins as areas where fish aggregate, thereby putting the latter at risk of bycatch by the former. In July and August, the mesoscale dynamic activity of the Bay of Biscay is rather different than in winter. In summer, there are mainly fronts due to tidal flow [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Summer tidal fronts are quite consistent from 1 year to the next because they are correlated to a repetitive process (i.e., tides) [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. During summer, the main frontal activity is a seasonal tidal front, called the Ushant Front and located in front of the French Finistère county [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. Its activity peaks in July and August [START_REF] Yelekçi | Spatial and seasonal distributions of frontal activity over the French continental shelf in the Bay of Biscay[END_REF]. We can speculate that the years associated with a high bycatch risk were also those when oceanographic processes favouring the spatial overlap (mediated by fish species; Spitz et al., 2013;[START_REF] Astarloa | The role of climate, oceanography, and prey in driving decadal spatiotemporal patterns of a highly mobile top predator[END_REF] between fisheries and common dolphins were particularly operant.

Stranding records are an independent source of data for estimating the number of bycaught dolphins (Peltier et al., 2016). Reverse drift modelling allows the death location of each stranded dolphin showing bycatch evidence for each month between 1990 and 2020 to be inferred. Observed stranding tallies for each month can be corrected for both stranding and buoyancy probabilities (Peltier and Ridoux, 2015). Reverse drift modelling cannot disaggregate estimates by métiers or fisheries but provides an independent estimates of total mortality due to bycatch in the study area: bycatch mortality due to PTM should be lower than the total estimated from strandings. This was verified for all years save for 2010, but uncertainties were large and credibility intervals had a large overlap. While the correlation between model-based and stranding-based estimates was modest at the year level, it was larger at the within-year level, especially after accounting for a lag due to drift (Figure 4). The magnitude of the within-year correlation decreased between 2005 and 2020. One interpretation is that of a change in the relative contribution of PTM in total dolphin mortality over time, with PTM having a lesser impact on common dolphins in recent years compared to the 2000s.

Limitations and Improvements

The model used to estimate the bycatch of common dolphins in PTM has been developed to address the issue of nonrepresentative sampling [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. It relies on a poststratification step that requires accurate effort data at the scale of the whole fleet. The effort measurement retained was that of DaS as in international working groups (e.g., ICES WGBYC; ICES, 2019). Leveraging this important piece of information required the joint modelling of risk at the haul level, haul duration, and that of the average number of hauls per DaS. This modelling choice proved successful for PTM but need not be so for other métiers, in particular for passive gears such as gillnets and setnets. In the later case, a better measure of effort at haul level is soak time, taking into account net length and height, and possible mesh size. These pieces of information may be difficult to collect and retrospectively obtain for post-stratification. Any method seeking to scale up a sample from onboard observer to the whole fleet must confront the difficult issue of accurate measurement and quantification of effort. The model developed for PTM may not necessarily transfer seamlessly to other gears or métiers.

Since 2021, PTM flying the French flag are required to use deterrent acoustic devices (pingers5 ). If these devices are efficient to reduce bycatch risk, this may be taken into account in the model, by adding a covariate in Equation ( 1). Doing so requires on the other hand to post-stratify on that covariate, which is likely to be a major hurdle. Ignoring the deployment of pingers need not be problematic as the model allows for between-and within-year variations in bycatch risk. Large-scale deployment of pingers in 2021, if effective in reducing risk, will manifest itself in an estimated risk lower compared to previous years. In other words, the model does not have to necessarily take into account all haul-level covariates as long as the aim is prediction rather than explanation [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. Taking explicitly into account the pinger effect is only required to make sense of the between-and within-year variations in risk, but not necessarily to estimate those variations.

While [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF] concluded on increased accuracy of using regularised multi-level regression with post-stratification to estimate bycatch with observer data, they also found that estimated precision was low. This was also the case in this study (Table 5). A simple way to increase precision is to include selfdeclared positive bycatch events from fishermen in Equation ( 7) and (8). Doing so provides a strong incentive for compliance on self-declaration and would result in increased precision as a greater number of hauls (and possibly DaS) would be monitored. Ultimately, full compliance would render modelling moot as bycatch would be perfectly known, if all events were properly recorded (e.g., with Electronic Remote Monitoring) or reported systematically and accurately in logbooks.

Implications for Common Dolphin Conservation

The common dolphin is one of the most abundant delphinid species within the North-East Atlantic [START_REF] Hammond | Estimates of Cetacean Abundance in European Atlantic Waters in Summer 2016 from the SCANS-III Aerial and Shipboard Surveys[END_REF]. This species may be described as a "keystone species" and an "umbrella species" considering its ecological importance [START_REF] Murphy | Conservation management of common dolphins: lessons learned from the North-East Atlantic[END_REF]. The large additional mortality due to anthropogenic activities on this species triggered a dedicated working group on emergency measures in 2020: the workshop on fisheries emergencies measures to minimise bycatch of short-beaked common dolphins in the Bay of Biscay and harbour porpoises in the Baltic Sea (WKEMBYC) took place remotely in spring 2020 (ICES, 2020d) and informed an ICES advice that same year (ICES, 2020c). This advice led to an infringement procedure issued in July 2020 against France for failing its obligations under the Habitats Directive, which lists the common dolphin as a species requiring full protection on its Annex IV. The same day, the Paris Administrative Court of Justice condemned the French government for failing to transpose and apply in a timely manner the dispositions of the Habitats Directive and Technical Measures regulating fisheries 6 (in French). Following the unprecedented number of strandings in 2017, a national working group with fishermen, their representatives, government officials, Non-Governmental Organizations, and academics was initiated to address the bycatch issue (Peltier et al., 2021). One recommended action was to improve estimates of bycatch due to high-risk métiers, and to develop adequate methodologies to analyse data from nonrepresentative samples [START_REF] Authier | Estimating Bycatch from Non-Representative Samples with (Regularized) Multilevel Regression with Post-Stratification[END_REF]. The present work reports on a case study on PTM and operating for a large part in the Bay of Biscay, and to a lesser extent in the Celtic seas. The model-based estimates (i) can inform on pressures acting on common dolphins as required by the Marine Strategy Framework Directive (EU 2008/56) and (ii) heed ICES recommendation to develop estimation methods to make the best use of already collected data to inform management in a timely manner (ICES, 2020c).

Using a Potential Biological Removal (PBR) approach (Wade, 1998), ICES (2020d) estimated a removal limit of common dolphin for the whole North-East Atlantic of 4, 926 individuals. An annual bycatch no greater than PBR would allow the population of common dolphins to recover to or be maintained at or above 50% of carrying capacity with a probability of 0.95 [START_REF] Wade | Best practices for assessing and managing bycatch of marine mammals[END_REF]. This conservation objective is, however, different from the ASCOBANS interim objective "to restore and/or maintain stocks/populations to 80% or more of the carrying capacity. " Genu et al. (this issue) tuned a modified PBR to a quantitative interpretation of the ASCOBANS interim objective: "a population should be able to recover to or be maintained at 80% of carrying capacity, with probability 0.8, within a 100-year period." The removals limit computed using the modified PBR was down to 985 animals (that is, one fifth of PBR; Genu et al., this issue): in 2017, the estimated bycatch due to PTM and operating the Bay of Biscay amounted to more than 60% of this limit (Table 5). In recent years, the estimated contribution of this métier relative to the modified PBR remained large according to our results. Other fishing métiers could potentially impact the common dolphins in the Bay of Biscay resulting in mortality exceeding the threshold inferred by both modified and non-modified PBR. Regarding vessels flying the French flag, gill trammel netters (GTR), gillnetters (GNS), and pair trawlers were potentially associated with common dolphin mortality in ICES divisions 8.a and 8.b for different years (regarding the co-occurrence of mortality and fishing effort) (Peltier et al., 2021). Estimating the contribution of each métiers to overall mortality remains a difficult endeavor. Regarding the PBR removals limit used in WKEMBYC (ICES, 2020d), the overall mortality considering all the fishing métiers exceed PBR, notably from 2016 to 2019, years 6 http://paris.tribunal-administratif.fr/content/download/172866/1715763/ version/1/file/1901535.pdf associated with the suspected highest contribution for the métiers listed above.

CONCLUSION

We have provided a case study on estimating bycatch of common dolphins by PTM and operating in the Bay of Biscay from a non-representative sample of bycatch events collected by nondedicated onboard observers. Leveraging recent methodological developments in statistical modelling, we have illustrated how to use imperfect but currently available data to inform management. Our contribution thus heeds two recent recommendations: to use adequate estimation methods on existing data and to gauge the resulting estimates against threshold values for incidental bycatch, tuned to relevant conservation objectives. We evidenced a substantial contribution of PTM to common dolphin bycatch in the Bay of Biscay, especially in 2017. Considering the entire time series and the correlations with the estimates made from strandings, it is possible that other métiers than PTM were associated with bycatch, especially in recent years. Currently, the main mitigation measures recommended are spatio-temporal closures and the widespread use of acoustic deterrent devices on PTM/OTM and PTB to repel dolphins (ICES, 2020b). Spatio-temporal closures were not implemented in 2021 but systematic and mandatory deployment of pingers on trawls were 7 . Relevant to management in broadening the scope of potential measures is the evidenced correlation between bycatch risk and haul duration: further studies should investigate limiting haul duration (for example, below 5 h) as a complementary mitigation strategy, especially in winter.

Figure

  Figure . : Schematic representation of the keystone role of predatory starfish Pisaster ochraceus in an intertidal ecosystem in Washington. (a) Pisaster ochraceus predation maintains a diverse community. (b) removal of Pisaster ochraceus allows mussels to dominate, and reduces species diversity. Based on (Paine ).

Figure

  Figure . : Conceptual definition of megafauna based on size and importance for the ecosystem. a. Large animals have an high impact on local ecosystems whereas the impact of smaller ones is more variable.There is a challenge to correctly define the size-effect relationship. b. Two-dimensional qualitative disruption of animal species. The twodimensional space is defined by body size and ecosystem effect. Keystone species have an high effect on ecosystems and only the largest keystone species are defined as megafauna. Figure from Moleón et al..

Figure . :

 . Figure . : Typology of pressures based on their effects on marine megafauna. Figure from Authier et al..

Figure . :

 . Figure . : Survival probability of killer whales along time. Nondepredating killer whales are represented with empty circles whereas depredating killer-whales are represented with black circles. Error bars are the % CI. From Tixier et al. ().

Figure

  Figure . : Vaquita (Phocoena sinus, Norris and McFarland ). Picture from

Figure .

