REMERCIEMENTS

Je tiens à remercier toutes les personnes qui m'ont aidée lors de la rédaction de ce rapport, ainsi que toutes celles qui m'ont accompagnée de près comme de loin.

Je

tiens dans un premier temps à remercier vivement mes maîtres de stage. Merci à Mr Vincent Devictor pour son accueil au sein de l'équipe, son temps, sa bienveillance ainsi que le partage de son expertise au quotidien. Merci aussi à Mr Vasilis Dakos, qui, même à distance, a su m'accorder du temps et m'accompagner tout du long avec pertinence. C'est notamment grâce à leur confiance et leurs précieux conseils qui j'ai pu mener mon projet à terme. Merci également à Stanislas Rigal pour son aide précieuse et ses conseils avisés tout au long de mon projet, cela m'a grandement aidée. Je remercie d'une manière générale toute l'équipe BioDICée pour leur accueil, leur bonne humeur, leur écoute et tous les moments partagés. Ce fut un réel plaisir de travailler à leur côté dans un environnement stimulant et riche de partages.

Merci à tous mes amis, qui sont là depuis la prépa ou l'école, pour tous les moments passés ensemble et toutes les discussions qui ont nourrit mes réflexions et m'ont amenée là où je suis.

Enfin, je tiens à remercier très chaleureusement ma famille. Merci à mes parents de m'avoir encouragée à m'engager dans cette voie même lorsque je n'y croyais pas, de m'avoir toujours soutenue et cru en moi, c'est grâce à vous si j'en suis là. Merci à ma cousine Enora pour avoir réussi à égayer la fin de mes études malgré un contexte difficile. Un merci tout particulier à ma soeur, Eléa, pour sa présence et son soutien indéfectible, qu'importent les kilomètres qui nous séparent.

Trajectoires non linéaires et stabilité des dynamiques de populations à l'échelle mondiale

GLOSSARY

Population/time series: in this report we will refer to the terms 'population' and 'time series' (sometimes also called 'population time series') in the same way. This refers to a series of abundance data for a given species at a given location.

Record: a record is the fundamental unit of observation in the BioTIME database. It is the observation of a given species at a given location in a given sample.

Sample: a sample groups together all the records observed at the same location and on the same date. It can be a transect or a quadrat for example.

Study: A study groups together records belonging to the same data source and having a regular and consistent sampling methodology. Sources of data in which the sampling methodology have changed during the course of the study belong to separate studies.

Trajectory: the trajectory is defined here as the description of changes in abundance over time in terms of direction, velocity, and curvature.

Trend: we will speak of trend in the generic sense as defined for linear models, i.e. the overall change in terms of direction (growth or decline).

Velocity: velocity is considered to be the non-linear analogue of linear trend, i.e. a measure of growth or decline. The two terms are sometimes combined under the unique term trend.

"Fake no trend": We use the term "fake no trend" to refer to populations that are considered "constant" by the usual linear approach -i.e. whose trend is not significantly different from zerobut which are not considered so by a non-linear approach. We speak of "fake no trend" in the sense that a classical approach would consider those populations to be constant (no significant trend) but only because the model is limited to a linear one.

Stability: In this report, stability refers to the absence of variability in the number of individuals, whether considered in total or over time. It should not be confused with populations that are sometimes defined as "stable" when their trend is not significantly different from zero. It is considered here that a constant trend does not necessarily imply the stability of a population since it does not provide any information on fluctuations.

Winners/losers: Populations are sometimes referred to as 'winners' or 'losers'. The classic approach is to describe populations with a growing trend as winners and those with a decreasing trend as losers [START_REF] Bowler | Mapping human pressures across the planet uncovers anthropogenic threat complexes[END_REF]. The use of these terms here also implies the dimension of stability, considering winners as populations that grow and stabilise, and losers as populations that fall and whose instability increases over time. [START_REF] Benton | Diversification and Extinction in the History of Life[END_REF], et on estime que cette crise anthropique pourrait à terme éliminer 75% des espèces (IPBES, 2019). Cette situation a même conduit les scientifiques à appeler "Anthropocène" cette nouvelle ère géologique dominée par les impacts humains [START_REF] Crutzen | The "Anthropocene[END_REF]. L'explosion du développement des activités humaines entraîne la dégradation des espaces naturels et l'érosion de la biodiversité à un rythme alarmant. En moins de 50 ans, la taille moyenne des populations de vertébrés terrestres a diminué de 68% [START_REF] Wwf | Living Planet Report 2020 -Bending the curve of biodiversity loss[END_REF]. Sur les 128 500 espèces inscrites sur la liste rouge de l'UICN, 35 500 sont menacées d'extinction (UICN, 2019). Si le réchauffement climatique se poursuit, près de 50 % des espèces vivant actuellement dans les régions les plus riches en biodiversité pourraient être menacées d'extinction d'ici 2080 [START_REF] Warren | The implications of the United Nations Paris Agreement on climate change for globally significant biodiversity areas[END_REF].

| Enjeux et controverses autour de la crise de la biodiversité

Par conséquent, la description et la compréhension de l'état et de la dynamique mondiale de la biodiversité constituent des défis scientifiques et politiques majeurs. En principe, une quantification correcte du déclin de la biodiversité devrait s'appuyer sur des données temporelles et spatiales couvrant la plupart des taxons et reparties dans tous les biomes. En pratique cependant, les études actuelles se concentrent principalement sur des données spatiales ou temporelles [START_REF] Wolkovich | Temporal ecology in the Anthropocene[END_REF], ou sur des taxons spécifiques tels que les oiseaux (e.g. Donald et al., 2001 ;[START_REF] Julliard | Common birds facing global changes: what makes a species at risk?[END_REF][START_REF] Inger | Common European birds are declining rapidly while less abundant species' numbers are rising[END_REF], les poissons (e.g. Christensen et al., 2014) ou les insectes (e.g. [START_REF] Lister | Climate-driven declines in arthropod abundance restructure a rainforest food web[END_REF][START_REF] Sánchez-Bayo | Worldwide decline of the entomofauna: A review of its drivers[END_REF]Seibold et al., 2019). Ainsi, l'ampleur et l'étendue de la crise de la biodiversité restent un sujet de discussion dans la communauté scientifique [START_REF] Thomas | Local diversity stays about the same, regional diversity increases, and global diversity declines[END_REF][START_REF] Cardinale | Overlooked local biodiversity loss[END_REF][START_REF] Gonzalez | Estimating local biodiversity change: a critique of papers claiming no net loss of local diversity[END_REF][START_REF] Cardinale | Is local biodiversity declining or not? A summary of the debate over analysis of species richness time trends[END_REF]. Des études récentes ont même suggéré que la dynamique de la biodiversité n'est pas vraiment en déclin en moyenne. Elles postulent que les espèces en péril tirent les indices mondiaux vers le bas [START_REF] Leung | Clustered versus catastrophic global vertebrate declines[END_REF], mais qu'il y a au total autant d'espèces "gagnantes" que "perdantes" [START_REF] Bowler | Mapping human pressures across the planet uncovers anthropogenic threat complexes[END_REF]. Plutôt que de se fier à l'augmentation ou à la diminution du nombre d'espèces ou d'individus, il est donc également nécessaire de suivre les changements dans la composition des assemblages locaux [START_REF] Dornelas | Assemblage Time Series Reveal Biodiversity Change but Not Systematic Loss[END_REF][START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF]. Dans l'ensemble, l'utilisation de mesures agrégées de la biodiversité mondiale pour quantifier la perte de biodiversité tend à masquer d'importantes hétérogénéités spatiales, temporelles et spécifiques.

Une partie de la difficulté à saisir correctement les tendances de la biodiversité réside dans les méthodes et les données considérées [START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF]. En ce qui concerne les données, cela se reflète dans les nombreux résultats contradictoires générés par deux ensembles de données mondiaux sur lesquels se basent la plupart des analyses mondiales de la dynamique de la biodiversité, à savoir BioTIME [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF] principales limites de ce modèle statistique très simple est la forte hypothèse concernant l'erreur de mesure (inexactitude de l'observation elle-même pendant les relevés sur le terrain), qui n'est pas séparée de l'erreur de processus (mécanismes qui ne sont pas mis en oeuvre dans le modèle) [START_REF] Dornelas | Quantifying temporal change in biodiversity: challenges and opportunities[END_REF]. Des modèles dits « state-space models », plus avancés, ont ainsi été utilisés dans des méta-analyses récentes [START_REF] Leung | Trends in mean growth and stability in temperate vertebrate populations[END_REF][START_REF] Daskalova | Rare and common vertebrates span a wide spectrum of population trends[END_REF], car ils permettent de rendre compte explicitement de la différence entre ces deux composantes d'erreur (Humbert et al., 2009). Pourtant, ces premiers types de modèles (state-space et linéaires) considèrent toujours une dynamique linéaire bien que les exemples de dynamiques de population non linéaires soient nombreux [START_REF] Clark | Nonlinear population dynamics are ubiquitous in animals[END_REF]. L'utilisation de modèles simples ou de state-space models peut donc manquer une partie de la dynamique et conduire à une mauvaise interprétation.

Par exemple, une courbe en cloche est interprétée comme une tendance constante par une approche linéaire alors qu'elle peut refléter un fort déclin récent. Il peut être crucial d'étudier non seulement la direction, mais aussi la vitesse et la courbure des tendances. L'adoption de techniques de modélisation mieux à même de décrire des formes non linéaires est donc importante (Rigal et al., 2020).

En outre, si les moteurs anthropiques de la perte de biodiversité sont largement documentés, les liens de causalité, voire de corrélation, entre les pressions anthropiques et les réponses de la biodiversité restent difficiles à établir à l'échelle mondiale. Quel est l'effet global du changement climatique sur la dynamique de la biodiversité ? Ou de la conversion des terres à l'agriculture industrielle ou de l'urbanisation sur la biodiversité ? Il existe de nombreux exemples locaux dans lesquels les liens pressions-réponses ont été établis [START_REF] Kohsaka | Local Assessment of Tokyo: Satoyama and Satoumi -Traditional Landscapes and Management Practices in a Contemporary Urban Environment[END_REF][START_REF] Liu | Local Assessment of Shanghai: Effects of Urbanization on the Diversity of Macrobenthic Invertebrates[END_REF][START_REF] Sudhira | Local Assessment of Bangalore: Graying and Greening in Bangalore -Impacts of Urbanization on Ecosystems, Ecosystem Services and Biodiversity[END_REF][START_REF] Phillips | Land-use effects on local biodiversity in tropical forests vary between continents[END_REF], mais la manière dont les pressions se transforment à l'échelle globale et se propagent sur la dynamique spatio-temporelle de la biodiversité reste mal comprise.

| Objectifs et questions de recherche

Dans ce mémoire, nous testons si et comment des modèles conçus pour suivre la non-linéarité et la stabilité des populations pourraient aider à fournir une évaluation plus complète de l'état de la biodiversité à l'échelle globale. Nous nous concentrons sur trois étapes particulières :

1. Nous explorons si les populations sont gouvernées par des trajectoires non linéaires significatives, et si ces trajectoires sont vécues de manière disproportionnée dans les différents groupes taxonomiques.

2. Nous cherchons à savoir si les populations deviennent de plus en plus instables, en particulier celles qui sont également en déclin. [START_REF] Dornelas | Quantifying temporal change in biodiversity: challenges and opportunities[END_REF].

Pour chaque série temporelle, les coordonnées géographiques résultantes ont été calculées en utilisant les centroïdes des enregistrements assemblés.