  Figure .: Relation between per capita growth rate, abundance and carrying capacity. When abundance (N) is low, the per capita growth rate is high enough to allow the population to produce individuals and increase abundance to the carrying capacity which is the stable equilibrium point. In the case of a population subject to the Allee effect, a too low abundance below a critical threshold is associated with a decline in growth rate. It is no longer high enough to support mortality and the population disappears.

Figure . :

 . Figure . : Conceptual model a agebased-matrix model. Individuals can reach the maximum age of . Each age class contributes to the renewal of the population with its associated fecundity rate . Each individual advances in age in the population with a probability equal to the age-specific survival rate.

Figure . :

 . Figure . : Exemples of possible individual-based long term monitorings. A. Ear-tags. B. Natural marks of rare animals observed thanks to camera traps. C. Bird ringing. D. Recovery of bands.

Figure . :

 . Figure . : Exemples of possible individual-based long term monitorings for cetaceans. E. Callosities on Right-Whales. F. Caudal fin markings on Humpback whales. G. Dorsal fin shape on bottlenose dolphins. H. Dorsal fin shape and natural markings on killer whales.

Figure . :

 . Figure . : Exemples of possible cross-sectional monitorings. I. Length-age relation for reptiles from https://www.mcgeesswamptours.com/. J. Otholits for fishes from Knopp et al. ( ). K. Growth layer groups in tooth from Martin et al. ( ). L. Endosteal layers in tibia for birds from Klomp and Furness ( ). M. Claws growth layer groups from E. O. Ferreira, Loseto, and Ferguson ().

Figure

  Figure . : Diagram of Bay of Biscay location. Picture from

Figure . :

 . Figure . : Main historical events in common dolphin bycatch management in the Bay of Biscay. DCF, Data Collection Framework; EC, European Commission; UME, unusual mortality event. From Peltier et al..

Figure . :

 . Figure . : Flowchart of the thesis project.

Figure . :

 . Figure . : Common dolphin morphological recognition traits and morphological parameters.Figure from the International Whaling Commission handbook https://wwhandbook.iwc.int/en/species/

  Figure . : Common dolphin morphological recognition traits and morphological parameters.Figure from the International Whaling Commission handbook https://wwhandbook.iwc.int/en/species/ common-dolphin

Figure

  Figure . : Common dolphin oceanic subspecies. Figure from Perrin ( ), illustration by Uko Gorter.

Figure . :

 . Figure . : Global distribution of the common dolphin. The species is distributed in temperate and tropical waters. It is distributed in several populations.

Figure . :

 . Figure . : Common dolphin density within the North-East Atlantic.The density of common dolphins is expressed as animals/km 2 . The highest densities are recorded for the continental shelf areas associated with the Bay of Biscay. The Celtic Seas and the oceanic part of the Bay of Biscay also have high densities, as well as the Spanish-Portuguese and Galician coasts. The English Channel and the North Sea, on the other hand, have low densities. These estimates are made in the framework of the SCANS III monitoring programme. Figure fromHammond et al. ( ).

Figure . :

 . Figure . : Common dolphin age-atdeath data. These age-at-death were obtained on stranded dolphins. Data are from Saavedra (). Data are not published and do not have associated survival estimates. However, it is possible to estimate the survival, mortality and viability of the population based on the protocol detailed in the study.

Figure

  Figure . : Common dolphin strandings in France from to . Data from Observatoire Pelagis.

Figure

  Figure . : Projected population trend. From the vital rates obtained in their study, Mannocci et al. ( ) projected the evolution of the population from the growth rate estimated by Leslie's matrix model. It appears that the population will become extinct within the next years. Figure from Mannocci et al. ().

Figure . :

 . Figure . : Common dolphin stranding records in Europe from to . Only three countries are represented as these are the ones with the most strandings of common dolphins in Europe. France is the country with the most. It can be seen that the trend has been upwards since the early s for all three countries, particularly France. Figure from Murphy et al. ().

Figure . :

 . Figure . : Ecoregions and statistical areas defined by ICES within the OSPAR area of the North-East Atlantic. The OSPAR area is divided by ICES into several ecoregions defined on the basis of biogeographic and oceanographic features and existing political, social, economic, and management divisions. Another division is made by taking into account statistical areas. These area are linked to ICES fishing areas are developed from historical links between an area and the collection of fisheries statistics. Figure from https://www.ices.dk/Pages/default.aspx.

Figure

  Figure . : ASCOBANS Marine area. Originally, the area was restricted to English Channel, North Sea and Baltic Sea. From the February of , the area now takes into account Bay of Biscay, Celtic seas and offshore Portugal waters. Figure from https://www.ascobans. org/en/legalinstrument/ ascobans.

Figure . :

 . Figure . : OSPAR Maritime Area. This area is divided into five Regions for assessment purposes: Arctic Waters (I), Greater North Sea (II), Celtic Seas (III), Bay of Biscay and Iberian Coast (IV) and Wider Atlantic (V).Figure from www.ospar.org

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Figure . :

 . Figure . : Sampling plans of the SCANS and CODA surveys. Transect solid lines are the covered effort. Blue areas were surveyed by boat and pink areas were surveyed by planes. Campaigns made were conducted for SCANS in (topleft), SCANS-II in (topright), CODA in (bottom left) and SCANS-III in (bottom right). Figure from Hammond et al. ().

  The area of the proposed management unit for North-East Atlantic common dolphins covers OSPAR Regions II (Greater North Sea), III (Celtic Sea) and IV (Bay of Biscay and Iberian coast) (Margin-Figure.regions along with SCANS III and ObSERVE surveys blocks).

Figure

  Figure . : Sampling plans of the SAMM surveys. Surveys were conducted between November and February (winter part) and between May and August (summer part). Figure from Laran et al. ().

Figure . :

 . Figure . : Dr. Duguy Raymond. Photo from https://www. observatoire-pelagis.cnrs. fr/dr-raymond-duguy/. : Link to a historical video on the stranding of pilot whales at "Ile d'Yeu" in https://dai.ly/xp3zp7. More informations on Dr. Duguy Raymond in https://www.observatoire-pelagis.cnrs.fr/dr-raymond-duguy/

Figure . :

 . Figure . : External examinations of the common dolphin. The pictures show different cases of mortality assumed by external examinations (which sometimes have to be confirmed on the basis of internal examinations or even biological analyses in the case of pathologies). : Amputated tail (Bycatch evidence). : Cut pectoral fin (Bycatch evidence). : Rope hanging from the tail (Bycatch evidence). : Trace of fishing nets (Bycatch evidence). : Fractured rostrum (Possible bycatch evidence). : Pathological necrosis (Pathology). : Possible bacterial infection (Pathology).: Unknown cause with scoliosis on calf (Unknown). For information purposes, in % of stranded dolphins are due to bycatch (according to external reviews). % are stranded without the cause being known. % from pathological reasons and % for unknown reasons. These percentages vary slightly after internal review. To see more detailed data: rapportecouhage . ©Observatoire Pelagis.

Figure . :

 . Figure . : Influence of decomposition states on the demographic data collection and on bycatch estimates. The collection of teeth and gonads is carried out to obtain biological data on the population and to estimate its vital rates in the context of the MSFD (Mannocci et al.). It is also possible to estimate the drift time of individuals based on their state. This drift time is then corrected by the probability of floating, the probability of running aground and finally the probability of being discovered. This correction makes it possible to estimate the number of dolphins captured by fishing gear each year(Peltier et al. ). ©Observatoire Pelagis.

Figure . :

 . Figure . : Lexis diagrams for each monitoring. With the longitudinal monitoring, an individual born in is monitored through each year as it ages. With the cross-sectional monitoring, an individual is sampled in and its age (here years) is determined at the time of its death.

Figure

  Figure . : Example of Kaplan-Meier survivorship curve.

  Examples of survival models commonly used with associated distribution is provided in Figure.. For Realistic parametric model for longlived mammals = bathtub shape mortality or hazard rate curve.

Figure . :

 . Figure . : Survival parametric models commonly used. For each survival model and distribution, the cumulative survival (survivorship l(x), the hazard rate curve and the age-specific survival (S(x)) are depicted. a. Uniform distribution. Hazard mortality curve grows slowly and a little more at the end. b. Exponential distribution. Hazard mortality curve remains constant over lifetime. c. Weibull distribution. Hazard mortality curve increases after a short plateau at . d. Gompertz distribution. Hazard mortality curve increases after a long plateau near . e. Siler distribution. Hazard mortality curve is in U-shape or so called-Bathtub shape with a decrease, plateau and increase. Figure from Skalski, Ryding, and Millspaugh .

Figure

  Figure . : Biological scenarios considered in the Monte Carlo simulation study.: low juvenile mortality, a high adult mortality and no senescence. a: low juvenile mortality and a higher adult mortality plateau.b: high juvenile mortality and a lower adult mortality that decreases in a linear fashion.: high juvenile mortality, a lower adult mortality and senescence (bathtub shaped hazard).: high juvenile mortality and a high adult mortality, with a transient dip in mortality risk between these two life-stages.

:

  SI text and Figure S . : SI text and Figure S . : SI text .

  Figure . It consisted in a comprehensive factorial design crossing (a) sample size (100, 200, 300, 400, 500 and 1000), (b) mortality patterns (5 scenarios), (c) parametric models (3 models) and (d) individual covariate inclusion. Two covariates ( 1 , 2 ) were generated by sampling from a Bernoulli distribution with probability 0.5. These : Figure S and Figure S a.

  covariates could represent for example sex or two sub-populations in different geographic areas.

Figure . :

 . Figure . : Monte Carlo study flowchart. We considered five biological scenarios, six different sample sizes for data, and three models for analysis. For each combination, we assessed model fit and the accuracy of parameter estimates. Model selection is done with Watanabe-Akaike information criterion (WAIC) and root-mean-square error (RMSE).

:

  Figure S to S .

Figure . :

 . Figure . : Estimated hazard curves confidence intervals for each combination of scenarios (rows), models (columns), and sample size (facets). Estimated confidence intervals are depicted in each panel and compared to the true hazard curves (in black). Even-numbered scenarios ( a, b and ) are on the left, and odd-numbered ones ( and ) on the right.

Figure . :

 . Figure . : Estimated survivorship confidence intervals for each combination of scenarios (rows), models (columns) and sample size (facets). Estimated confidence intervals are depicted in each panel and compared to the truth (in black). Even-numbered scenarios ( a, b and ) are on the left, and odd-numbered ones ( and ) on the right.

Figure . :

 . Figure . : Estimated survivorship difference under each scenario. The covariate effect could correspond to that of sex (e.g., x = for females and x = for males). Ten estimated difference curves are depicted in each panel and compared with the truth.

Figure . :

 . Figure . : Survivorship curves estimated from published data sets. Each subpanel corresponds to a case studies in Table . Data set size is represented horizontally. Panels are associated to the following data sets: a. Murphy et al. b. Slooten c. Rodriguez-Caro et al. d. Murphy et al. e. Kesselring et al. f. Saavedra g. Barlow and Hohn .