Aucune transformation de données n'a été effectuée sur la LPD car elle est déjà structurée en séries temporelles.

Sélection des données

Afin de fusionner les deux bases de données (BioTIME et LPD), nous avons réalisé une sélection en quatre étapes (voir fig. 1).

| Modélisation des tendances à l'aide de state-space models

Afin de saisir les informations supplémentaires fournies par l'utilisation de modèles polynomiaux (MP), nous avons comparé les classifications entre les MP et une approche linéaire adaptée. La méthode prédominante d'estimation des tendances (modèles linéaires) ne sépare pas les erreurs de mesure et de processus, bien que celles-ci soient deux des quatre facteurs expliquant les changements temporels observés dans les données d'abondance, avec l'autocorrélation temporelle et les changements systémiques [START_REF] Dornelas | Quantifying temporal change in biodiversity: challenges and opportunities[END_REF]. Les state-space models (SSM) tiennent compte à la fois de l'erreur de processus et de l'erreur de mesure ; et il a été démontré qu'ils fournissent de meilleures estimations de la tendance sur de larges gammes de longueur de séries temporelles et de sources de variation que les modèles linéaires habituels [START_REF] Dennis | Estimating Density Dependence, Process Noise, and Observation Error[END_REF]Humbert et al., 2009). Ainsi, nous avons également modélisé les estimations d'abondance loge transformées en utilisant des SSM. Voir détails dans la partie 2.

| Quantification de la stabilité à partir des fluctuations des populations

Pour quantifier la stabilité, nous avons utilisé cinq métriques différentes, à savoir la variabilité globale (nommée ci-après 𝑀𝑆𝐸 pour variabilité globale et 𝐶𝑉 pour coefficient de variation), le changement de variabilité dans le temps (nommé ci-après 𝜂 1 ), l'asymétrie des résidus, la "fréquence d'extinction locale" et le "syndrome de colonisation ou d'extinction". Voir détails dans la partie 2.

Cette partition de la stabilité a ensuite été utilisée pour examiner les corrélations entre les différentes métriques quantitatives et la vélocité en calculant les coefficients de corrélation de

Pearson. Une ACP a également été réalisée pour voir si ces métriques pouvaient être utilisées pour partitionner les populations en différents groupes et les caractériser.

| Etude des sources potentielles d'hétérogénéité des tendances et de la stabilité des populations

Nous avons étudié l'influence des caractéristiques biogéographiques, des taxons, des traits au niveau des espèces et des pressions anthropiques sur les tendances et la stabilité des populations.

Pour ce faire, nous avons utilisé des modèles linéaires généralisés à effets mixtes (GLMM). Nous avons pris soit la vélocité (c'est-à-dire la tendance de la population), la MSE (c'est-à-dire la variabilité globale de la population) ou 𝜂 1 (c'est-à-dire le changement de la variabilité dans le temps) comme variables de réponse, et les coordonnées géographiques, les taxons, les grands domaines (terrestre, marin ou eau douce), la taille des espèces, le statut de conservation ou les pressions anthropiques comme variables explicatives dans des modèles séparés (fig. 3). Chaque modèle incluait l'espèce comme effet aléatoire pour tenir compte de la corrélation possible entre les populations d'une même espèce.

Pour évaluer les tendances géographiques et taxonomiques, nous avons modélisé les tendances et les paramètres de stabilité en fonction de la latitude, du domaine et des taxons séparément, ce qui a donné lieu à trois modèles pour chaque paramètre. Pour évaluer l'impact des pressions (catégorie de la Liste rouge, GHF) et des traits au niveau de l'espèce (taille et masse du corps) sur les vertébrés terrestres, nous avons modélisé les tendances et les mesures de stabilité par taxa en fonction des trois prédicteurs séparément, ce qui a donné lieu à 12 modèles pour chaque mesure.

Voir détails dans la partie 2.

__________________________________________________________________________________

| RÉSULTATS

| Distribution des trajectoires parmi les classes non linéaires

Nous avons constaté que parmi les 2 786 trajectoires de population considérées, 610 -soit 22%présentaient une dynamique non linéaire (fig. 4). Les trajectoires concaves et convexes représentaient même un quart des cas constants.

Nous avons également constaté que la proportion de dynamiques non linéaires varie selon les taxons (fig. 5). Si nous ne considérons que les vertébrés terrestres, qui sont les groupes les plus étudiés en matière de dynamique de la biodiversité, nous trouvons près de 30% de trajectoires non linéaires. Ce chiffre monte à 40% pour les mammifères mais aussi pour les reptiles.

Si l'on compare ces résultats avec ceux des SSM, on note des différences nettes (tableau 1). Les sorties des SSM suggèrent que 2 415 populations sont constantes, alors que l'approche non linéaire identifie 1 428 populations comme constantes. Cela signifie que l'approche habituelle présente un faux pourcentage de tendances constantes de plus de 40%.

Focus sur les vertébrés terrestres

En examinant l'impact de la taille et de la masse corporelle sur les mesures des tendances et de la stabilité des populations de vertébrés terrestres, nous avons trouvé des associations entre les tendances et la variabilité globale chez les mammifères, mais pas chez les autres taxons.

Contrairement à nos attentes, aucun lien clair n'a pu être établi entre les tendances ou les fluctuations de nos populations et leur catégorie sur la liste rouge de l'UICN. De manière générale, il semble que la catégorie de la liste rouge de l'UICN n'explique pas l'hétérogénéité des tendances ou de la stabilité des populations que nous avons étudiées, mais ces populations sont également pour la grande majorité d'entre elles classées "least concern" (n=759 ; 84% des 902 populations de vertébrés terrestres).

Enfin, nous n'avons pas observé d'impact significatif du taux de changement du « Global Human Footprint » (GHF) entre 1993 et 2009 sur les tendances des populations de vertébrés terrestres.

Le taux de changement moyen sur les sites considérés était de +1%. Même si nous nous attendons à ce que les pressions augmentent à l'échelle mondiale, les sites considérés ne reflètent peut-être pas le taux de changement moyen mondial, ce qui pourrait expliquer pourquoi nous n'avons pas détecté d'impact. L'investigation de l'impact du GHF avec une approche qualitative, en utilisant différentes classes de pressions, n'a pas montré de résultats significatifs. between 50% and 95% of species [START_REF] Benton | Diversification and Extinction in the History of Life[END_REF], and it is estimated that this anthropogenic crisis could eventually eliminate 75% of species (IPBES, 2019). This situation has even led scientists to call "Anthropocene" this new geological era dominated by human impacts [START_REF] Crutzen | The "Anthropocene[END_REF]. The explosion in the development of human activities is resulting in the degradation of natural areas and the erosion of biodiversity at an alarming rate. In less than 50 years, the average population size of terrestrial vertebrates has fallen by 68% (WWF, 2020). Over the 128,500 species listed in the IUCN Red List, 35,500 are threatened by extinction (IUCN, 2019). If global warming continues, nearly 50% of the species currently living in the most biodiverse regions could be threatened by extinction by 2080 [START_REF] Warren | The implications of the United Nations Paris Agreement on climate change for globally significant biodiversity areas[END_REF].

__________________________________________________________________________________

| DISCUSSION

| Issues and controversies surrounding the biodiversity crisis

As a result, describing and understanding the status and global dynamics of biodiversity are major scientific and political challenges. In principle, a proper quantification of biodiversity decline should be based on temporal and spatial data covering most of Earth's taxa and distributed across biomes. In practice, however, current studies have mainly focused on data in space or in time [START_REF] Wolkovich | Temporal ecology in the Anthropocene[END_REF], or on focal taxa such as birds (e.g. Donald et al., 2001;[START_REF] Julliard | Common birds facing global changes: what makes a species at risk?[END_REF][START_REF] Inger | Common European birds are declining rapidly while less abundant species' numbers are rising[END_REF], fish (e.g. Christensen et al., 2014) or insects (e.g. [START_REF] Lister | Climate-driven declines in arthropod abundance restructure a rainforest food web[END_REF][START_REF] Sánchez-Bayo | Worldwide decline of the entomofauna: A review of its drivers[END_REF]Seibold et al., 2019). Thus, the magnitude and extent of the biodiversity crisis remains a matter of discussion in the scientific community [START_REF] Thomas | Local diversity stays about the same, regional diversity increases, and global diversity declines[END_REF][START_REF] Cardinale | Overlooked local biodiversity loss[END_REF][START_REF] Gonzalez | Estimating local biodiversity change: a critique of papers claiming no net loss of local diversity[END_REF][START_REF] Cardinale | Is local biodiversity declining or not? A summary of the debate over analysis of species richness time trends[END_REF]. Recent studies have even suggested that the dynamics of biodiversity are not really declining on average. They postulate that species at risk are dragging down global indices [START_REF] Leung | Clustered versus catastrophic global vertebrate declines[END_REF], but that there are as many "winners"

as "losers" species in total [START_REF] Bowler | Mapping human pressures across the planet uncovers anthropogenic threat complexes[END_REF]. Rather than relying on the increase or decrease in species numbers or individuals, tracking changes in the composition of local assemblages is therefore also necessary [START_REF] Dornelas | Assemblage Time Series Reveal Biodiversity Change but Not Systematic Loss[END_REF][START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF]. Overall, using aggregated metrics of global biodiversity to quantify biodiversity loss tend to hide major spatial, temporal, and specific heterogeneities.

Part of the difficulty to correctly capture trends in biodiversity lies in the methods and data considered [START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF]. Regarding the data, this is mirrored in the many contradictory results generated by two alternative global datasets on which most global analysis of biodiversity dynamics are based, namely BioTIME [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF] and the Living Planet Database (LPD) (LPI, 2016). While the LPD suggests that terrestrial vertebrates have declined sharply since 1970, BioTIME shows much more contrasting results. Moreover, the models adopted to estimate the extent of biodiversity loss can also be confusing. Usually, trends are measured by applying a linear model to the data to estimate the rate of change of a variable (e.g. a number of species, individual or any aggregated metrics) over a period of time. This is a widely used approach already applied to many groups such as birds (e.g. [START_REF] Inger | Common European birds are declining rapidly while less abundant species' numbers are rising[END_REF], fish (e.g. Christensen et al., 2014) or insects (e.g. Seibold et al., 2019). One of the major limits of this very simple statistical model, however, is the strong assumption regarding the measurement error (inaccuracy in the observation itself during the field surveys), that is not separated from the process error (mechanisms that are not implemented in the model) [START_REF] Dornelas | Quantifying temporal change in biodiversity: challenges and opportunities[END_REF]. More advanced state-space models, which are sort of improved linear models, have thus been used in recent meta-analyses [START_REF] Leung | Trends in mean growth and stability in temperate vertebrate populations[END_REF][START_REF] Daskalova | Rare and common vertebrates span a wide spectrum of population trends[END_REF], as they allow to account explicitly for the difference in those two error components (Humbert et al., 2009). Yet, these first types of models still consider linear dynamics

although the examples of non-linear population dynamics are numerous [START_REF] Clark | Nonlinear population dynamics are ubiquitous in animals[END_REF].

Using either simple or state-space models may therefore miss part of the dynamics and lead to misinterpretation. For example, a bell curve is interpreted as a constant trend by a linear approach when it may reflect a recent sharp decline. Investigating not only the direction but also the velocity and curvature of the trends can be crucial. Adopting modelling techniques best able to describe non-linear shapes is therefore straightforward (Rigal et al., 2020).