Figure . :

 . Figure . : Estimated survivorship difference for both case studies.

NoDC.

  The demographic characteristics (e.g. size or age structure, sex distribution, fertility rates, survival rates) of the species' populations indicate a healthy population, unaffected by anthropogenic pressures Yes through the production of life-tables and vital rates using new crosssectional methodologies D C The range of the species and, where appropriate, their pattern of distribution within that range, is consistent with prevailing physiographic, geographic and climatic conditions No D C The habitat for the species has the necessary extent and condition to support the different stages in the life history of the species No Elements of marine food webs occur at normal abundance D C -Primary Diversity (species composition and relative abundance) of the trophic guild is not affected by anthropogenic pressures No D C -Primary: The balance of total abundance between trophic guilds is not affected by anthropogenic pressures No D C -Secondary The size distribution of individuals within a trophic guild is not affected by anthropogenic pressures No D C -Secondary The productivity of the trophic guild is not affected by anthropogenic pressures

Figure

  Figure . : Strandings records on the Atlantic French seashore recorded by the "Observatoire Pelagis" between and . The number of stranded dolphins has been increasing since with a record number of strandings in . In the project presented, the time series extends to part of . The whole of as well as and are presented to show the latest stranding estimates. Data from Observatoire Pelagis website:

Figure . :

 . Figure . : Diagram of the stratified random sampling. In this figure, the stratified random sampling method is illustrated. From the stranded population, each individual is assigned a value for the criteria of interest. These values are used to stratify the individuals. Within each stratum, random sampling is then carried out. The sample is obtained. The strata presented here and their associated criteria values are schematic. The number of strata is determined by the number of criteria values.

Figure

  Figure . : Map of all the stranded common dolphins records from to . It can be seen that all strandings are recorded on the Bay of Biscay but none are recorded in the English-Channel. Data from Observatoire Pelagis website: https: //www.observatoire-pelagis. cnrs.fr/?lang=en

Figure . :

 . Figure . : Main steps in stratified random sampling. This figure complements the diagram. It describes the important conceptual steps to recreate the sampling protocol.

Figure . :

 . Figure . : Growth layer groups (GLGs) in a common dolphin teeth. Years are depicted with numbers and neonatal line with "n". Photo from Murphy and Rogan ().

Figure . :

 . Figure . : Tooth sections from three individuals in the age-atdeath dataset. For each individual (a, b and c), Growth Layer Groups and accessory striae can be seen. The reading is done from the left or right side of the tooth. The age can be read on three areas for confirmation: the apex ( ), the middle ( ) and the base ( ).

Figure . :

 . Figure . : Histological sections of common dolphin gonads.The female gonads (ovaries) are indicated by the letter f. De Graaf follicles can be seen in the sections. The male gonads (testicles) are indicated by the letter m. m. : immature individual with small underdeveloped seminiferous tubules and a relatively high degree of interstitial tissue. m. : Mature but regressed testis. Small tubules and high amount of interstitial tissue. m. : mature in production with increased tubule size and few interstitial tissues. m. : higher enlargement of the mature tubule showing signs of spermatogenesis. Black scale bar = m

  Figure . : Comparisons of proportions of each criterion value for the stranding database and the two study datasets.The stranding database is shown in yellow. The age-at-death dataset in dark blue. The reproductive-status-at-death dataset in blue-green. Stratified random sampling allows for similar proportions between the age-at-death dataset and the stranding database, including cause of death, years, size and sex. Within the reproductive-status-at-death dataset, there is a large proportion of samples from the Biocet a program conducted in . There is also a greater proportion of females than males, contrary to what is seen in strandings. Individuals of -cm also seem to be over-represented. The causes of death are globally balanced but not representative of the stranding signal.

a

  EU-funded BIOCET project (Bioaccumulation of persistent organic pollutants in small cetaceans in European waters: transport pathways and impact on reproduction, EVK --). See Pierce et al. ( ) for a study within this program.

  Figure .: age-at-death data as an age pyramid by sex. The maximum age class is -years with some females of this age. The data seem to be characterised by a low representation of age groups -, -and sometimes -and -. Over-representation is also possible for ages to . For comparisons, see Chapter with two age distributions on the same population. In total, there are individuals.

Figure . :

 . Figure . : Distribution of reproductive status by age (regardless of sex). No immature individuals were founded after -years. Pubescent individuals are found from -years to -years of age with the largest proportion between and years. The first mature individuals are found at -years of age. For comparisons with other datasets from this population, see Chapter . In total, there are individuals.

Figure . :

 . Figure . : Conceptual model of the Female leslie matrix modelling approach.age classes are modelled. After age class , the females die, so there is no survival rate for this class. Before age class , the gestation rate is . No offspring are produced by females strictly under the age of .

Figure . :

 . Figure . : Survivorship estimation for each age-class on the two dataset types with Reed and Siler models. The estimation realised with the Siler model is displayed in black dashed line whereas the estimation using the Reed model is displayed in black solid line. For the dataset built using all the age-at-death available (Mannocci et al.), there is a difference in survivorship estimates. This difference is much less marked, or even almost non-existent, for the dataset constructed from the stratified-random-sampling.

Figure . :

 . Figure . : Survivorship estimated from the age-at-death dataset for each age-class. The blue dotted line corresponds to the observed survivorship estimated from the Kaplan-Meier estimator. The black dashed line is made from the Reed model. Uncertainty is displayed as the % uncertainty interval (i.e. Bayesian credibility interval).

Figure . :

 . Figure . : Proportion of mature females along age-class obtained through logistic regression. Black dots are the observed proportion for each age-class. The black solid line is the model fit. Data showed a null proportion of mature females from to years and a full proportion of mature females from to . From to , the proportion increases but the pattern in the data is noisy. Female Age at Sexual Maturity (ASM) is of . years (e.g. % of the population is estimated as mature).

  The proportion of mature females, estimated by the joint model using pubescent status as a reference, according to each age-class is highlighted in Figure . . For comparison, the proportion of mature females estimated by logistic regression from the proportions of mature females at each age is shown in Margin-Figure . . In Figure . , availability of data is displayed in rug line. Black rugs are associated with immature and mature individuals (both sexes). Orange rugs are associated with pubescent individuals (both sexes). Raw proportions of mature females are displayed in both Figure . (grey dots) and Margin-Figure .

Figure . :

 . Figure . : Proportion of mature females estimated with joint model for each age-class. The real data are displayed as grey dots and the estimated proportion done with the AFT Weibull approach is displayed as black dashed line. The % uncertainty around the mean estimated is displayed as a grey area.

  Figure . horizontal panel A. The percentage of change between the two estimated survivorship Bycatch, reference: Natural death Sex,

  Figure . : Percentage change between survivorship and fertility patterns for each covariate being tested.A: Effect of covariates on survivorship. The difference in survival between captured and uncaptured individuals increases with age. It is about % at years old and reaches % at years old. Females seem to survive slightly better than males at the end of their lives, particularly from the age of . Their survival is about % better at years. B: Effect of covariates on proportion of mature individuals. The proportion of mature individuals appears to be higher in the younger age classes among the individuals caught. At years of age, there are % more mature individuals. The females also seem to mature faster than the males. At years of age, females are % more mature than males.

Figure . :

 . Figure . : Cohort effect on proportion of mature females. The black dashed line corresponds to the mean proportion of mature females. Each coloured solid line of is a deviation around this mean (i.e. random effect) associated to a cohort. The cohorts share death as a common event. For example, the cohort includes all individuals who died in regardless of age. This cohort is associated with the earliest Age of Sexual Maturity. The latest Age of Sexual Maturity is estimated for the cohort.

  Figure . : Cohort effect on population growth fromto . The population model admits a exponential geometric growth. The lowest asymptotic growth rate is observed for the cohort whereas the highest is observed for the cohort.

  Figure . and Diagram . ): ) Define the need for representativeness by choosing relevant criteria and calculate the proportion of each in the stranded population ) Stratify by assigning each individual a value for each criterion ) Random sample individuals within each stratum to match proportions of the stranded population

Figure . :

 . Figure . : Death origin of the dolphins used in this study. The blue dots represent the age-at-death dataset. The red dots represent the reproductive-status-at-death dataset. The origin at death was obtained by using the MOTHY inverse drift model. The great majority of individuals died on the continental shelf of the Bay of Biscay.

Figure . :

 . Figure . : Stage-specific elasticities for mammal populations, grouped by age at maturity and ordered by increasing generation time. A. Age at first maturity = yr; no juvenile stage. B. Age at first maturity = yr; fertility elasticity = juvenile survival elasticity. C. Age at first maturity > yr. This groups correspond to species that exhibit a similar life history as the common dolphin. With delphinids in it and marine mammals. Fertility elasticities for those species are almost equals to the one that we founded for the common dolphin.Figure from Heppell, Caswell, and Crowder ().

Figure . :

 . Figure . : Ekman pumping. Horizontal divergence of the integrated Ekman transports gives rise to a vertical velocity at the base of the Ekman layer (i.e. "Ekman "pumping"): A. Cyclonic winds drive upwelling (Ekman suction). B. Anticyclonic winds drive downwelling (Ekman pumping).Figure from Colling (). The book Fieux () also provides information on this subject.

Figure . :

 . Figure . : Drawings of fish important to the feeding ecology of the common dolphin.

Figure . :

 . Figure . : Scheme of pelagic pair trawler net. Figure from https://sng.ie/ fishing/pelagic-old/.

Figure . :

 . Figure . : Diagram of trophic chain associated with common dolphin.

Figure . :

 . Figure . : Seasonal variation in the main hydrographic structures in the Bay of Biscay. With: ( ) winter warm slope current, ( ) swoddies, ( ) river plumes, ( ) cold waters, ( ) upwellings, ( ) warm waters of Bay of Biscay, ( ) slope current, ( ) tidal currents. Figure from Koutsikopoulos and Le Cann ().

Figure . :

 . Figure . : Study area in the North-East Atlantic ocean, with ICES divisions overlayed.

Figure . :

 . Figure . : Total weekly PTM fishing effort in Days at Sea for each year in division . .a. The raw data suggest significant inter-annual variation but a common pattern. Total effort appears to increase between week and week before declining.

Figure . :

 . Figure . : Diagram of bycatch risk per weeks. This diagram shows how sparse the data available on the collection by observers is.

Figure . :

 . Figure . : Diagram of stratification. This diagram shows the stratification of the data. The dots represent trawl hauls. Red dots are those with at least one bycatch event. The blue dots are those with no bycatch events. The lines are stratified by week, division and year.