Moreover, while the anthropogenic drivers of biodiversity loss are widely documented, causal, or even correlative links between anthropogenic pressures and biodiversity responses remain difficult to establish at a global scale. What is the overall effect of climate change on biodiversity dynamics? Or of land use conversion for industrial agriculture or urbanization on biodiversity?

There are many local examples in which the pressure-response links have been established [START_REF] Kohsaka | Local Assessment of Tokyo: Satoyama and Satoumi -Traditional Landscapes and Management Practices in a Contemporary Urban Environment[END_REF][START_REF] Liu | Local Assessment of Shanghai: Effects of Urbanization on the Diversity of Macrobenthic Invertebrates[END_REF][START_REF] Sudhira | Local Assessment of Bangalore: Graying and Greening in Bangalore -Impacts of Urbanization on Ecosystems, Ecosystem Services and Biodiversity[END_REF][START_REF] Phillips | Land-use effects on local biodiversity in tropical forests vary between continents[END_REF], but the way pressures scale up to global scale and propagate on the spatio-temporal dynamics of biodiversity remains poorly understood.

| Objectives and research questions

In this thesis, we test whether and how models designed to track the non-linearity and stability of populations could help to provide a more complete assessment of the state of biodiversity at a global scale. We focus on three particular steps:

1. We explore whether populations are governed by significant non-linear trajectories, and if these trajectories are experienced disproportionately across taxonomic groups

We aim at assessing the existence and distribution of non-linear trajectories in populations on a global scale. We assume that studying population dynamics through a linear approach masks important trajectories that could be highlighted with a non-linear approach. We seek to determine the proportion of non-linearity in the trajectories of the populations studied, as well as the proportion of fake "no trend" that results from a classical linear approach. We also ask if there is a shared fate between different groups. In particular, we analyse whether amphibians experience greater declines than other groups as suggested by many studies [START_REF] Adams | Trends in Amphibian Occupancy in the United States[END_REF][START_REF] Houlahan | Quantitative evidence for global amphibian population declines[END_REF][START_REF] Stuart | Status and Trends of Amphibian Declines and Extinctions Worldwide[END_REF].

2. We investigate whether populations are becoming increasingly unstable, especially those that are also declining

Stability is another overlooked aspect of biodiversity dynamics at global scale which is not fully encompassed by the detection of non-linear trajectories. Ecological stability is a multifaceted concept and has received many definitions, including absence of temporal variability, resistance to environmental change and resilience [START_REF] Pimm | The complexity and stability of ecosystems[END_REF][START_REF] Mccann | The diversity-stability debate[END_REF][START_REF] Donohue | Navigating the complexity of ecological stability[END_REF]. Here we consider stability as the change in temporal variance in the number of individuals. Increase in variability has been long identified as loss of stability in ecological systems [START_REF] Grimm | Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion[END_REF] or even signature of abrupt ecological collapse [START_REF] Scheffer | Early-warning signals for critical transitions[END_REF][START_REF] Hughes | Multiscale regime shifts and planetary boundaries[END_REF]. We therefore investigate whether populations in general are becoming increasingly unstable by measuring changes in the variability of their fluctuations. We also ask if there is a link between trend, abundance and stability. For example, are small populations also those exposed to stronger demographic and environmental stochasticity [START_REF] Marsh | Fluctuations in amphibian populations: a meta-analysis[END_REF][START_REF] Green | The ecology of extinction: population fluctuation and decline in amphibians[END_REF]?

3. We explore whether population trajectories and stability are influenced by specific and anthropogenic factors and exhibit particular biogeographic patterns

Finally, we propose models that could link the responses of biodiversity in terms of dynamics and stability to anthropogenic pressures. We investigate if trajectories are influenced by humaninduced pressures and/or species-specific characteristics like body size or IUCN Red List category.

Also, we assume that there will be biogeographic patterns in population trends and stability across the planet's realms and taxa. This hypothesis is supported by macro-ecological reasons such as the existence of biodiversity hotspots (e.g. tropical regions (Almeida-Cortez and Santos, 2008)), suggesting a latitudinal gradient. Studies also show a non-random distribution of the pressures assumed to drive population dynamics (Bowler et al., 2020). We also expect populations of threatened species and experiencing a higher rate of change in their pressures to be more likely to decline and fluctuate than the populations of least concern species and those exposed to a decreasing amount of pressure.

N.B.: for all words in bold, please refer to the glossary.

| MATERIALS AND METHODS

| Population data

To quantify population trends and stability and classify their trajectories, we used two databases, namely BioTIME [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF] and the Living Planet Database (LPD) [START_REF] Mcrae | The Diversity-Weighted Living Planet Index: Controlling for Taxonomic Bias in a Global Biodiversity Indicator[END_REF].

BioTIME contains species abundance records for assemblage that have been sampled through time with a consistent methodology. The data were recorded since 2010, using bibliography, already established data bases, ecological data, personal networking and broadcasted data (from conferences and social media) [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF]. Similarly, the LPD contains time series of either population size, density, abundance but also proxy of abundance. Time series information was collated from published scientific literature, online databases and grey literature using the same method on the same population throughout the time series. We merged the abundance data of population time series from both BioTIME [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF] and the LPD [START_REF] Mcrae | The Diversity-Weighted Living Planet Index: Controlling for Taxonomic Bias in a Global Biodiversity Indicator[END_REF], yielding to a final dataset of 2,786 population time series. In doing so, we should generate the most comprehensive database on population dynamics at global scale ever produced.

Moreover, working with this combined database should overcome some of the limitations (e.g. spatial or group coverage) associated with BioTIME or LPD if analysed separately.

Data transformation

First, a preliminary transformation of the BioTIME database was necessary to account for the different sampling effort in time. BioTIME originally contains raw records from populations worldwide, coming from 361 different studies. A rarefaction code based on the samples was applied to recreate time series. For a given year and species within a study, abundance data were summed independently of samples. If there are several transects for example, the abundance of the species from the different transects in that year is summed. This sample-based rarefaction process can be visualized in the supplementary figure S1. The sampling effort may vary through time, affecting the abundance estimates. Thus, the minimum number of samples in a year was extracted within a time series, and this number of samples was randomly selected from each of the other years [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF]. In doing so, we ensured that each time series is based on a constant sampling effort [START_REF] Dornelas | Quantifying temporal change in biodiversity: challenges and opportunities[END_REF]. For each time series, the resulting geographic coordinates were calculated using the centroids of the assembled records.

No data transformations were conducted on the LPD as it is already structured in time series.

Data selection

In order to merge the two databases (BioTIME and LPD), we performed a selection framework in four steps (fig. 1).

1. Avoid overlapping data: First, as the two databases are compilations of population time series, we had to make sure that no time series was counted twice, i.e. that there was no overlap between the two databases. For this purpose, we excluded from the LPD all the data that concerned species present in both databases. As BioTIME contained 44,175 time series at this stage of the selection and LPD 11,067, we have chosen to delete the species common to both databases in LPD in order to keep a maximum of data in total.

Avoid anachronisms:

The BioTIME database covers a period from 1874 to 2016, though most data come from the last 50 years; the LPD covers a period from 1970 to 2015. We thus focused on the shared period of time and selected only data collected between 1970 and 2015 in both datasets.

Focus on abundance and density data:

Populations were surveyed using different methods (e.g., expressed as counts, estimates, indices or densities), but we only included populations that were monitored using number of individuals (abundance) or densities, as being the most relevant measures to answer our questions.

Time series length and composition:

We considered time series running from the first nonzero population to the last non-zero population, as we want to calculate population dynamics for species that are actually present (note that zero values can still exist between the first and the last non-zero values). We omitted populations which had less than fifteen time points of monitoring data. Previous studies have found that shorter time series might not adequately assess population trends, and suggest to consider six points at least (Humbert et al., 2009;Rigal et al., 2020). By keeping a minimum of 15 time points, we aimed at retaining sufficient power to detect different types of trajectories. Population time series for which 50% or more of the dataset is represented by zero counts have also been omitted, to avoid for the presence of false zeros that are attributable to observation error. In particular, this allows the exclusion of the rarest species from the analysis, which usually have an excess of zeros, creating difficulties to estimate reliable trends [START_REF] Potts | Comparing species abundance models[END_REF].

| Body size and mass, conservation status and anthropogenic pressure data

We retrieved body size (and mass), conservation status and pressures that could influence trends and stability in the populations studied. Such information is not available for all groups, so we restricted this part of the study to terrestrial vertebrates, a group that has been widely documented and studied [START_REF] Titley | Scientific research on animal biodiversity is systematically biased towards vertebrates and temperate regions[END_REF]. This restricted our dataset to 902 time series, considering 691 species of amphibians, reptiles, birds and mammals at 357 geographical locations. We considered three variables as potential predictors of population trends and variations, including body size, IUCN Red List Categories and the Global Human Footprint Index.

Red List Category data for each species are easily available from the International Union for Conservation (IUCN, 2020). To characterize species body size, we used the median body mass for birds, reptiles and mammals from a recent normalized and consolidated database of amniotes lifehistory parameters (Myhrvold et al., 2015). For amphibians, we used the body length from an extensive database containing history traits of amphibian species globally (Oliveira et al., 2017).

In all cases, the body size was log-transformed and standardized within taxa (by subtracting the mean and dividing by the standard deviation) to allow comparisons. Anthropogenic pressures were assessed through the Global Human Footprint (GHF) which corresponds to the combined effect of eight human pressures globally (the extent of built environments, crop land, pasture land, human population density, night-time lights, railways, roads, and navigable waterways). It is to date the most complete and highest-resolution global dataset on human pressures (Venter et al., 2016a). The link between pressure and populations was established through their location. We used two maps, from 1993 and 2009, of globally standardized measure of the cumulative human footprint on the terrestrial environment at 1 km 2 resolution (Venter et al., 2016b). In order to ensure a most accurate value of the GHF measurement, GHF values were averaged over 200 points spread over an area of 4km² around the location of each of the populations studied. We chose an area of 4km² as this is the average size of the studies contained in BioTIME [START_REF] Dornelas | BioTIME: A database of biodiversity time series for the Anthropocene[END_REF], and as the information was not specified for the LPD. We tested the robustness of our conclusions to changes in the size of the buffer considered (from 4km² to 100km²) (see supplementary figure S2). The percentage of change in GHF between 1993 and 2009 was used as the predictor value. We also tested the impact of the pressure classes, classified as followed: 'no pressure, mean human footprint=0'; 'low pressure, human footprint=1-2; 'moderate pressure, human footprint=3-5'; 'high pressure, human footprint=6-11'; and 'very high pressure, human footprint=12-50', following Venter et al. (2016). This classification was shown to bin land areas into five classes of human pressure such that each bin encompasses a similar proportion (∼20%) of the planet (Venter et al., 2016a).

| Modelling population trends and trajectories using polynomial models

In order to study trends and trajectories, each population time series was fitted with a second order polynomial model to quantify the trend and classify the trajectory (Rigal et al., 2020) through:

𝑋(𝑡) = 𝛼 0 + 𝛼 1 𝑡 + 𝛼 2 𝑡 2 + 𝜀(𝑡) (1)
Where 𝑋 is the loge transformed abundance estimate (we added 1 to every population count, to account for estimates of zero abundance in a given year), 𝑡 the time in years, with process error represented by 𝜀 ~ 𝑁(0, 𝜎 ² ). The qualitative characterisation of the trajectory led to the identification of nine classes (fig. 2).