Figure . :

 . Figure . : Diagram of the estimation process. This diagram is based on the example of the bycatch risk. It could also be based on the other two estimated parameters, which are: The Haul and the number of Hauls per Days at Sea. The raw bycatch risk data are logistically transformed (logit function) in order to place the estimate on a continuous scale and not between and . Each data point corresponds to a haul. A first estimate of the risk in one year (week to ) is modelled by a random walk of order .The first stratification is performed and the value of each risk profile for each year is estimated from a Gaussian process whose mean is the previous weekly risk profile. The second estimation then takes place and the value of the risk profile for each division within each year is estimated by another Gaussian process whose mean is the previous Gaussian profile.

Figure . :

 . Figure . : Diagram of the interaction between the equations in the estimation. The value of gives the intercept for the weekly risk profile.The vector corresponding to the weekly risk profile. This vector is the mean estimate of the first Gaussian process for each year . The estimate provides a risk profile for each year . From this annual vector, a final Gaussian process takes it as an average to estimate the profile of each division within each year :. The values of this vector are then used to inform the estimation of the bycatch parameters.

Figure . :

 . Figure . : Diagram of the deviation when estimating the profile of the parameter of interest. What causes the profile to deviate is the variance due to the year effect ( year ) and the variance due to the division effect ( zone ). These two variances are included in the variance covariance matrices Δ.

Figure . :

 . Figure . : Diagram of autocorrelation independance. The omegacovariance function admits independence after weeks.

  -) 2 where ( -) = | -| is the temporal distance (in weeks) between weeks and . The choice of the range parameter induces a temporal correlation of 0.05 after 4 weeks (Margin-Figure . and Figure . ; Authier, Rouby, and Macleod ).

Figure . :

 . Figure . : Graphical representation of the assumed correlation matrix Ω. The correlation is equal to after weeks.

Figure . :

 . Figure . : Directed Acyclic Graph of the methodology.The observed data are the basis of the statistical inference. Each parameter can be a hyperparameter of the others. For example, is a hyperparameter of the whole process while is a simple parameter. Each parameter, or hyperparameter, is estimated from the previous one and a small deviation induced by stratification.

.

  Figure . : Model-based estimates (posterior medians) regardless of division. Estimates were made for bycatch risk, haul duration and number of hauls per DaS of pair-trawlers flying the French flag operating in the study area. Each colour represents a different year (parameters in Eq. . ) and the dotted black line the yearly average (parameters in Eq. . ).

Figure . :

 . Figure . : Temporal patterns for division . .a. The temporal pattern is displayed for variables: haul duration (hours), bycatch events (n), bycatch risk per haul (probability), common dolphins bycaught (n).The estimated haul duration is the highest from week to . The lowest at week and then it remains stable from to the end. The estimated bycatch risk is highest between weeks and and is highest for years beyond .

Figure

  Figure . : Temporal patterns for division . .b. The temporal pattern is displayed for variables: haul duration (hours), bycatch events (n), bycatch risk per haul (probability), common dolphins bycaught (n).The estimated haul duration is the highest from week to . The lowest at week and then it remains stable from to the end. The estimated bycatch risk is highest between weeks and and is highest for years beyond and the year .

Figure . :

 . Figure . : Comparison of model-based (orange solid line) and stranding-based estimates (dark violet solid line) of common dolphin bycatch. Upper panel: raw estimates with uncertainty intervals ( % for model-based estimates and % for stranding-based estimates); middle panel: standardized (mean centered and unit variance) estimates. Bottom panel: year-level and month-level (within each year) correlation were computed. At the within year (between month level), correlations were computed with and without a two-weeks lag. Change in the magnitude of the Pearson correlation coefficient is shown with an arrow.

Figure

  Figure . : Simple diagram of bycatch phenomenon components.

Figure . :

 . Figure . : Photo of Pinger used in Bay of Biscay. ©Parc marin d'Iroise.

Figure . :

 . Figure . : Flowchart of the whole MSE and Application case approaches. MSE runs foryears and allows the selection of a quantile. The application case runs for year using real historical data and allows to obtain a distribution of removals. The quantile selected in the previous MSE framework is applied on the distribution to select a removal value.

Figure

  Figure . : Classical conservation approach. The classical approach involves steps (A. J. Read). This approach is not proactive and may lead to viability problems for dolphin populations(Wade ).

A

  Conservation and management objectives should not be confused. A conservation objective is a goal to be achieved. It does not change. A management objective is a means to achieve the conservation objective. It is adaptable and can be reevaluated. management strategy is an agreed-upon set of rules for determining thresholds beyond which a CO runs the risk of not being met with unacceptably high probability (Winship ; Punt ; Bunnefeld, Hoshino, and Milner-Gulland ; I. C.Kaplan et al.

  Figure . : Age-specific survival rates obtained taking into account the constant effect on survival.

  Figure . : Remaining population overyears with constant effect on survival. The constant effect was applied to found the value for which the population remains stationary. With a survival intercept to which is added a constant effect equal to . , the population remains stationary over time. This effect is then used in the estimation of the survival model to obtain the age-specific survival rates of a stationary population which will be used to inform the Pella-Tomlinson population dynamics model.

Figure . :

 . Figure . : Diagram of the Management Strategy Evaluation associated with common dolphin population. The diagram shows both process (here blocks) that make up the Management Strategy Evaluation. The first is the operating model. This process involves a population dynamic that runs every year for years. The second is monitoring in relation to the definition of the management strategy. Abundance monitoring is carried out every years and bycatch monitoring every year. For each -year cycle a new bycatch limit is defined.

Figure . :

 . Figure . : Diagram of tuning. The frequency of removals thresholds is depicted as a posterior distribution.The choice of quantile affects the definition of the management objective. Given the posterior distribution of RLA t , a quantile value of % involves a tolerable bycatch threshold of individuals per year. A quantile value of % involves a tolerable bycatch threshold of individuals per year. The choice of the quantile value therefore defines the management objective. This choice is made in order to achieve the CO.

  Figure . : Survey platform and survey blocks of abundance surveys campaigns. SCANS I is not shown here as it was not used to obtain the abundances. Between campaigns, the blocks sampled are different. A recalculation of the abundances was necessary. This calculation was carried out within the ICES divisions. It took into account the density obtained in each block and the surface of the ICES division.

Figure . :

 . Figure . : Common dolphin abundance used for this study depending on the ICES divisions. Abundances were recalculated using both ICES divisions specific surface and the estimated density in the SCANS II + CODA and SCANS III block.

Figure . :

 . Figure . : Base case scenario abundance and removals over years associated with quantiles. Only quantiles . , . , and. are depicted here. These are the most interesting to look at because it is at this point that the % threshold defined for K is reached. All quantiles below . meet the objective after years. Here it is the quantile at . that should be retained. Because it is the one that best fits the conservation objective. It is associated with annual removals that fluctuate around at the end of the -year simulation.

Figure

  Figure .: Robustness trials depletion levels for each quantile. When we start the simulation, we do not know the depletion state of the population with respect to K. To cover this uncertainty, initial depletion levels are simulated for each scenario and quantile. In the base case, the quantile to be retained is . . In the case of the underestimation of abundance by a factor of , the quantile to be retained is . . In the case of an underestimation of the bycatch by a factor of , the quantile to be retained is . . The absence of curves for the last quantiles of the robustness trials is due to numerical problems.

  Figure . : Final removals posterior distribution. The posterior distribution obtained in the Real Application case is depicted here. The maximum removal value is about .The minimum is about . The quantiles that were obtained previously in the MSE approach involving the robustness trials allows to select removal value in the distribution.

  Figure . : Cause of death for .

Figure . :

 . Figure . : Diagram of qualitative management strategy. Rather than consider a quantitative management strategy, strategies may be defined taking into account several socio-economic variables. A. Strategy which is more concerned with the sustainability of the fishing economy. B. Strategy which is more concerned with the viability of dolphin and fish populations.

Figure . :

 . Figure . : Period of activity of the networks that provided the starting date of their activity and the perceived stability of effort over time (as indicated by the respondents of the WGMME questionnaire). Very few European countries exhibit a unique and long-term stranding network. Island, United-Kingdom, Belgium and France have the most structured stranding networks. Figure from ICES ( a).

Figure . :

 . Figure . : Relative percentage difference of cohort vital rates around the mean profile from -. Relative percentage difference in survivorship. From, the survivorship of adults seems reduced compared to the mean survivorship over the period.

Figure . :

 . Figure . : Diagram of Atlantis model. Models blocks are depicted such as relation between blocks. The model takes into account the whole ecosystem from the oceanography to the top of marine foodweb and human activities.Figure from https://niwa.co.nz/

  Figure . : Diagram of Atlantis model. Models blocks are depicted such as relation between blocks. The model takes into account the whole ecosystem from the oceanography to the top of marine foodweb and human activities.Figure from https://niwa.co.nz/ ecosystem-modelling-at-niwa/ atlantis-ecosystem-model.

Figure . :

 . Figure . : Possible management units for the North-East Atlantic common dolphin. The current management unit recognises a panmictic population by taking into account the Bay of Biscay, Iberian coast, Celtic seas and North sea. This management unit could be divided into three separate units. It could also be extended to a larger scale, including much of the North Atlantic. Figures from Perrin ( ), ICES ( a) and Caurant et al. ( ).

Figure . :

 . Figure . : Common dolphin sightings from the SAMM surveys. SAMM surveys were conducted in Winter (November th to February th ) and Summer (May th to August th). There were more common dolphins observed in summer on the oceanic part of the Bay of Biscay than in winter. Figure from(Laran et al. )

Figure

  Figure . : Marine biology Stommel Diagram. Emphasizes factors in marine biology, with an emphasis on biological productivity, here labeled "biomass variability". Figure from Haury, McGowan, and Wiebe ( )

Figure

  Figure . : Rough Stommel diagramfor common dolphin. This raw diagram presents processes associated with the population at three spatiotemporal scales. The population can be resident of the Oceanic or Shelf parts for about two months. Individuals may migrate within the Bay of Biscay each year. The population may change its distribution on a larger spatial and temporal scale later. This diagram is only a rough and conceptual view and should not be considered as completed.

Figure . :

 . Figure . : Density of Harbour porpoise estimated from the SCANS and CODA campaigns. The time period under consideration ranges from to with main estimates. The density of harbour porpoises in the North-East Atlantic appears to have possibly changed over the period. The population now seems to be denser in the Western Channel, Celtic seas and Bay of Biscay than before. Figures from Hammond et al. ( , ).

Figure . :

 . Figure . : Abundance estimates of common dolphin by ICES divisions. The abundance estimates seems to be slightly higher for SCANS III than for SCANS II + CODA. Estimates from Hammond et al. ().

Figure . :

 . Figure . : Correlation between common dolphin abundance and ecosystem variables. Abundance estimates predicted by the baseline spatio-temporal model (black line) and by the covariates-based model (with no random effects, colored line) so that the contribution made by each variable (A-F) can be visualized. Figure from Astarloa et al. ().

  Figure .: Percent of stocks within four ranges of statistical power. Statistical power is the probability of correctly rejecting the null hypothesis that a population is not declining (i.e., one-tailed t-test) when the stock is experiencing a precipitous decline ( % over yr). Results are summarized for six categories of marine mammals: large whales, beaked whales, dolphins and porpoises, pinnipeds on ice, pinnipeds on land, and polar bears and sea otters. Figure from (B. L. Taylor et al.).

Figure . :

 . Figure . : Aerial photograph of a purse-seine set on a school of tuna and dolphins. The purse-seine vessel is deploying the net in a large circle around the entire school while a skiff holds the end of the net in place. In this photograph the net is not yet closed; four speedboats are driving in tight circles near the opening to prevent the dolphins (and tuna) from escaping. Photo from Ballance et al. ().

Figure . :

 . Figure . : Estimated number of dolphins killed annually in the eastern tropical Pacific tuna purse-seine fishery. Total for all dolphins and separately for the stocks of the two dolphin species with the highest number killed. The inset graph has an expanded vertical scale to show details from to (change of scale on the Y-axis). Figure from Ballance et al. ().

Figure . :

 . Figure . : raphical representation of predicted changes in optimal reproductive effort (RE) of a generalized life history. RE is defined as the proportion of resources allocated to reproduction as opposed to investment in growth or survival. The age at which RE increases from zero indicates age at maturity. The linearly increasing depiction of RE with age is arbitrary, and plot A is the baseline iteroparous life history. Plots B-D give predictions for unregulated populations growing exponentially after: (B) mortality increases only for older age classes (earlier age at maturity and higher RE at each age), (C) mortality increases only for younger age classes (later age at maturity and lower RE at each age), and (D) mortality increases uniformly across all ages (no change from baseline). Plots E and F give predictions for regulated populations subjected to density dependence: the effect of a uniform increase in mortality when density dependence affects (E) only juveniles (note the increase in RE over baseline and earlier age at maturity) vs. (F) all ages (no change from baseline).Figure from Reznick, Bryant, and Bashey ().

Figure . :

 . Figure . : Cumulative number of studies published prior to . . containing animal matrix population models (MPMs). Some Important events in the development of animal MPMs: (a, b) applications of matrix population models in demography, (f) nonlinear, density-dependent MPMs for animals, (g) sensitivity analysis for stage-classified MPMs and calculation of selection gradients for animals, (i) calculation of the stochastic growth rate from an animal MPM, (k) application of elasticity analysis to conservation biology and Life Table Response Experiment analysis, (n) presentation of multistate mark-recapture methods for estimating stage-structured MPMs in animals, (o) development of MPM from photograph identification data. Figure from Salguero-Gómez et al. ().

Figure . :

 . Figure . : Diagram of the proposed framework along with the example in our case. The steps are defined to allow for a better consideration of representativeness bias. This framework is primarily a point of discussion and not definitive.

Figure . :

 . Figure . : Age-frequency distribution and sample sizes of female and male harbor porpoises sampled within the North Sea MU and Celtic and Irish seas MU during the two time periods, -and -. These data could be analysed with our joint modelling approach. This would highlight possible differences in vital rates values depending on area and/or sex. Figure from Murphy et al. ( )

Figure

  Figure . : Map showing Recommended Management Units for Harbour Porpoise in the ASCOBANS Agreement Area and Environs. Figure from Evans and Teilmann ().

Figure . :

 . Figure . : Hector's dolphin draw and distribution. Hector's dolphins are found along the east coast of the South Island. They are found down the West Coast to Jackson Bay and Fiordland. On the south coast, two bays east and west of Invercargill have Hector's dolphins. Draw from https:// www.fisheries.noaa.gov/ species/hectors-dolphin.Distribution fromhttps: //teara.govt.nz/en/map/4675/ distribution-of-hectors-and-mauis-dolphins.

(

  pollution disruption)(Murphy et al.

Figure :

 : Figure : Estimated abundances of common dolphin and stripped dolphin for the SCANS II, CODA, SAMM and SCANS III surveys. Abundances are provided by survey cruise, by block and by species.Table formatted by Olivier Van Canneyt, Observatoire Pelagis.

Figure :

 : Figure : Marine regions as identified by the European Marine Strategy Framework Directive. Figure from http: //www.eea.europa.eu/about-us/countries-and-eionet/marine-regions.

  2. It consisted in a comprehensive factorial design crossing (a) sample size (100, 200, 300, 400, 500 and T A B L E 1 Design of the Monte Carlo study: model ℳ 2 acts as a negative control as it was never used to simulate data, hence it should not be selected as the best model as it is misspecified

F

  I G U R E 2 Monte Carlo study flowchart. We considered five biological scenarios, six different sample sizes for data, and three models for analysis. For each combination, we assessed model fit and the accuracy of parameter estimates. Model selection is done with Watanabe-Akaike information criterion (WAIC) and root-mean-square error (RMSE) [Color figure can be viewed at wileyonlinelibrary.com] 1000), (b) mortality patterns (five scenarios), (c) parametric models (three models) and (d) individual covariate inclusion. Two covariates (x 1 ,

  Data sets were selected based on the availability of published (raw) data, species and sample size.

F

  I G U R E 3 Models parameters convergence assessment for each sampling size with n(warm-up) = 1000 iterations and a total of N = 2000 iterations [Color figure can be viewed at wileyonlinelibrary.com]

F

  I G U R E 4 Estimated hazard curves confidence intervals for each combination of scenarios (rows), models (columns), and sample size (facets). Estimated confidence intervals are depicted in each panel and compared to the true hazard curves (in black). Even-numbered scenarios (2a, 2b and 4) are on the left, and odd-numbered ones (1 and 3) on the right [Color figure can be viewed at wileyonlinelibrary.com]

F

  I G U R E 5 Estimated survivorship confidence intervals for each combination of scenarios (rows), models (columns) and sample size (facets). Estimated confidence intervals are depicted in each panel and compared to the truth (in black). Even-numbered scenarios (2a, 2b and 4) are on the left, and odd-numbered ones (1 and 3) on the right [Color figure can be viewed at wileyonlinelibrary.com] et al.

F

  I G U R E 7 Survivorship curves estimated from published data sets. Each subpanel corresponds to a case studies in Table 2. Data set size is represented horizontally. Panels are associated to the following data sets: (a) Murphy et al., (2012); (b) Slooten, (1991); (c) Rodriguez-Caro et al., (2019); (d) Murphy et al., (2009); (e) Kesserling et al., (2018); (f) Saavedra, (2018); (g) Barlow and Hohn, (1984) [Color figure can be viewed at wileyonlinelibrary.com]

Fig. 1 .

 1 Fig. 1. Study area showing bathymetry and geographic references mentioned in the text, and location of common dolphin stranded carcasses used for the study (red dots)

Fig. 2 .

 2 Fig. 2. Spatialized mortality index (MI) for March 2018. MI was computed from reverse drift modelling of stranded carcasses of common dolphins showing bycatch evidence. It allows us to locate potential mortality areas and to quantify the intensity of mortality events

199Fig. 4 .

 4 Fig. 4. Stranding probability in March 2018, estimated from simulated drifts of common dolphins, then smoothed with conditional autoregressive (CAR) model to account for spatial dependency between adjacent cells

Fig. 5 .

 5 Fig. 5. Means estimates from the 12 separated monthly models of intercepts and coefficients of oceanographic covariates, for each month and each year. Spatiotemporal monthly models were inferred with integrated nested Laplace approximations (INLA). eke: eddy kinetic energy; mean_sst_grad: mean sea surface temperature; sst: sea surface temperature

203Fig. 6 .

 6 Fig. 6. For each of the 12 spatiotemporal monthly models inferred with integrated nested Laplace approximations (INLA), proportion (transformed with a square root) of variance accounted for by each model component. Residual: residual component; Spatial: spatial component (defined by a Gaussian random field); Annual: annual intercepts and coefficients of covariates

FIGURE 5 |

 5 FIGURE 5 | Estimated temporal pattern in mean bycatch risk from the model-based approach. Left: data sampling was unbiased. Middle: Under-sampling. Right:Over-sampling. The model-based approach recovered the correct pattern overall, but overestimated risk in the over-sampling scenarios when risk was, in fact, nil but no data were collected.

FIGURE 6 |

 6 FIGURE 6 | Box plots of bias (in number of estimated bycatch events compared to the truth) in the weekly model-based estimates of bycatch events. Left: data sampling was unbiased. Middle: Under-sampling. Right: Over-sampling. Each row corresponds to data simulated for a different number of years.

FIGURE 1 |

 1 FIGURE 1 | Study area in the North-East Atlantic ocean, with ICES divisions overlayed.

  and λ jkt = e α 3

  The vector α jk are division-specific deviations from the mean yearly pattern δ k . Smoothness in α jk and δ k is controlled via the covariance matrices division = division division and year = year year .Matrices . have dimensions n week ×n week (53×53). These covariance matrices are decomposed into a product of a diagonal matrix . (of dimension 53 × 53) with the common scale parameter on the diagonal, and a correlation matrix (of dimension 53 × 53; Chen and Dunson, 2003): with the correlation between week t and week t ′ of dimensions n week × n week (53 × 53). A Matérn correlation function of order ν = 3

Figure 2 .

 2 Equations 4 and 5 allow modelling an

FIGURE 2 |

 2 FIGURE 2 | Graphical representation of the assumed correlation matrix .

  jkt

FIGURE 3 |

 3 FIGURE 3 | Model-based estimates (posterior medians) of bycatch risk, haul duration, and number of hauls per Days at Sea (DaS) of pair-trawlers (PTM) flying the French flag operating in the study area. Each colour represents a different year (parameters αjkt in Equation4) and the dotted black line the yearly average (parameters ǫt in Equation4).

FIGURE 4 |

 4 FIGURE 4 | Comparison of model-based (orange solid line) and stranding-based estimates (dark violet solid line) of common dolphin bycatch. Upper: Raw estimates with uncertainty intervals (80% for model-based estimates and 95% for stranding-based estimates); Middle: standardised (mean centred and unit variance) estimates. Bottom: Year-level and month-level (within each year) correlations were computed. At the within year (between month level), correlations were computed with and without 2-weeks lag. Change in the magnitude of the Pearson correlation coefficient is shown with an arrow.

Table . : Examples of population dynamic models. Other exists but these are the main used in the project.

 .