Quantitatively, we were interested in two characteristic metrics: direction, i.e. decline, In linear cases (𝛼 2 non significant), we simply considered 𝛼 1 , its sign indicating the direction and its absolute value the velocity. In non-linear cases (𝛼 2 significant), we placed ourselves at point 𝑋 𝑚 , the centre of the interval [𝑋 0 , 𝑋 𝑇 ] (corresponding to the whole time series length). We then study the tangent at this point (𝑌(𝑋 𝑚 ) ̇), the sign indicating the direction and the absolute value the velocity.

We will refer to this approach as the 'non-linear' approach as it allows populations to have nonlinear trajectories.

| Modelling population trends using state-space models

In order to capture the additional information provided by the use of polynomial models (PM),

we compared the classifications between PM and an adapted linear approach. The predominant method of estimating trends (linear models) does not separate measurement and process errors, although these are two of the four factors explaining the temporal changes observed in abundance data, along with temporal autocorrelation and systemic changes [START_REF] Dornelas | Quantifying temporal change in biodiversity: challenges and opportunities[END_REF]. Statespace models account for both process error and measurement error; and they have been shown

to provide better estimates of trend over wide ranges of time series length and sources of variation than the usual linear models [START_REF] Dennis | Estimating Density Dependence, Process Noise, and Observation Error[END_REF]Humbert et al., 2009).

Thus, we additionally modelled the loge transformed abundance estimates using state-space models (SSM). They partition the variance in abundance estimates into estimated process error (σ²) and measurement error (τ²) and estimates population trends (μ) through:

𝑋 𝑡 = 𝑋 𝑡-1 + µ + 𝜀 𝑡 (2)
where 𝑋 𝑡 and 𝑋 𝑡-1 are the loge transformed abundance estimates in the present and past year, with process noise represented by 𝜀 𝑡 ~𝑁(0, 𝜎 ² ). Measurement error was included following:

𝑌 𝑡 = 𝑋 𝑡 + 𝐹 𝑡 (3)
where 𝑌 𝑡 is the estimate of the true (unobserved) population abundance with measurement error:

𝐹 𝑡 ~𝑁(0, 𝜏 2 ). We substituted the estimate of population abundance (𝑌 𝑡 ) into Eq. 2:

𝑌 𝑡 = 𝑋 𝑡-1 + µ + 𝜀 𝑡 + 𝐹 𝑡 (4)

Given Eq. 3:

𝑌 𝑡 = 𝑌 𝑡-1 + µ + 𝜀 𝑡 + 𝐹 𝑡 -𝐹 𝑡-1 (5)
The trend, μ, was estimated by restricted maximum likelihood according to the methodology and R code provided by Humbert et al. (2009) for each population time series. We used Wald's 95%

confidence intervals to examine whether the trends differed from zero (either increasing or decreasing). We then compared the classifications resulting from the two models (polynomial and state-space) and quantified the percentage of fake "no trend".

| Quantifying stability based on population fluctuations

To quantify stability, we used five different metrics, namely the overall variability, the change in variability over time, the skewness of the residuals, the 'local extinction frequency' and the 'colonisation or extinction syndrome' (see definitions below).

1. Overall variability of a population was calculated both as the mean squared error (𝑀𝑆𝐸 = 𝜀 𝑖 2 𝑁 ) and as the coefficient of variation (𝐶𝑉), which is a relative measure of the dispersion of the data around the mean variability.

2. Change in variability over time for each population was calculated by regressing the squared residuals of the fitted polynomial model against time:

𝜀 2 (𝑡) = 𝜂 0 + 𝜂 1 𝑡 + 𝜔(𝑡) (6)
yielding to a regression coefficient (𝜂 1 ) for each population. We interpreted a significantly positive 𝜂 1 value as an increasing variability in time, and a significantly negative 𝜂 1 value as a decreasing variability in time.

3. We measured skewness of the residuals for each population. Skewness quantifies the symmetry of the distribution of the residuals.

4.

Local extinction frequency was the frequency of observed zero-abundance (number of times the population reaches zero abundance), that we interpreted as a metric of stability (high zero frequency, less stable).

5. Finally, we considered what we called the 'colonisation or extinction syndrome'. For this, we were interested in the location of the zeros within the time series. To do this, we split the time series in two and calculated the ratio (𝑅𝑎𝑡𝑖𝑜𝑍 = 𝐵𝑍+1 𝐸𝑍+1 ) between the number of zeros at the beginning (𝐵𝑍) and at the end (𝐸𝑍). Thus, a ratio greater than 1 indicates an aggregation of zeros at the beginning of the time series (as expected for colonizers), and a ratio less than 1 an aggregation at the end of the time series (as expected for populations experiencing a local extinction). The structure is also the same for all models with a quantitative fixed effect (Latitude, Body size, percentage of GHF change), with the exception of the identity of the fixed effect as well. For example, the latitude model for velocity is structured as follows:

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖,𝑗 = 𝛽 0 + 𝛽 1 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝑖,𝑗 + µ 0,𝑗 + Ɛ 𝑖,𝑗 where 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖,𝑗 is the estimate of the velocity for the i th population time series from the j th species, 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒 𝑖,𝑗 is the latitude of this population time series, 𝛽 0 the global intercept, 𝛽 1 the global slope estimates for the latitude (fixed effect), µ 0,𝑗 is the species-level departure from 𝛽 0 (random effect), and Ɛ 𝑖,𝑗 the random error.

We examined the significance of fixed effect terms in individual analyses (latitude, realm, taxa, body size, red list category and GHF) by examining the likelihood ratio of including each term versus a null model (containing only the random effects terms). We used Wald's 95% confidence intervals to determine whether velocity and change in variability over time (𝜂 1 ) differed significantly from zero (either increasing or decreasing), and to investigate significant differences

QUESTION

in overall variability (𝑀𝑆𝐸). All mixed-effect models were fit using maximum likelihood as implemented in the R package "lme4" [START_REF] Bates | Fitting Linear Mixed-Effects Models Using lme4[END_REF].

All data syntheses, visualisation and statistical analyses were conducted using R version 4.0.0.

| RESULTS

| Distribution of the population trajectories among non-linear classes

We found that among the 2,786 population trajectories considered, 610 -i.e. 22% -exhibited nonlinear dynamics (fig. 4). Concave and convex trajectories even represented a quarter of the constant cases. Also, we found that the proportion of non-linear dynamics varied between taxa (fig. 5). If we consider only terrestrial vertebrates -mammals, amphibians, birds, reptiles -which are the most studied groups when it comes to biodiversity dynamics, we find almost 30% of non-linear trajectories. This figure rises to 40% for mammals but also for reptiles. If we compare these results with those of state space models, we note clear differences (table 1).

The outputs of the state-space models suggest that 2,415 populations are constant, whereas the non-linear approach identifies 1,428 populations as constant. This means that the usual approach has a fake "no trend" percentage of over 40%. Among these misclassifications, 522 (239+46+201+36) time series are classified as constant when they are increasing or decreasing according to the PM, and 465 of them are concave or convex.

constant constant

Table 1. Comparison of the classifications according to the method used. Each cell represents the number of populations classified 'x' by the application of polynomial models (PM) in the column, and 'y' in the row by the application of state-space models (SSM). Impossible cases are marked with crosses (the same population cannot be described as decreasing by one model and increasing by the other).

| Characterisation of population dynamics according to their trends and stability

313 populations have significant changes in their variability over time (𝜂 1 significantly different from zero). Among these populations, 40% show an increasing variability over time (𝜂 1 > 0) and 60% show a decreasing variability (𝜂 1 < 0). 26% of the populations with an increasing variability in time have decreased (velocity < 0), 16% have increased (velocity > 0) and 58% have remained constant. 46% of the populations with decreasing variability in time have increased, 10% have decreased and the 44% have remained constant.

Velocity and the measure of change in variability over time (𝜂 1 ) are indeed negatively correlated (Pearson correlation coefficient = -0.39) (see supplementary figure S3), which means that declining populations become more unstable, while growing populations fluctuate less over time and stabilise. The change in stability over time is also negatively correlated with the ratio of zeros, so fluctuations are strong when populations have low abundances, either at the beginning of the time series for those that are growing or at the end for those that are declining. More generally, the frequency of zeros seems to be positively related to the metrics of overall variability (𝑀𝑆𝐸, 𝐶𝑉)

(see supplementary figure S3). We performed a PCA (using the calculated trends (velocity) and stability metrics i.e. 𝑀𝑆𝐸, 𝐶𝑉, 𝜂 1 , skewness of the residuals, local extinction frequency, and RatioZ, as well as latitude and longitude)

to investigate which metrics tend to better separate the population time series in similar groups.

The first two dimensions contain 45% of the total inertia (fig. 6a). The first dimension is driven by the frequency of zeros (contribution: 33.9%), the coefficient of variation (33.4%) and the skewness of residuals (13.8%). It contrasts time series populations with a high proportion of zeros and a high coefficient of variation with populations with few zeros and a low coefficient of variation (fig. 6a). As the frequency of zeros and the 𝐶𝑉 are highly correlated and the most contributing variable, we can consider that the local extinction frequency best separates the populations.

The second axis contrasts populations with increasing trends, with high overall variability but decreasing over time (e.g. fig. 6c), with populations that have lower overall variability but increasing over time, and that are in decline (e.g. fig. 6d). Following this approach, the factorial plane can be divided into four parts (fig. 6b). First, populations with an important number of zeros, positive trends with a decreasing variability in time, which corresponds to a colonizer profile (fig. 6c). Second, populations with an important number of zeros, negative trends with an increasing variability in time, which corresponds to an extinction profile (fig. 6d). Other populations are close to this profile, with a low proportion of zeros but decreasing trends, and correspond to declining populations but that are not close to extinction yet. Finally, populations with a low number of zeros and trends close to zero, gathering populations experiencing mainly constant dynamics.

The PCA suggests that populations differ mainly from their local extinction frequency, their velocity and their change in stability over time.

| Influence of biogeographic and species-level characteristics and anthropogenic pressures on population trends and stability

General patterns

We did not detect a latitudinal signature either in the distribution of trends (velocity), the overall variability (𝑀𝑆𝐸) (fig. 7a,b,c) or in the change in temporal variation (𝜂 1 ). In every realm, the trends were rather increasing, with a significantly positive mean in the marine domain (Velocity = 0.02, CI = [0.01, 0.03]; fig. 8a). The overall variability was less pronounced in the terrestrial than in the marine domain (terrestrial slope = 0.47, CI = [0.31, 0.63]; freshwater slope = 1.86, CI = [1.56, 2.18]; marine slope = 2.55, CI = [2.30, 2.79]; fig. 8b), but we did not detect a realm-related pattern in the case of change in variability over time (χ²=2.4, p=0.3).

a b c

Velocity

We found taxonomic patterns in both trends and overall variability (fig. 8c,d) but not in the change in variability over time (χ²=0.63, p=0.43). Amphibians emerged as the most endangered group, but also as the only group experiencing a net decline over time (slope = -0.03, CI = [-0.06, -0.001]).

In contrast, marine plants and invertebrates are the only groups that show net increases (respectively, slope = 0.04, CI = [0.003, 0.08]; slope = 0.08, CI = [0.05, 0.11]). All other groups are centred on zero with small differences in means, with confidence intervals that overlap zero (fig. 8c). Fluctuations were significantly higher for plant species and especially for marine plants, followed by the marine invertebrates, benthos and fishes' groups (fig. 8d).