	Model			Equation	Reference	Chapter
	Exponential			+1 =		Malthus, Winch, and P. James	Chapter in .
	Logistic	+1 =	+	× (	-1) 1 -	Hutchinson	-
	Generalized logistic	+1 =	+	× (	-1) 1 -	Pella and Tomlinson	Chapter in .

Table . : Pros and cons of both monitoring schemes.

 .

	Monitoring Strengths	Weaknesses		Key reference
		Underlying processes are
		highlighted	Costly	monitoring	Lebreton et
		Individual differences to	scheme		al.
	Longitudinal	be highlighted	May involve small sam-	Sandercock
		Population growth rate	ples
		are estimated directly
			Does not highlight the
			changes an individual
		Include large parts of the	undergoes over the
		population	course of its life	Caughley
	Cross-	Allows to apply simple	Low representativeness
	Sectional	growth models	of the data if the time se-	Deevey Jr
		Not very costly com-	ries is short
		pared to longitudinal	Assumes the population
			to be at stationary age dis-
			tribution

Table . : World common dolphin populations features. This

 . table summarises some important current knowledge about the global populations of common dolphins. Table from Murphy et al. ( ).

	Area	Climate	Sample pe-riod	Mating period	Age maturity sexual	at	Dataset
				May-September
	North-East Atlantic	Temperate	-		. years		Murphy et al. (	)
				(spring-summer)
				July-August (sum-			A. J. Westgate and A. J. Read
	North-West Atlantic	Temperate	-		. years
				mer)			(	)
				Calve all year
	Tropical-East Pacific	Tropical	-		. years		Danil and Chivers (	)
				round
	North Pacific	Temperate	-	May-June (spring)	years		Ferrero and Walker (	)
				Primarily austral
	New Zealand	Temperate	-		NA		Institute of Zoology (	)
				summer
	South Africa	Temperate	-	Austral summer	-years		Mendolia (	)

Figure . : Age at death and vital rates data available for the popula- tion. Three

	types of biodemographic
	parameters are presented. The first
	line is age-at-death. The second is sex-
	ual maturity profiles of females. The
	third is the estimated survivorship of
	the population which is only avail-
	able for study A.
	Column A corresponds to the study
	Mannocci et al. (	).
	Column B corresponds to the study
	Murphy et al. (	).
	Figures are extracted from these stud-
	ies.

Table . : Summary of tools, legislative instruments and organisations relevant to the conservation status of the North- East Atlantic common dolphin population.

 .

	Instrument/Tool/Organization		Type	Year	Application scale
				Non Gou-
	IUCN -International Union for Conservation of	verne-mental		Worldwide
	Nature Red List of Threatened Species
				Organiza-
				tion
				Intergovernmental
	ICES -The International Council for the Explo-	marine		North-
	ration of the Sea			science or-		Atlantic
				ganization
	CITES -Convention on International Trade in	International		Worldwide
	Endangered Species of Wild Fauna and Flora	agreement
	ASCOBANS -Agreement on the Conservation of	International		European
	Small Cetaceans of the Baltic, North East Atlantic,
				agreement		waters
	Irish and North Seas
	HD -Habitats Directive			European Directive		European waters
	OSPAR -Convention for the Protection of the Ma-	International legislative		European
	rine Environment of the North-East (Oslo-Paris)			waters
				instrument
	Bycatch Regulation no.	/	a	European regulation		European waters
	MSFD -Marine Strategy Framework Directive	European Directive		European waters
				Fisheries
	CFP -Common Fisheries Policy		policy of the European		European waters
				Union
	Bycatch Regulation no.	/		European regulation		European waters

a repealled in Figure . : SCANS-

III and Ob- SERVE surveys areas that approxi- mate the North East Atlantic Assess- ment

Unit. OSPAR Regions depict the AU proposed by ICES ( ). Figure from ICES ( a).

  several types of models, more or less complex. Here is a brief review of them. The first type of survival model is the single-age model. It considers a population in which individuals are considered to have a single age. The model traditionally used to estimate survival in this case is the Cormack-Jolly-Seber (CJS) of survival that can be estimated is apparent survival. The multiple-age model is an extension of the single-age model that allows capture and survival probabilities to vary with time and age. The model that generalises the single-age model to multiple-age modelling was developed by Pollock

				Longitudinal survival analysis mod-
				els can be distinguished in this
				way: single-age model, multiple-age
				model, multi-state model and inverse-
				time model.
	model (Seber	; Cormack	; Jolly	). With this model,
	the only component

are

Table . : Design of the Monte Carlo study. Model M 2 acts as

 .

	Our framework transcribed by the equation .	a flexible
	location-scale model and boils down to a generalized linear mixed
	model (GLMM; Bolker et al.
		a negative
		control as it was never used to sim-
		ulate data. Scenarios a, b and
		corresponds to data generated from
		mixture models not included in the
		set. They serve as tests of the perfor-
		mance of our framework. Scenarios
		and act as positive control as they
		correspond to data simulated under
		models M 1 and M 3 respectively.

Table . : Summary of the MSFD descriptors and how they relate to the project. Each

 .

			descriptor is summarised. For each,
	its relevance to marine mammals is addressed. As well as the associated demographic need. For a more precise description
	of the descriptors see M. B. Santos and Pierce (	) or European Parliament (	).
				Need for marine mam-
	Descriptor	Relevance for marine mammals	mals demographic infor-
				mation
		Marine mammals is an important functional
	. Maintain biological diversity	group as marine megafauna and sometimes key-	Yes
		stone species
	. Non-indigenous species	Not addressed		No
	. Populations of commercial fish/shellfish bio-	Not addressed		No
	logically safe
	. Elements of marine food webs occur at normal abundance	All marine mammals in european waters are top predators	Yes
	. Human-induced eutrophication is minimised Not addressed		No
	. Sea-floor integrity maintained	Not addressed		No
	. Hydrographical conditions not unfavourable	Not addressed		No
	. Concentrations of contaminants	Due to their trophic position, marine mammals bioaccumulate pollutants	No
	. Contaminants in fish and other seafood	Not addressed		No
	. Properties and quantities of marine litter do	Ingestion of plastics and entanglement in debris No
	not cause harm
	. Introduction of energy, including underwater	Underwater noise can have a range of effects on	No
	noise, non-disruptive	marine mammals

Table to

 to

	find out
	about surveys by campaign and by
	season.

Table . : Summary of descriptor criteria within which the production of demographic state indicators is required. See European Parliament (

 .

Table . : Characteristics of the dif- ferent reproductive stages.

 .

	Ovarian features	Reproductive state
	No mature follicle	Immature
	De Graaf follicle and no corpus albicans	Pubscent
	De Graaf follicle and one or some corpus albicans Resting mature
	corpus luteum with or without a fetus	Pregnant
	corpus luteum and productive mammary glands	Mature pregnant lactat-ing

Table . :

 . Models specification.Model and are not joint models. Simple: a null effect, Cov: Covariate effect, Years: Random effect (joint modelling on Years), Cor: Correlation between random effects, Trend: Trend effect on random effects. WAIC and LOOIC are depicted in Appendix Table .

	Model	Specification	Joint modelling
		simple	No
		cov	No
		years	Yes
		years + cor	Yes
		cov + years	Yes
		cov + years + cor	Yes
		cov + years + trend	Yes
		cov + years + trend + cor	Yes

  -since one of the mature females of this age was pregnant. It is equal to . for -since of the mature females of this age were pregnant. It is equal to . for -since of the mature females of this age were pregnant. And It is equal to . for -since of the mature females of this age were pregnant. Fertility rate ( ) at age . The proportion of mature females (estimated with the pubescent status in the AFT modelling framework) is multiplied by the gestation rate (divided by two assuming a sex ratio at birth of : ) and multiplied by the age-specific survival rate considering the transition to the next age -1 :

	Reproductive status
	Lactating Mature Pregnant Mature Pregnant Mature Lactating Resting Mature
	Figure . : Reproductive status of
	females given four age-class inter-
	val. The total number of female used
	is about . For a comparison with
	other dataset, see Chapter .

Gestation rate ( ) at age . The gestation rate was obtained considering age-class intervals ( -, -, -and -) as depicted in Figure

.

. Gestation rate before years is equal to since no female was pregnant before years. It is equal to . for

Table . : Descriptive statistics for Obsmer and SACROIS data displayed for each year.

 . The bycatch events observed seem to be recurrent since . The year associated with the highest level of observed bycatch is with dolphins. Total observer coverage of fishing effort is rarely more than %. The maximum PTM fishing effort covered in Days at Sea is . % in .

	Dataset				Obsmer				SACROIS
	Year	Hauls	Average	Bycatch	Median	Max. nb	DaS (Cover-	Total Effort
			Dura-	events	nb	of	of dol-	age %)	(DaS)
			tion		dol-		phins
			(hours)		phins
		4	2.80	0	-		-	4 (0.0)	8
		5	4.26	0	-		-	4 (0.0)	8
		122	4.62	0	-		-	90 (1.1)	7
		727	3.89	6	1.5		5	401 (6.4)	6
		554	4.81	6	1.5		4	328 (10.9)	3
		464	5.50	20	2		50	326 (7.4)	4
		305	3.52	1	4		4	159 (3.5)	4
		173	3.99	2	3		3	86 (2.1)	4
		210	3.58	4	4		8	96 (2.4)	4
		128	3.81	2	5.5		9	75 (1.8)	4
		114	4.44	0	-		-	78 (1.9)	4
		136	2.77	1	2		2	78 (1.7)	4
		156	4.75	5	3		10	106 (2.3)	4
		196	5.23	12	2		20	124 (2.6)	4
		184	3.85	1	1		1	102 (2.8)	3
		438	5.45	11	2		8	289 (7.4)	3
		123	3.69	2	2		3	70 (4.0)	1
								Model	likelihood	Specification
									for duration
								M6	gamma	ICES division × week × year
								M5	log-normal	ICES division × week × year
								M4	gamma	ICES division + week × year
								M3	log-normal	ICES division + week × year
								M2	gamma	ICES division + week + year
								M1	log-normal	ICES division + week + year

Table . : Models specification.

 .

Table . : Dummy dataset to illus- trate the typical sparseness in Ob- smer data.

 .

Table . :

 . Model

	-based estimates of
	common dolphin bycaught in PTM
	in the study area. Divisions .j and
	.d are labelled 'Oceanic', divisions
	.defgh are labelled 'Neritic ' and
	divisions .abc are labelled 'Neritic
	'. Estimates (posterior median) are
	reported with the lower and upper
	bound of a 80% credibility interval
	(Louis and Zeger	)

  . A simple way to increase precision is to include Electronic Remote Monitoring observations in Equations . and . . Doing so would result in increased precision as a hauls (and possibly DaS) would be monitored. Even if the estimate precision is low, it may provide a sufficient precision to inform management procedures algorithms conjointly to other data sources and biological indicators.