Focus on terrestrial vertebrates

Examining the impact of body size and mass on measures of trends and stability among terrestrial vertebrates' populations, we found associations between population trends and overall variability in mammals, but not in the other taxa. Velocity of mammals' populations was positively correlated with body size (slope = 0.008, CI = [0.0009, 0.01]), while their overall variability was negatively correlated with body size (slope = -0.22, CI = [-0.29, -0.15]).

Contrary to our expectations, no clear link could be established between the trends or fluctuations of our populations and their IUCN Red List Category. Only birds showed a higher mean variability in 'vulnerable' populations than in 'least concern' populations (respectively, slope = 0.61, CI = [0.38, 0.84]; slope = 0.31, CI = [0.26, 0.36]). For reptiles, variability increases significantly over time for populations classified as 'endangered' (slope = 0.27, CI = [0.16, 0.38]). However, in general, it seems that the IUCN Red List Category does not explain the heterogeneity of trends or stability in the populations we studied, but these populations are also for the vast majority of them classified as 'least concern' (n=759; 84% of the 902 terrestrial vertebrates' populations).

Finally, we did not observe any significant impact of the rate of change of the Global Human Footprint (GHF) between 1993 and 2009 on the trends of terrestrial vertebrates' populations. The average rate of change on the considered locations was of +1%. Even if we expect pressures to increase globally, the locations considered may not reflect the global average rate of change, possibly explaining why we did not detect any impact. In terms of variability, the amphibians were the only group being significantly driven by the GHF. But curiously they show a negative correlation between overall variability and rate of pressure change (slope = -004, CI = [-0.06, -0.02]), meaning that their variability decreases when the pressure increases. The investigation of the impact of the GHF with a qualitative approach, using different classes of pressures, showed no significant results (χ²=1.44, p=0.6).

| DISCUSSION

There are real scientific and political challenges in characterising the state of global biodiversity.

The LPI is emerging as the political indicator by excellence, thanks to its capacity to communicate and raise the alarm. From a more scientific point of view, however, its degree of uncertainty and its aggregated nature do not allow to provide a satisfactory picture of biodiversity. On the other hand, many studies have sought to characterise biodiversity more precisely, in all its characteristics [START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF]. Despite their scientific rigour, the complexity of such approaches makes it difficult to disseminate a clear message that can be understood by as many people as possible. Our objective was therefore, humbly, to ask what could be a simple solution to characterise the heterogeneity of biodiversity, while keeping the approach simple enough to be understood and shared. In this report, we developed such an approach by identifying non-linear trajectories and measuring stability in time series of globally distributed populations. To do so, we merged the BioTIME and the Living Planet Databases, two global datasets on which most global analysis of biodiversity dynamics are based, generating the most comprehensive database on population dynamics at global scale. Three main hypotheses were explored.

First, we wanted to investigate the non-linear character of population dynamics at a global scale.

We established that 22% of the trajectories of the populations studied were non-linear, and that the application of state-space models generated 40% of fake "no trend". We also measured the stability in population trajectories. The negative relationship between temporal change in stability and velocity provided evidence to support our intuition that declining populations are also those that are subject to increasing instability. In contrast, populations described as colonising tended to stabilise over time. Overall, we identified 'winners' as well as 'losers', with rare and common populations in each category. Our final objective was to look for potential drivers that could explain the observed heterogeneity among trends and stability. No latitudinal signatures related to conservation status or anthropogenic pressures were noted. On the other hand, marine populations, as well as plants (all realms), were found to show greater overall variability than other groups, even for populations with a constant trend. In addition, amphibians were found to be a particularly endangered group with significantly declining trends.

| Consequences of a non-linear approach for the global analysis of biodiversity dynamics

Our analysis has shown that a non-linear method for studying population dynamics seems to define more precisely the overall pattern of population dynamics. In particular, we found that 40% of the populations that were classified as constant by applying the state-space models turn out to reveal recently increasing or decreasing dynamics, which could lead to very different decisionmaking in the perspective of setting up conservation policies. 22% of the 2,786 studied populations showed non-linear trajectories, up to 40% depending on the group.

Such a non-linear approach of population dynamics offers several advantages. First, it better reflects the complexity of the populations studied. Indeed, a classic linear approach, based solely on whether the trends are increasing or decreasing, masks significant heterogeneities and stochastic processes and may lead to errors in the identification of the type of trajectory. Applying a non-linear approach could also help to better quantify the extinction debt and protect biodiversity. It is noted that responses may come late after pressures, but that this debt is difficult to detect [START_REF] Kuussaari | Extinction debt: a challenge for biodiversity conservation[END_REF]. This type of methodology, with its ability to anticipate early declines or recoveries, is a potential candidate to contribute to this kind of analysis.

Beyond the shape of population trajectories, the pressures on ecosystems themselves are also not linear in many cases. Most human-induced pressures (e.g. habitat destruction, human population growth, CO2 emissions) are even more close to exponential since 1970's [START_REF] Steffen | The trajectory of the Anthropocene: The Great Acceleration[END_REF].

Therefore, the type of approach developed in this study should also be useful for describing pressures and testing the synchrony or potential delay of responses. This would make it possible to study the role played by the pressures in greater detail and, for example, to assess a time lag between the implementation of a conservation policy and the responses at the population level (Rigal et al., 2020). Finally, not only is there non-linearity in the trends themselves in pressures and responses, but studies have also shown that the interactions between the two are also nonlinear (defined as non-proportional here) [START_REF] Clark | Nonlinear population dynamics are ubiquitous in animals[END_REF]. This argues once again for a more sophisticated approach that allows more complex relationships to be established.

| The importance of assessing measures of stability when studying population dynamics

We also showed that populations were distinguished on the one hand by their heterogeneity in their local extinction frequency (rare vs. abundant) and on the other hand in their trends and variations (winners vs. losers). This analysis invites us to nuance the approach that opposes generalist and specialist species. Biotic homogenisation describes the process that shows the decline of specialist species in favour of generalist species [START_REF] Mckinney | Biotic homogenization: a few winners replacing many losers in the next mass extinction[END_REF][START_REF] Olden | Ecological and evolutionary consequences of biotic homogenization[END_REF]. Generalists seem to benefit from global changes by colonising new areas of distribution or increasing their population size, resulting in an overall decline in biodiversity in the genetic sense [START_REF] Fisher | The comparative method in conservation biology[END_REF]. The literature therefore suggests that specialists are the losers in this biodiversity crisis, with rare and declining populations, while generalists are the winners, with larger population sizes and increasing trends [START_REF] Clavel | Worldwide decline of specialist species: toward a global functional homogenization?[END_REF]. Our conclusions lead to a re-reading of this opposition, since it would seem that the level of disturbance is such that all scenarios are now possible. We identify both rare and abundant populations among winners and losers. We have not quantified the relative proportions of these different scenarios, but the disjunction is by no means simply dichotomous. Again, nuance is required, since rare or common winners do not reflect the same thing, nor do rare or commons losers. These measurable changes in assemblage structure in the recent past highlight the need to use multiple measures of biodiversity to understand changes in biodiversity.

The relation between the declining trends and the proximity with zero is also in line with what Caughley pointed out in his 1994 review 'Directions in conservation biology'. He stated that there are two paradigms in conservation biology; one approach is to focus on population decline, and the other on small populations. The smallness of populations has laid some of the theoretical foundations for conservation biology, for instance small populations are more exposed to environmental stochasticity and driven to extinction following small environmental changes. Yet, the paradigm suffers, according to Caughley, from focusing on situations already too late for active policy making. This also argues in favour of the added value of a non-linear approach in that it constitutes a form of anticipation: it allows an early diagnosis of a rarity syndrome where detection is usually too late. On the contrary, declines of large populations are in practice often used to detect responses to environmental pressures but suffer from the absence of clear theoretical foundations: the decline of a given population being often caused by multiple stressors acting simultaneously at different spatial and temporal scales. In this case, policy-making is not too late but too hazardous to prevent population declines. As our results show, these two types of dynamics (of rare versus abundant) are combined together in global datasets. This undermines even more the possibility to clearly identify univoqual trajectories for biodiversity conceived as an integrated category. In addition to this, population fluctuations also seem to play an important role in the decline of populations and could therefore contribute to the identification of endangered species. Therefore, instead of relying on unified approaches of population trajectories for policy making, it's more likely that, as Caughley had anticipated, "cautious intermixing of the two [paradigms] might well lead to a reduction in the rate at which species are presently going extinct" [START_REF] Caughley | Directions in Conservation Biology[END_REF].

Our results therefore show that the characterisation of populations by a single approach, whether it be the opposition winners/losers via trends or rare/common via abundance, lacks nuance. In fact, patterns of combination between trends, smallness and stability can be observed; these three characteristics must therefore be considered simultaneously.

| Distribution of trends and stability across space, realms and taxonomic groups

Concerning the investigation of biogeographic hotspots of biodiversity losses or gains, we found that declines, increases and variability occur across all latitudes. However, we have shown that populations in the marine realm experienced pronounced increases in their trends, accompanied by strong overall variability indicating a certain instability. This can be explained by several things. Firstly, climate change is positively correlated with other drivers in the marine realm (Bowler et al., 2020), and it is also known that marine species tend to be more sensitive to climate change [START_REF] Sunday | Thermal tolerance and the global redistribution of animals[END_REF][START_REF] Pinsky | Greater vulnerability to warming of marine versus terrestrial ectotherms[END_REF]. Monitoring species in marine areas is also more difficult as populations are more mobile and thus exposed to more fluctuations in the sampling [START_REF] Cassie | The Sampling Problem, with particular reference to Marine Organisms[END_REF], which could also explain the observed variability. On the other hand, the ecological characteristics of the marine environment, notably the fact that there are few barriers to dispersal and that colonisation rates are higher [START_REF] Lessios | Massive gene flow across the world's most potent marine biogeographic barrier[END_REF][START_REF] Waples | Separating the wheat from the chaff: patterns of genetic differentiation in high gene flow species[END_REF], are factors that could justify the increasing trends in this environment. Moreover, studies have demonstrated that marine communities experience much more rapid changes in compositions than terrestrial or freshwater ones [START_REF] Bowler | Mapping human pressures across the planet uncovers anthropogenic threat complexes[END_REF], which is consistent with the idea that marine assemblages may show increasing trends but are highly unstable.

Our work also joins a list of previous studies that have raised the alarm about the fate of amphibians [START_REF] Leung | Trends in mean growth and stability in temperate vertebrate populations[END_REF]Daskalova et al., 2020b). Whatever the model, whatever the data, this group does seem to be endangered and should therefore be a prime target for conservation policies. However, even though the other groups do not experience net declines, many of them are experiencing temporal variations, marine ones and plants in particular. This is expected to be a marker of instability or even an alarm to ecological collapse, being either at the ecosystem [START_REF] Scheffer | Early-warning signals for critical transitions[END_REF] or the global level [START_REF] Hughes | Multiscale regime shifts and planetary boundaries[END_REF]. This is also consistent with previous studies showing that the species composition of natural ecosystems is changing at an unprecedented rate: populations with increasing but unstable trends may be succeeding each other if they fail to settle permanently [START_REF] Dornelas | Assemblage Time Series Reveal Biodiversity Change but Not Systematic Loss[END_REF].

| Examining links between pressures and biodiversity responses

When looking for causes of heterogeneity in trends and variations in terrestrial vertebrates, we did not detect a significant impact of either the IUCN category or anthropogenic pressures across the GHF. Also, trends and variations were not found to be associated with body size, except for mammals for which a larger body size seems to be favourable, body size being positively correlated with the velocity and negatively correlated with the evolution of fluctuations. However, this stage of the analysis was more exploratory and deserves consolidated and larger datasets to be really conclusive. As pointed out above, the diversity of IUCN categories is very weakly represented in the data we used (84% of the species included being considered as Least Concern).