	: http://paris.
	tribunal-administratif.fr/
	content/download/172866/
	1715763/version/1/file/1901535.

pdf greater number of

Table . : Difference between the threshold

 .

		of	removals and es-
	timated bycaught dolphins. Model
	based estimates corresponds to the
	Chapter	estimates with the
	Bayesian Credibility Interval. Cells
	are depicted in red if the difference is
	in favour of an excess of the bycatch
	threshold. Model based estimates are
	made taking into account PTMs.

  Tech. rep. International Council for the Exploration of the Sea.

						: https:
	//doi.org/10.17895/ices.pub.5563 (cited on pages ,	,	,	,	).
	-(	a). ICES Fisheries Overviews, Bay of Biscay and Iberian Coast ecoregion. Tech. rep. (cited on
	page	).
	-(	b). Road map for ICES bycatch advice on protected, endangered, and threatened species. Tech. rep.,
	section . (cited on pages	,	).
	-(	c). Workshop on fisheries Emergency Measures to minimize BYCatch of short-beaked common
	dolphins in the Bay of Biscay and harbour porpoise in the Baltic Sea (WKEMBYC). Research rep. : .
	pp. International Council for the Exploration of the Sea.	: http://doi.org/10.17895/
	ices.pub.7472 (cited on pages	,	,	,	).
	-(	a). ICES Technical Guidelines, ecosystem overview. Tech. rep. (cited on page	).
	-(

Report of the Working Group on Bycatch of Protected Species (WGBYC), p. (cited on page ). -( ). Working Group on Bycatch of Protected Species (WGBYC). ICES Scientific Reports. : . pp. Faro, Portugal. b). Working Group on Marine Mammal Ecology (WGMME). Tech. rep., pp. -(cited on pages , , ).

  Table formatted by Olivier Van Canneyt, Observatoire Pelagis.

Table 2

 2

	are

Table :

 : Model selection. Models are ordered in increasing order of WAIC (the smaller, the better the fit). stands for 'standard error'.

	Model	Specification	Joint modelling WAIC	LOOIC
		cov + years + trend + cor	Yes	733 13	1169 16
		cov + years + trend	Yes	730 13	1150 16
		cov + years + cor	Yes	782 13	1195 16
		cov + years	Yes	780 13	1194 16
		years + cor	Yes	781 13	1192 16
		years	Yes	777 13	1190 16
		cov	No	922 14	1307 17
		simple	No	925 14	1318 17

Table 1 .

 1 Root mean squared error (RMSE) for cross-validation (CV) and repetition scenarios (RS)

	Grey dotted lines: 95%CI of fitted values

TABLE 2 |

 2 Statistical properties of estimates from the model-based approach.Several years of data were simulated a 100 times. Bias of point estimate (in number of bycatch events), coverage of (Bayesian 80%) credible interval (CI) and precision (as CI width relative to the point estimate) are reported. The last column indicates the average number of bycatch events (nobs = E ijkl yijkl ) that were recorded by onboard observers during data sampling.

	Method	Uncertainty	Data	nyears	Bias	Coverage	Width of CI	nobs
			sampling		(bycatch events)	(%)	(%)
	Model-based	Bayesian	Unbiased	3	3.0	100.0	91.1	45
	Model-based	Bayesian	Unbiased	5	2.1	100.0	76.3	75
	Model-based	Bayesian	Unbiased	10	1.1	100.0	59.1	150
	Model-based	Bayesian	Unbiased	15	1.9	100.0	50.9	225
	Model-based	Bayesian	Under-	3	10.0	100.0	164.6	15
	Model-based	Bayesian	Under-	5	6.4	100.0	142.0	25
	Model-based	Bayesian	Under-	10	8.3	100.0	112.9	50
	Model-based	Bayesian	Under-	15	5.3	100.0	97.8	75
	Model-based	Bayesian	Over-	3	7.4	100.0	53.2	63
	Model-based	Bayesian	Over-	5	4.8	100.0	42.6	126
	Model-based	Bayesian	Over-	10	3.5	100.0	32.6	630
	Model-based	Bayesian	Over-	15	3.3	100.0	27.7	756

  1,3,7 * 1 Université de La Rochelle, La Rochelle, France, 2 UMR7372 Centre d'Etudes Biologiques de Chizé (CEBC), Villiers En Bois, France, 3 Observatoire PELAGIS, UMS3462, CNRS-La Rochelle Université, La Rochelle, France, 4 Ifremer, Laboratoire Ressources Halieutiques de Port en Bessin, Port-en-Bessin-Huppain, France, 5 Ifremer, Laboratoire de Biologie Halieutique de Brest, Brest, France, 6 Joint Nature Conservation Committee, Aberdeen, United Kingdom, 7 ADERA, Pessac, France

TABLE 1 |

 1 List of acronyms. fisheries Emergency Measures to minimise BYCatch of short-beaked common dolphins in the Bay of Biscay and harbor porpoise in the Baltic Sea which strandings of common dolphins with evidence of bycatch have increased in recent years (ICES, 2020d), could have been implemented as emergency measures under the provisions of the Common Fisheries Policy. For 2021, France instead required the mandatory use of acoustic repulsive devices (pingers) on all pair trawlers flying the French Flag (code métier Pair trawlers and hereafter referred to as PTM) operating in the Bay of Biscay 2 , a technical mitigation measure whose efficiency was found wanting (Ulrich and Doerner

	Acronym	Meaning
	ASCOBANS	Agreement on the Conservation of Small Cetaceans of the
		Baltic, North East Atlantic, Irish and North Seas
	DCF	Data collection framework
	DPMA	"Direction des pêches maritimes et de l'aquaculture"
	GNS	Gillnetters
	GTR	Gill trammel netters
	ICES	International Council for the Exploration of the Sea
	Ifremer	Institut Français de Recherche pour l'Exploitation de la Mer
	ObsMer	Observation des captures en Mer (French national observer
		scheme for monitoring fisheries)
	PBR	Potential Biological Removal
	PTM	Pair trawlers
	PTB	Bottom pair trawlers
	VAST	Vector-Autoregressive Spatio-Temporal
	WGBYC	ICES Working Group on Bycatch of Protected Species
	WKEMBYC	ICES Workshop on

TABLE 4 |

 4 Model selection.

	Model	Likelihood for duration	Specification	W ÂICse	W ÂIC	Computation time (h)
	M6 M5 M4 M3 M2 M1	Gamma Log-normal Gamma Log-normal Gamma Log-normal	ICES division × week × year ICES division × week × year ICES division + week × year ICES division + week × year ICES division + week + year ICES division + week + year	18, 265169 18, 746185 19, 065151 19, 475167 21, 553133 21, 886148	0 481 800 1, 210 3, 288 3, 621	50 47 10 11 4 3

TABLE 5 |

 5 Model-based estimates of common dolphin bycaught in PTM in the study area.

MarEcol Prog Ser 679: 195-212, 2021

Frontiers in Marine Science | www.frontiersin.org

October 2021 | Volume 8 | Article 719956

induce moderate heterogeneity on a logit scale;Authier et al., 2017a).Frontiers in Marine Science | www.frontiersin.org

Genu, M., Gilles, A., Hammond, P., Macleod, K., Paillé, J., Paradinas, I. A., et al. (in preparation). Evaluating strategies for managing anthropogenic mortality on marine mammals: an R Implementation with the Package RLA.Frontiers in Marine Science | www.frontiersin.org

Rouby, E., Dubroca, L., Cloâtre, T., Demanèche, S., Genu, M., Macleod, K., et al. (in preparation). Estimating cetacean bycatch from non-representative samples (II): a Case Study on Common Dolphins in the Bay of Biscay.Frontiers in Marine Science | www.frontiersin.org

https://www.nytimes.com/2019/05/02/world/europe/france-dolphins-fishing. html

https://www.mer.gouv.fr/protecting-cetaceans-annick-giradin-presents-7commitments-made-french-state-fishermen-and Frontiers in Marine Science | www.frontiersin.org

January 2022 | Volume 8 | Article 795942

https://sih.Ifremer.fr/Ressources/ObsMer Frontiers in Marine Science | www.frontiersin.org

https://www.cetambicion-project.eu/ Frontiers in Marine Science | www.frontiersin.org

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Remerciements

ACKNOWLEDGMENTS

We would like to thanks Pr. Elisabeth Slooten, Dr. Aleta Hohn, Dr. Ursula Siebert, Dr. Sinéad Murphy, Pr. Jay Barlow, Dr. Leszek Karczmarski and Dr. Camilo Saavedra for sharing their data. Also, we would like to thanks the two anonymous reviewers for their critical comments that led to many improvements in the manuscript.

Acknowledgements. We warmly thank all members of the French and UK stranding networks for their continuous effort in collecting stranding data. We thank Marie-Pierre Etienne and Etienne Rivot for their insightful comments on our work. We also thank the community participating in the development of INLA, and especially Havard Rue for his devotion towards helping INLA users. Finally, we thank the reviewers for their insightful comments which helped to significantly improve the quality of the manuscript. The Observatoire PELAGIS is funded by the ministry in charge of the environment, and by Communauté d'Agglomération de la Ville de La Rochelle, with support from Région Poitou-Charentes and the European Union. EOVs were computed by the Shom and partially funded by the ministry in charge of the environment under the Marine Strategy Framework Directive (MSFD) monitoring program.

ACKNOWLEDGMENTS

We thank two reviewers from critical and constructive comments that help in improving the present manuscript. MA thanks Amaterasu for support when carrying the simulations.

ACKNOWLEDGMENTS

We thank the reviewers, Emilie Leblond, Thomas Rimaud, and Julien Lamothe for their constructive comments. We are indebted to Mikaël Guichard and the dedicated staff at the Mésocentre de Calcul de Poitou Charentes.

CITES

-Convention on International Trade in Endangered Species of Wild Fauna and Flora CMR -Capture-Mark-Recapture CO -Conservation Objective DCF -Data collection framework DPMA -Direction des pêches maritimes et de l'aquaculture EC -European Commission EU -European Union GLG -Growth Layer Group GNS -Gillnetters GTR -Gill trammel netters ICES -International Council for the Exploration of the Sea Ifremer -Institut Français de Recherche pour l'Exploitation de la Mer IUCN -International Union for Conservation of Nature mPBR -modified Potential Biological Removal MMPA -Marine Mammal Protection Act MSFD -Marine Strategy Framework Directive NAO -North Atlantic Oscillation NEA -North East Atlantic NGO -Non Governmental Organisation ObsMer -Observation des captures en Mer OMMEG -OSPAR Marine Mammal Expert Group OSPAR -Oslo-Paris Commission PBR -Potential Biological Removal PTM -Pair trawlers

Flowchart of the study presented in this chapter.

Publication in prep.

This chapter is associated with one publication which is under preparation: Rouby, E., Authier, M., Mauchamp, A., Mendez-Fernandez, P., Dabin, W., Thobois, J. and Ridoux, V. North-East Atlantic common dolphin population viability and vital rates variations. In prep.

FUNDING

This work was supported in the form of salaries from (ADERA to MA) and (HiDef to KM) but did not have any additional role in the study design, data collection, and analysis, decision to publish, or preparation of the manuscript.

DATA AVAILABILITY STATEMENT

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: https://gitlab.univ-lr. fr/mauthier/regularized_bycatch.

DATA AVAILABILITY STATEMENT

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of this article.

Appendix : Insights of the demographic approach

Convergence

To estimate vital rates from from previously acquired data, we built models combining both a survival and a reproduction part. The survival part was built from (Rouby, Ridoux, and Authier ) framework and the reproduction component was built from an Accelerated failure time framework. The two components may share a part of the overall variance. Estimation was carried out in a Bayesian framework using programming language Stan (Carpenter et al. b) called from R v. . . (R Core Team ) with library Rstan (Stan Development Team ). Stan uses Hamiltonian dynamics in Markov chain Monte Carlo (MCMC) to sample values from the joint posterior distribution (Carpenter et al. b). Four chains were initialised from diffuse random starting points and run for a total of 20, 000 iterations, discarding the first 10, 000 as warm-up. Default settings for the No-U-Turn Sampler (NUTS) were changed to 0.90 for adapt delta and 15 for max treedepth (Hoffman and Gelman ).

Siler and Heligman-Pollard

Complementary to the use of this model, we also used the models of Siler (Siler ) and Heligman-Pollard (Heligman and Pollard ) which are the conventional methods in crosssectional survival analysis on delphinids (Mannocci et al. ; Barlow and Hohn ; Stolen and Barlow ; Barlow and Boveng ; Secchi and Fletcher ). The Siler model is a five-parameter model ( 1 , 1 , 2 , 3 and3 ). Siler parameter estimations are realised using the Nelder & Mead optimization function (Nelder and Mead ) method and optim function of the package stats in R as detailed in the strandCet package (Saavedra ). The survivorship ( ) and the total mortality ( ) are expressed as the product of the three competing risks processes:

with the exponential decreasing risk due to the juveniles:

with the constant risk experienced by each age-class:

with the exponential risk due to senescent ages:

The Heligman-Pollard model used here is suited to admit cetacean bycatch at age as it is explained in (Saavedra ). The mortality function at age is expressed as it follows:

It considers three mortality components: the young mortality curve ( + ) , the "accident hump" in adult life (for exemple due to bycatch) × -2 and the adult mortality curve (1+ ) . The parameter allows to control the starting point of the mortality curve, making possible to represent bycatch at age . Both Siler and Heligman-Pollard models were computed using the strandCet R package and are used for comparison with the Reed survival model (Reed ). The use of Siler and Heligman-Pollard is more consistent with the literature and acts here as a sensibility analysis on the survival estimation for the common dolphin population.

We applied the Heligman-Pollard model on the dataset used in this study to compute mortality depending on the three life-components detailed in (Heligman and Pollard ). By using the methodology developed by (Saavedra ), it was possible to take into account a bycatch effect on age-class. The overall mortality curve showed an increasing mortality from to years (figure ) with a "plateau" from to years. The hump from to is associated to the additional mortality model component whereas the increasing along the lifetime is due to the senescent component of the model.

Logistic regression for Age at Sexual Maturity

To be consistent with the literature, we also used a logistic regression approach to compute the proportion of mature females and the age at sexual maturity (ASM) along the ages (Mannocci et al. ; Murphy et al. ; Danil and Chivers ). The logistic regression was fitted using a generalized linear model approach with the glm() function of the package stats on R. It is also possible to use the dose.p() function of the package MASS on R to determine the ASM using the previously fitted GLM. One year of data was simulated a 100 times. Bias of point estimate, coverage of (frequentist 95%) confidence or (Bayesian 80%) credible interval (CI) and precision (as CI width relative to the point estimate) are reported. The last column indicates the average number of bycatch events (nobs = E ijk yijk ) that were recorded by onboard observers during data sampling.

AUTHOR CONTRIBUTIONS

MA: led the analyses, the conception, and writing of the paper. ER and KM: support in analyses, paper conception, and writing. All authors contributed to the article and approved the submitted version.

FUNDING

ADERA and JNCC provided support in the form of salaries for MA and KM, respectively, but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Conflict of Interest: MA is employed by the commercial company ADERA which did not play any role in this study beyond that of employer.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Copyright

Methods

Modelling Bycatch Risk and Duration of Hauls

Observation des captures en Mer data allow both bycatch risk and haul duration to be modelled. The two may be correlated as a longer towing time may result in an increased likelihood of bycatch, all else being equal. Bycatch risk is defined at the level of a haul. Hauls can differ in duration as skippers may target different commercial species at different times of the year. However, the population-level data on effort is aggregated and available as DaS, the metric currently used in international fora (e.g., ICES Working Group on BYCatch, WGBYC). The number of hauls per DaS was also modelled from the ObsMer dataset in order to scale up bycatch risk per haul by the number of hauls per DaS. We modelled jointly bycatch risk, fishing duration of hauls, and the number of hauls per DaS of pair-trawlers flying the French flag at the week-level for each year between 2004 and 2020 (Table 2) and each ICES division (Figure 1). The goal of the approach is to model bycatch rates at the weekly scale for each year within each ICES division using a simple autoregressive model.

To smooth the fluctuations of estimated bycatch rates in weekly estimates we constrained estimation using Gaussian Process structured priors. These priors allow (i) to estimate an average bycatch risk profile at the weekly scale and from this weekly average, (ii) to estimate year-and divisionlevel deviations.

Notations

Let and covariance matrix depending only on the distances of the point x from each other (Gelman et al., 2021, page 465). In the following, we will drop the x and write in a shorthand manner θ ∼ GP (m, S) to mean that the vector θ of n parameters has a Gaussian process prior and follows a multivariate normal distribution whose mean vector m is equal to m(x) and whose covariance matrix S is defined for any pairs (x, x ′ ) as S x, x ′ = c(x, x ′ ), where c is the covariance function of the Gaussian process prior.

Joint Modelling Approach

Let i denotes the i th haul (fishing operation) happening in ICES statistical division j in week t of year k. Let y ijkt , d ijkt , and n jkt denote, respectively, bycatch event (0 or 1), fishing duration (in hours, d ijkt > 0), and the number of hauls per DaS jkt . Bycatch risk p jkt is estimated from

Estimating Common Dolphin Bycatch

Smoothness of Matérn correlation function

D() denotes the Dirichlet distribution for modelling proportions (such that 3 l=1 propl = 1) and GG() the Gamma-Gamma distribution for scale parameters (Griffin and Brown, 2017;Pérez et al., 2017).

ETHICS STATEMENT

Ethical review and approval was not required for the animal study because this study uses data that is collected as part of the Data Collection Framework under the Common Fishery Policy of the European Union.

AUTHOR CONTRIBUTIONS

ER and MA led the analyses, the conception, and writing of the paper. LD, TC, and SD curated and prepared the data. MG designed the shiny application. All authors support in analyses, paper conception, and writing, contributed to the article, and approved the submitted version.

Conflict of Interest: MA and HP are employed by the commercial company ADERA which did not play any role in this study beyond that of employer.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Dynamique des populations d'espèces élusives : Le cas du dauphin commun dans l'océan Atlantique Nord-Est

Résumé :

Les espèces élusives sont difficiles à suivre individuellement et il est difficile d'estimer leur mortalité ce qui freine la mise en place de politiques de gestion. En tant qu'espèce élusive, le dauphin commun, Delphinus delphis présente des enjeux de conservation importants. La population Atlantique Nord-Est est affectée par les captures accidentelles qui menacent sa viabilité. Les informations démographiques sur cette population sont insuffisantes et notre compréhension de la structure spatio-temporelle des captures ne permet pas de fixer des objectifs de gestion appropriés. Ce projet vise à déterminer le statut actuel de la population au regard des objectifs de conservation Européens. Premièrement, une approche flexible d'estimation des taux vitaux a été développée afin de mettre en évidence les variations temporelles et les effets de covariables en tenant compte de l'hétérogénéité individuelle. Ensuite, l'utilisation de cette méthode, combinée à un suivi transversal, nous a permis de mettre en évidence les variations temporelles des taux vitaux de la population et l'impact des captures sur sa démographie. La population n'est pas viable et son taux de croissance semble s'être détérioré depuis le milieu des années 2000. Troisièmement, en utilisant les données biaisées d'observateurs embarqués, le nombre de captures accidentelles a été quantifié sur une base hebdomadaire pour dix divisions CIEM sur seize ans. Quatrièmement, les taux vitaux obtenus combinés aux estimations de captures permettent de produire des seuils de captures soutenables. Cette évaluation a été effectué par une approche d'évaluation de stratégies de gestion en accord avec l'objectif de conservation européen.

Mots clés : démographie, mammifères marins, mégafaune, dynamique des populations, captures accidentelles, gestion, statistiques Bayésiennes, suivi transversal, âge à la mort, biologie de la conservation, échouages.

Population dynamics of elusive species:

The case of the common dolphin in the North-East Atlantic Ocean

Summary :

Elusive species are notoriously difficult to monitor individually and therefore it is difficult to estimate their anthropogenic mortality and to implement management policies. As an elusive species, the common dolphin, Delphinus delphis is a good example of these conservation challenges. The North-East Atlantic population is affected by bycatch that threatens its viability. However, demographic information on this population is very limited and our understanding of the spatio-temporal structure and dynamics of bycatch is insufficient to set appropriate management objectives. This project aims to determine the current status of this common dolphin population with regard to conservation objectives in European waters with a focus on the Bay of Biscay. First a flexible vital rates estimation approach has been developed in order to highlight variations in vital rates over time and the effects of covariates while taking into account individual heterogeneity. Second, using such a methodological approach through a cross-sectional monitoring, allowed us to highlight both the temporal variations of the population vital rates and the impact of bycatch on the common dolphin population demography. The population is not viable and the population growth rate seems to have deteriorated since the mid 2000's. Third, using biased data from on-board observers, the number of incidental catches on a weekly basis was quantified for ten ICES sub-area over sixteen years. Fourth, the resulting vital rates combined with bycatch estimates finally allowed the production of sustainable bycatch thresholds through a Management Strategy Evaluation approach in line with the European conservation objective. Keywords: demography, marine mammals, megafauna, population dynamics, bycatch, management, Bayesian statistics, cross-sectional monitoring, age-at-death, conservation biology, strandings.