Concerning the GHF, only amphibians showed increasing trends for larger changes in pressures, but this result may be linked to the low abundance of data concerning amphibians (n=31 populations). Also, we considered the change in pressures and not the total level of pressure. It is possible that there is an anti-correlation between the GHF and its change. There are places where the GHF has decreased but because it was very high, while elsewhere the pressure may have increased but from a very low base. If it is the total level of pressure rather than the change that is affecting the trends, then the relationship may be the opposite of what would be expected by focusing on the impact of the change in pressure. As for the GHF itself, it is a quantitative data item that sums up various pressures but ignores the relationships between them. Pressures may have links between them, and their effects may add up, multiply or cancel each other out (Bowler et al., 2020). The relative importance of the various drivers should be taken into account in a more detailed analysis. It would also be interesting to directly link the time series concerning pressures with the time series of responses by use of specific methods such as convergent cross mapping [START_REF] Clark | Spatial convergent cross mapping to detect causal relationships from short time series[END_REF], for example, rather than studying the impact of a change in pressure between two time points on measures characteristic of our populations.

| Caveats and perspectives

Data selection

In our study, we chose to focus on very limited data through the application of numerous selection filters. This resulted in a very substantial loss of the originally available data (fig. 1). In particular, the selection of time series containing a minimum of fifteen points was perhaps too strong. There may be time series containing only two points, but which are very informative if the monitoring carried out is precise. In addition to the loss of data to be analysed caused by making a too drastic selection, this also results in the accentuation of taxonomic and geographical biases by concentrating our data on areas concentrating high levels of biodiversity, or high effort of species monitoring for example.

These taxonomic and geographical biases are present in our data anyway, even before any filter was applied. For example, there are very few population records from tropical and polar regions, and an overrepresentation of Europe and North America. Moreover, birds alone represent a quarter of the dataset, compared to only 1% for amphibians. The lack of tropical data, in particular, should be noted as it represents one of the most biodiverse regions [START_REF] Almeida-Cortez | Origins and maintenance of tropical biodiversity[END_REF], but also as recent intensifying pressures (e.g., mining and deforestation) affect this region [START_REF] Caballero Espejo | Deforestation and Forest Degradation Due to Gold Mining in the Peruvian Amazon: A 34-Year Perspective[END_REF]. In any case, these issues are pervasive across biodiversity datasets [START_REF] Godet | What Conservation Does[END_REF]. Future data compilations and new monitoring programs should focus on capturing under-represented geographic regions, taxa, and ranges of human impacts.

Comparison of modelling approaches

When comparing the non-linear and state-space model approaches, the sensitivity to the type of data used was not taken into account. The results of the state-space models on our data are insignificant compared to other studies that have been conducted. For example, Daskalova et al.

(2020) studied time series populations from the LPD and found that 15% of the populations were decreasing and 18% were increasing. Looking at the classification percentages for our subset of time series data from the LPD only (446 out of 2,786 in total), we find 18% of significant increasing trends and 6% of significant decreasing trends for the state space model. It is therefore likely that this type of model is not the most appropriate for studying the populations present in the BioTIME database, and that the percentage of fake "no trend" due to a linear approach has been overestimated. In fact, BioTIME is highly biased towards the marine realm, and there are many populations of small invertebrates or mollusc that vary greatly in abundance from one year to the next. For example, one study records no individual of Artica islandica in 1992, 416 individuals the next year, and again no individual in 1994 [START_REF] Liehr | The ocean quahog Arctica islandica L.: a bioindicator for contaminated sediments[END_REF]. The risk is that state-space models, by trying to consider measurement errors, erase all variability and therefore give insignificant results. If we compare velocity measurements with growth rates (µ), we observe a very strong link that seems linear (see supplementary figure S4). We also notice that the results overlap more for the data coming from the LPD, so that the SSM and non-linear approaches tend to give the same results for these data. This confirms the conclusion that state-space models are less suitable for studying BioTIME data.

Temporal and geographic limitations

The non-significance of the results in the study of the impact of drivers is potentially linked to the temporal variations of our data, some time series start in the 1970s and end in the 1990s for example. Necessarily, a change in anthropogenic pressures between 1993 and 2009 will not justify the trends and fluctuations observed, and could even lead to spurious results, as we have seen with amphibians. Similarly for the IUCN categories, the most recent version was used, dating from last year. The IUCN Red List is updated regularly, in order to monitor changes in biodiversity status. This reassessment can lead to a species being moved to another Red List category for several reasons (e.g. if the main threats have disappeared, if conservation measures have improved the status of the species; on the contrary if the main threats have persisted or increased, leading to a deterioration of the status of the species). Thus, considering the most updated category of the species considered, it is possible that this is not in line with the time period over which we have examined their trends and stability.

In addition, there are time gaps between the moment when a pressure is applied and the moment when it shows effects on populations. Such gaps have been observed, for example, in the evolution of forest populations following disturbances [START_REF] Daskalova | Rare and common vertebrates span a wide spectrum of population trends[END_REF]. It is also possible that monitoring only appears late in relation to the peak of disturbance. This suggests that the time series population data may not necessarily be temporally consistent with the periods of change, or that responses have not yet occurred if the pressures have only recently manifested themselves.

Also, the precise coordinates of the locations where individual populations were monitored are not available. Instead, we have considered the centroids of the original records as a proxy (for BioTIME). Although the average monitoring area per study is 4km², some are larger than this and therefore the centroids may not be representative of the monitoring locations. As the GHF is averaged around these centroids, this may have impacted our results as well.

| Conclusion

In this report, we explored non-linearity, stability and distribution between groups and locations as key aspects of global biodiversity dynamics. We created the largest database of population abundance data by merging the two most widely used databases in this field. Our analysis showed the ubiquitous nature of population dynamics on a global scale. We have shown that the pressures are such that we observe simultaneously declining populations, marked by increasing instability, growing populations, but also a whole range of non-linear trajectories. These dynamics are unevenly distributed among taxonomic groups, with amphibians being particularly at risk. The study of the stability revealed, however, that although there are no significant declines in most groups, or even increasing trends, this is by no means indicative of a good state of biodiversity, as it may mask strong variations in population composition, which may ultimately prove to be just as worrying as declines in specific species. We conclude that the use of a non-linear approach seems particularly promising and suitable for conservation use; and that in addition to trends, other populations' characteristics should be considered, such as their stability.

Our results come at the advent of a renewal of conservation issues made possible by the aggregation of precise data, on a global scale, on the fate of numerous populations, as well as on pressures of anthropic origin on the same scale. Previously, biodiversity analyses were carried out either in space but without temporal depth, or in time but without satisfactory spatial coverage. A spatio-temporal ecology of biodiversity is now possible on a large spatial scale since the creation of new databases and the implementation of new methods. So far unexplored predictions could be tested for the first time on such datasets, such as the search for empirical links between major pressures of anthropogenic origin, the development of the global network of protected area and biodiversity dynamics. For example, has the establishment of protected areas led to an increase in trends? Can we link changes in conservation policies and biodiversity responses, can we quantify the time lag between the implementation of measures and their impact? At the level of pressures, we can ask whether some have a stronger impact. It would also be interesting to establish links between the pressures themselves, perhaps to identify antagonistic driver complexes and counter the devastating effects of some. On the contrary, to identify complexes which, by acting in synergy, would be all the more dangerous for biodiversity. Nous supposons que les populations sont régies par des trajectoires non linéaires significatives, et que l'étude de la stabilité et de la distribution entre les groupes et les lieux pourrait contribuer à une caractérisation plus fine de l'état de la biodiversité. Nous avons analysé 2 786 séries chronologiques d'abondance en fusionnant les bases de données BioTIME et Living Planet Database. Nous avons estimé les dynamiques globales des populations en appliquant des modèles non linéaires et en cherchant des mesures de stabilité adaptées. Ensuite, des modèles linéaires à effets mixtes ont été utilisés pour tester les sources potentielles de l'hétérogénéité observée dans les tendances et la stabilité des populations étudiées. Nous avons constaté que 22% des populations présentaient des dynamiques non linéaires, voire 30% pour les vertébrés terrestres. Malgré le manque de signature géographique, nous avons trouvé des marqueurs taxonomiques montrant un déclin prononcé des amphibiens et une instabilité importante des populations marines. Ces changements mesurables dans la structure des populations soulignent la nécessité d'utiliser des approches permettant de caractériser la complexité des dynamiques de la biodiversité, comme la prise en compte de l'ensemble des trajectoires et de la stabilité. Cela permettra d'établir des liens plus précis entre les pressions et les réponses de la biodiversité.

Mots-clés : non-linéarité, stabilité écologique, dynamique des populations, analyse des tendances, crise de la biodiversité Abstract (1600 characters maximum): There are many questions and controversies surrounding the biodiversity crisis. General indices to describe the state of biodiversity, as well as classical linear approaches to describe population trends, are no longer satisfactory. We assume that populations are governed by significant non-linear trajectories, and that the study of stability and distribution between groups and locations could contribute to a finer characterisation of biodiversity status. We analysed 2,786 abundance time series by merging the BioTIME and Living Planet Databases. We estimated global population dynamics by applying non-linear models and searching for suitable stability measures. Next, linear mixed-effects models were used to test potential sources of observed heterogeneity in trends and stability of the study populations. We found that 22% of the populations showed non-linear dynamics, up to 30% for terrestrial vertebrates. Despite the lack of a geographical signature, we found taxonomic markers showing a pronounced decline in amphibians and significant instability in marine populations. These measurable changes in population structure underline the need to use approaches that characterise the complexity of biodiversity dynamics, such as taking into account the full range of trajectories and stability. This will allow more accurate links to be made between pressures and biodiversity responses.

Key Words: non-linearity, ecological stability, population dynamics, trend analysis, biodiversity crisis

  Par : Maëlys BOËNNEC Sous la direction de Monsieur Vincent DEVICTOR et Monsieur Vasilis DAKOS Equipe BioDICée, Institut des Sciences de l'Evolution Université de Montpellier -163 rue Auguste Broussonnet -Les analyses et les conclusions de ce travail d'étudiant n'engagent que la responsabilité de son auteur et non celle d'AGROCAMPUS OUEST Ce document est soumis aux conditions d'utilisation «Paternité-Pas d'Utilisation Commerciale-Pas de Modification 4.0 France» disponible en ligne http://creativecommons.org/licenses/by-nc-nd/4.0/deed.fr ingénieur d'AGROCAMPUS OUEST (École nationale supérieure des sciences agronomiques, agroalimentaires, horticoles et du paysage), école interne de L'institut Agro (lnstitut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement) de master d'AGROCAMPUS OUEST (École nationale supérieure des sciences agronomiques, agroalimentaires, horticoles et du paysage), école interne de L'institut Agro (lnstitut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement) ☐ de Montpellier SupAgro (étudiant arrivé en M2) ☐ d'un autre établissement (étudiant arrivé en M2)

  ....................................................................................................... Causes et chiffres clés de la crise de la biodiversité .......................................................................... 7 1.2. Enjeux et controverses autour de la crise de la biodiversité ............................................................. 7 1.3. Objectifs et questions de recherche .................................................................................................... 9 2 | Matériels et méthodes ......................................................................................................... 10 2.1. Données sur les populations ............................................................................................................. 10 2.2. Données sur la taille et la masse corporelles, le statut de conservation et les pressions anthropiques ................................................................................................................................................ 12 2.3. Modélisation des tendances et des trajectoires des populations à l'aide de modèles polynomiaux ....................................................................................................................................................................... 12 2.4. Modélisation des tendances des populations à l'aide de state-space models ............................. 13 2.5. Quantification de la stabilité à partir des fluctuations ..................................................................... 13 2.6. Etude des sources potentielles d'hétérogénéité des tendances et de la stabilité des populations .....................................................................................................................................................................Distribution des trajectoires parmi les classes non linéaires ........................................................... 14 3.2. Caractérisation des dynamiques de population en fonction de leurs tendances et de leur stabilité ....................................................................................................................................................................... 14 3.3. Influence des caractéristiques biogéographiques et spécifiques et des pressions anthropiques sur les tendances et la stabilité des populations ....................................................................................: issues and controversy on the measurement of the global biodiversity crisis ...................................................................................................................................................................... 1.1. Causes and key figures of the biodiversity crisis ........................................................................... 1.2. Issues and controversies surrounding the biodiversity crisis ...................................................... 1.3. Objectives and research questions ................................................................................................ 2 | Materials and methods .................................................................................................................

3. 2 |

 2 Caractérisation des dynamiques de populations en fonction de leurs tendances et de leur stabilité 313 populations présentent des changements significatifs de leur variabilité dans le temps. Parmi ces populations, 40% présentent une variabilité croissante dans le temps et 60% présentent une variabilité décroissante. 26% des populations présentant une variabilité croissante dans le temps ont diminué, 16% ont augmenté et 58% sont restées constantes. 46 % des populations présentant une variabilité décroissante dans le temps ont augmenté, 10 % ont diminué et 44 % sont restées constantes.La vélocité et la mesure du changement de la variabilité dans le temps sont en effet négativement corrélées (coefficient de corrélation de Pearson = -0,39) (voir figure supplémentaire S3), ce qui signifie que les populations en déclin deviennent plus instables, tandis que les populations en croissance fluctuent moins dans le temps et se stabilisent. L'évolution de la stabilité dans le temps est également corrélée négativement avec le ratio de zéros, de sorte que les fluctuations sont fortes lorsque les populations ont de faibles abondances, soit au début de la série temporelle pour celles qui sont en croissance, soit à la fin pour celles qui sont en déclin (voir figure supplémentaire S3).Nous avons effectué une ACP (en utilisant les tendances (vélocité) calculées et les métriques de stabilité) pour étudier quelles métriques tendent à mieux séparer les séries temporelles en groupes similaires.Les deux premières dimensions contiennent 45% de l'inertie totale (fig.6a). La première dimension est déterminée par la fréquence des zéros (contribution : 33,9%), le coefficient de variation (33,4%) et l'asymétrie des résidus (13,8%). Elle oppose les populations de séries temporelles présentant une forte proportion de zéros et un coefficient de variation élevé aux populations présentant peu de zéros et un faible coefficient de variation (fig.6a). Comme la fréquence des zéros et le CV sont fortement corrélés et constituent les variables les plus contributives, nous pouvons considérer que la fréquence d'extinction locale sépare le mieux les populations.Le deuxième axe oppose les populations dont la tendance est à la hausse, avec une variabilité globale élevée mais qui diminue dans le temps (par exemple, fig.6c), aux populations dont la variabilité globale est plus faible mais qui augmentent dans le temps, et qui sont en déclin (par exemple, fig.6d).Suivant cette approche, le plan factoriel peut être divisé en quatre parties (fig.6b). L'ACP suggère que les populations diffèrent principalement par leur fréquence d'extinction locale, leur vélocité et leur changement de stabilité dans le temps.3.3 | Influence des caractéristiques biogéographiques et spécifiques et des pressions anthropiques sur les tendances et la stabilité des populationsTendances généralesNous n'avons pas détecté de signature latitudinale ni dans la distribution des tendances (vélocité), ni dans la variabilité globale (fig.7a, b, c), ni dans le changement de la variation temporelle.Dans tous les domaines, les tendances étaient plutôt à la hausse, avec une moyenne significativement positive dans le domaine marin.Nous avons trouvé des marqueurs taxonomiques à la fois dans les tendances et la variabilité globale (fig.8c,d) mais pas dans le changement de la variabilité dans le temps. Les amphibiens sont apparus comme le groupe le plus menacé, mais aussi comme le seul groupe connaissant un déclin net au fil du temps. En revanche, les plantes marines et les invertébrés sont les seuls groupes qui présentent des augmentations nettes. Tous les autres groupes sont centrés sur zéro avec de faibles différences de moyennes, et des intervalles de confiance qui recouvrent zéro (fig.8c). Les fluctuations sont significativement plus élevées pour les espèces végétales et surtout pour les plantes marines, suivies par les groupes des invertébrés marins, des benthos et des poissons (fig.8d). Les populations marines semblent donc se distinguer par des tendances à la hausse, mais de fortes fluctuations par rapport aux autres groupes.

Figure 1 .

 1 Figure 1. Data selection framework. Temporal coverage, number of time series, species and unique geographical locations according to the different steps of the selection. Each circle corresponds to a data selection step. The selection criteria are indicated in bold. In each circle, the number of time series, species and geographical locations corresponding to the sub-dataset is specified. The map represents the locations of the data from the final database. Each point represents a population time series and is coloured according to its original dataset (i.e. BioTIME in green and Living Planet Database in brown) and sized according to its duration.

Figure 2 .

 2 Figure 2. Conceptual diagram of the classification of the population trajectories and the quantification of their trends and stability. 𝑋 𝑖 represents the loge transformed abundance of the i th population. The illustration of velocity and stability is taken from an example in the dataset (Alcyonidium hirsutum), with grey intervals representing the standard error. The midpoint Xm at which the velocity is calculated is marked by a vertical grey line. Trajectories classes are from Rigal et al., 2020.

1

  st stage analyses: Quantify population trends and stabilityQUESTIONMODELSHow can we classify the population trajectories and measure their stability?Step 1: fitting polynomial models. Classification of the trajectory, calculation of the velocity.𝑋 𝑖 = 𝛼 0,𝑖 + 𝛼 1,𝑖 𝑡 + 𝛼 2,𝑖 𝑡 2 + 𝜀INPUT OUTPUTAbundance data from population timeseries with more than 15 survey points and less than 50% of null abundances points (BioTIME, Living Planet Database)Population trends (velocity), population trajectories, measures of stabilityStep 2: fitting linear models to the squared residuals to evaluate the change in stability over time.𝜀 2 = 𝜂 0 + 𝜂 1 𝑡 + 𝜔 𝑀𝑆𝐸 = significant change (constant cases); and velocity, i.e. the magnitude of change.

Figure 3 .

 3 Figure 3. Conceptual diagram of the investigation of geographic, taxonomic, species-level traits and threats patterns within populations trends and stability. Trends (velocity) and stability metrics (𝑀𝑆𝐸, 𝜂 1 ) estimated from the first stage (Figure 2) are modelled using a generalized linear mixed-effect model framework.

  (GHF), species' IUCN Red List Category X ~ GHF, random = species X ~IUCN Red List Category, random = species

Figure 4 .

 4 Figure 4. Classification of population trajectories. 'n' indicates the number of populations. Classification of the 2,786 population time series trajectories into the nine possible linear (A, B, C) and non-linear (D, E, F, G, H, I) classes. Loge transformed abundances (black dots) with standard error (grey intervals) are shown for one species of each class. Second order polynomials are shown by a bold coloured line and the midpoint Xm at which the velocity is calculated is marked by a vertical grey line.

Figure 5 .

 5 Figure 5. Percentage of each type of trajectories according to each taxonomic group. 'n' indicates the number of populations.

Figure 6 .

 6 Figure 6. Characterisation of the different population dynamics according to the stability metrics, trends and geographical locations. a. PCA graph of variables. We consider the stability metrics (𝑀𝑆𝐸, 𝐶𝑉, 𝜂 1 (eta), skewness of the residuals (Skew), local extinction frequency (Freq), and RatioZ), the trends (velocity) and the geographical locations (latitude and longitude) as the variables. b. PCA graph of individuals. The point labels are shown for the 30 most contributing individuals. c. Example of a time series with an increasing trend and decreasing variability. d. Example of a time series with a decreasing trend and increasing variability. Grey intervals in c and d represent the standard error.

Figure 7 .

 7 Figure 7. Population declines, increases and variability occur across all latitudes. a and c show the distribution of population trends (velocity) and overall variability (𝑀𝑆𝐸) across latitude. b shows the trends on the map, colors referring to the velocity and size to the duration of the time series.

Figure 8 .

 8 Figure 8. Population trends (velocity) and overall variability (𝑴𝑺𝑬) vary across realms and taxa. a and b show the distribution of population trends and fluctuations respectively across realms including raw values (coloured points) and errorbars (including the mean estimated by the GLMM and the confidence interval in black). c and d are constructed the same way but according to taxa.

  Spécialisation / option : Master de Modélisation en écologie de l'Université de Rennes 1 Enseignant référent : Frédéric HAMELIN Auteur(s) : Maëlys BOËNNEC Organisme d'accueil : Institut des Sciences de l'Évolution de Montpellier Maîtres de stage : Vincent DEVICTOR, Vasilis DAKOS Année de soutenance : 2020 -2021 Titre français : Trajectoires non linéaires et stabilité des dynamiques de populations à l'échelle mondiale Titre anglais : Non-linear trajectories and stability in population dynamics at a global scale Résumé (1600 caractères maximum) : Les questions et les controverses entourant la crise de la biodiversité sont nombreuses. Les indices généraux pour décrire l'état de la biodiversité, ainsi que les approches linéaires classiques pour décrire les tendances des populations, ne sont plus satisfaisants.

TABLE OF CONTENTS

 OF Remerciements ............................................................................................................................................ 1 Glossary ......................................................................................................................................................... 2

List of abbreviations and acronyms ..

INTRODUCTION : ENJEUX ET CONTROVERSES SUR LA MESURE DE LA CRISE MONDIALE DE LA BIODIVERSITÉ

  ..... 2.1. Population data ................................................................................................................................ 2.2. Body size and mass, conservation status and anthropogenic pressure data .......................... 2.3. Modelling population trends and trajectories using polynomial models ...............................

	1.1 | Causes et chiffres clés de la crise de la biodiversité
	Comme l'illustrent les derniers rapports mondiaux sur la biodiversité, la diversité biologique est
	confrontée à une crise majeure (WWF, 2020 ; IPBES, 2019). L'IPBES estime que 75 % de
	l'environnement terrestre et 66 % de l'environnement marin sont affectés par les activités
	humaines, et que la dégradation des sols a réduit la productivité mondiale des terres de 23 %
	(IPBES, 2019). Le dernier rapport du WWF indique que 70% de la perte de biodiversité observée
	est due à la production agricole (WWF, 2020). Des études ont montré que le taux d'extinction de

2.4. Modelling population trends using state-space models .......................................................... 2.5. Quantifying stability based on fluctuations ............................................................................... 2.6. Testing potential sources of heterogeneity among population trends and stability .............. 3 | Results ................................................................................................................................................... 3.1. Distribution of population trajectories among non-linear classes .............................................. 3.2. Characterisation of population dynamics according to their trends and stability .................... 3.3. Influence of biogeographic and species-level characteristics and anthropogenic pressures on population trends and stability .............................................................................................................. 4 | Discussion ............................................................................................................................................. 4.1. Consequences of a non-linear approach for the global analysis of biodiversity dynamics .... 4.2. The importance of assessing measures of stability when studying population dynamics ....... 4.3. Distribution of trends and stability across space, realms and taxonomic groups..................... 4.4. Examining links between pressures and biodiversity responses ................................................ 4.5. Caveats and perspectives ............................................................................................................... 4.6. Conclusion ........................................................................................................................................ References ................................................................................................................................................. Supplementary materials........................................................................................................................ Abstract ...................................................................................................................................................... Part 1: Detailed summary in French _________________________________________________________________________________ 1 | plusieurs groupes ou la réduction d'écosystèmes spécifiques ont été exceptionnellement élevés depuis la présence de l'homme par rapport aux fluctuations attendues en dehors des grands changements géologiques. Les causes immédiates qui ont été identifiées sont principalement la perte, la dégradation et la fragmentation des habitats, la surexploitation, la pollution et le changement climatique. Bien que les crises de la biodiversité ne soient pas nouvelles, l'ampleur de la crise actuelle et sa rapidité particulière par rapport aux précédentes reflètent l'extrême impact des activités humaines (Walters et Scholes, 2017). Les cinq précédentes crises majeures ont éliminé entre 50% et 95% des espèces

1. Éviter le chevauchement des données : Comme les

	deux bases de données sont des
	compilations de séries temporelles de population, nous devions nous assurer qu'aucune série
	temporelle n'était comptée deux fois, c'est-à-dire qu'il n'y avait pas de chevauchement entre les
	deux bases de données. Pour cela, nous avons exclu de la LDP toutes les données qui concernaient
	des espèces présentes dans les deux bases de données.
	2. Eviter les anachronismes : La base de données BioTIME couvre une période allant de 1874 à
	2016, bien que la plupart des données proviennent des 50 dernières années ; la base LPD couvre
	une période allant de 1970 à 2015. Nous nous sommes donc concentrés sur la période commune
	et avons sélectionné uniquement les données collectées entre 1970 et 2015 dans les deux
	ensembles de données.
	3. Nature des données sélectionnées : Les populations ont été étudiées à l'aide de différentes
	méthodes (par exemple, exprimées sous forme de comptages, d'estimations, d'indices, de
	biomasses ou de densités), mais nous n'avons inclus que les populations suivies à l'aide du
	nombre d'individus (abondance) ou des densités, car il s'agit des mesures les plus pertinentes
	pour répondre à nos questions.

4.

Longueur et composition des séries temporelles : Nous avons

	2.2 | Données sur la taille et la masse corporelles, le statut de conservation et les
	pressions anthropiques
	Nous avons récupéré la taille (et la masse) corporelle et le statut de conservation des espèces ainsi
	que les pressions qui pourraient influencer les tendances et la stabilité des populations étudiées.
	Ces informations n'étant pas disponibles pour tous les groupes, nous avons limité cette partie de
	l'étude aux vertébrés terrestres, un groupe qui a été largement documenté et étudié (Titley et al.,
	2017). Cela a restreint notre ensemble de données à 902 séries temporelles, en considérant 691
	espèces d'amphibiens, de reptiles, d'oiseaux et de mammifères à 357 emplacements
	géographiques. Nous avons considéré trois variables comme des prédicteurs potentiels des
	tendances et des variations de la population, notamment la taille corporelle, les catégories de la
	liste rouge de l'UICN et le « Gobal Human Footprint » (GHF). Voir détails dans la partie 2.
	2.3 | Modélisation des tendances et des trajectoires des populations à l'aide de
	modèles polynomiaux
	Afin d'étudier les tendances et les trajectoires, chaque série temporelle de population a été ajustée
	avec un modèle polynomial du second ordre pour quantifier la tendance et classifier la trajectoire
	(Rigal et al., 2020) à travers :	considéré les séries temporelles
	allant de la première population non nulle à la dernière population non nulle, car nous voulons
	calculer la dynamique des populations pour les espèces qui sont réellement présentes (notez que 𝑋(𝑡) = 𝛼 0 + 𝛼 1 𝑡 + 𝛼 2 𝑡 2 + 𝜀(𝑡) (1)
	des valeurs nulles peuvent encore exister entre la première et la dernière valeur non nulle). Nous Où 𝑋 est l'estimation de l'abondance loge transformée (nous avons ajouté 1 à chaque comptage de
	avons omis les populations pour lesquelles les données de suivi étaient inférieures à quinze points population, pour tenir compte des estimations d'abondance nulle pour une année donnée), 𝑡 le
	dans le temps (c.f. partie 2 pour de plus amples informations). Les séries temporelles de temps en années, avec l'erreur de processus représentée par 𝜀 ~ 𝑁(0, 𝜎 ² ).
	population pour lesquelles 50 % ou plus de l'ensemble de données est représenté par des
	abondances nulles ont également été omises, afin d'éviter la présence de faux zéros attribuables La caractérisation qualitative de la trajectoire a conduit à l'identification de neuf classes (fig. 2).
	à une erreur d'observation. Sur le plan quantitatif, nous nous sommes intéressés à deux métriques caractéristiques : la
	direction, c'est-à-dire le déclin, l'accélération ou l'absence de changement significatif (cas
	constants) ; et la vitesse, c'est-à-dire l'amplitude du changement. Dans les cas linéaires (𝛼 2 non
	Il en résulte un ensemble final de données comprenant 2 786 séries temporelles (allant de 15 à significatif), nous avons simplement considéré 𝛼 1 , son signe indiquant la direction et sa valeur
	45 ans, avec une durée moyenne de 23,7 ans), représentant 2 320 espèces et réparties sur 835 absolue la vélocité. Dans les cas non linéaires (𝛼 2 significatif), nous nous sommes placés au point
	lieux géographiques différents. Elle contient des données parmi les domaines terrestre, d'eau 𝑋 𝑚 , centre de l'intervalle [𝑋 0 , 𝑋 𝑇 ] (correspondant à toute la longueur de la série temporelle). Nous
	douce et marin ; et considère les populations parmi les amphibiens, les benthos, les oiseaux, les ̇), le signe indiquant la direction et la valeur absolue étudions alors la tangente en ce point (𝑌(𝑋 𝑚 ) poissons, les invertébrés d'eau douce, les plantes d'eau douce, mammifères, invertébrés marins, la vitesse.
	plantes marines, reptiles, invertébrés terrestres et plantes terrestres. Nous appellerons cette approche l'approche "non linéaire" car elle permet aux populations d'avoir
	des trajectoires non linéaires.

Original report (English version) 1 | INTRODUCTION: ISSUES AND CONTROVERSY ON THE MEASUREMENT OF THE GLOBAL BIODIVERSITY CRISIS 1

  La caractérisation de l'état de la biodiversité mondiale présente de réels enjeux scientifiques et politiques. L'Indice Planète Vivante du WWF s'impose comme l'indicateur politique par excellence, grâce à sa capacité à communiquer et à tirer la sonnette d'alarme. D'un point de vue En revanche, les populations marines, ainsi que les plantes (tous domaines confondus), ont montré une plus grande variabilité globale que les autres groupes, même pour les populations ayant une tendance constante. En outre, les amphibiens se sont révélés être un groupe particulièrement menacé, avec des tendances significativement à la rapport, nous avons exploré la non-linéarité, la stabilité et la distribution entre les groupes et les lieux comme des aspects clés de la dynamique de la biodiversité mondiale. Nous .1 | Causes and key figures of the biodiversity crisis

	baisse. As the latest global reports on biodiversity illustrate, biological diversity is facing a major crisis
	(WWF, 2020; IPBES, 2019). IPBES estimates that 75% of the terrestrial environment and 66% of
	Voir partie 2 pour plus d'éléments de discussions. the marine environment are affected by human activities, and that soil degradation has reduced
	global land productivity by 23% (IPBES, 2019). The last WWF report indicates that 70% of the
	observed loss of biodiversity is due to agricultural production (WWF, 2020). Studies have shown
	that the rate of extinction of several groups (e.g. birds (Pimm et al., 2006), mammals (Ceballos et
	al., 2015)) or the reduction of specific ecosystems (e.g. tropical environments (Laurance, 2007))
	La relation négative entre l'évolution de la stabilité dans le temps et la vélocité a permis de
	confirmer notre intuition selon laquelle les populations en déclin sont aussi celles qui sont
	soumises à une instabilité croissante. En revanche, les populations décrites comme colonisatrices
	ont eu tendance à se stabiliser dans le temps. Dans l'ensemble, nous avons identifié des "gagnants"

plus scientifique, cependant, son degré d'incertitude et son caractère agrégé ne permettent pas de fournir une image satisfaisante de la biodiversité. A l'inverse, de nombreuses études ont cherché à caractériser plus précisément la biodiversité, dans toutes ses caractéristiques

[START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF]

. Malgré leur rigueur scientifique, la complexité de ces approches rend difficile la diffusion d'un message clair et compréhensible par le plus grand nombre. Notre objectif était donc, humblement, de nous demander quelle pourrait être une solution pour caractériser l'hétérogénéité de la biodiversité, tout en gardant l'approche suffisamment simple pour être comprise et partagée. Dans ce rapport, nous avons développé une telle approche en identifiant des trajectoires non linéaires et en mesurant la stabilité dans des séries temporelles distribuées sur tout le globe. Pour ce faire, nous avons fusionné les bases de données BioTIME et Living Planet Database, deux ensembles de données mondiales sur lesquelles reposent la plupart des analyses globales de la dynamique de la biodiversité, générant ainsi la base de données la plus complète sur la dynamique des populations à l'échelle mondiale. Trois hypothèses principales ont été explorées.

Premièrement, nous avons voulu étudier le caractère non linéaire de la dynamique des populations à l'échelle mondiale. Nous avons établi que 22% des trajectoires des populations étudiées étaient non linéaires, et que l'application de state-space models générait 40% de fausses tendances constantes. Nous avons également mesuré la stabilité des trajectoires des populations. et des "perdants", avec des populations rares et communes dans chaque catégorie. Notre objectif final était de rechercher les moteurs potentiels qui pourraient expliquer l'hétérogénéité observée entre les tendances et la stabilité. Aucune signature latitudinale liée au statut de conservation ou aux pressions anthropiques n'a été relevée.

Conclusion

Dans ce avons créé la plus grande base de données sur l'abondance des populations en fusionnant les deux bases de données les plus utilisées dans ce domaine. Nous avons montré que les pressions sont telles que nous observons simultanément des populations en déclin, marquées par une instabilité croissante, des populations en croissance, mais aussi toute une gamme de trajectoires non linéaires. Ces dynamiques sont inégalement réparties entre les groupes taxonomiques, les amphibiens étant particulièrement menacés. L'étude de la stabilité a cependant révélé que, bien qu'il n'y ait pas de déclin significatif dans la plupart des groupes, voire des tendances à la hausse, cela n'est en aucun cas indicatif d'un bon état de la biodiversité, car cela peut masquer de fortes variations dans les populations, qui peuvent finalement s'avérer tout aussi inquiétantes que les déclins d'espèces spécifiques. Nous concluons que l'utilisation d'une approche non linéaire semble particulièrement prometteuse et adaptée à la biologie de la conservation ; et qu'en plus des tendances, d'autres caractéristiques des populations devraient être prises en compte, comme leur stabilité.

Nos résultats interviennent à l'aube d'un renouveau des problématiques de conservation rendu possible par l'agrégation de données précises, à l'échelle globale, sur le devenir de nombreuses populations, ainsi que sur les pressions d'origine anthropique à la même échelle. Auparavant, les analyses de la biodiversité étaient réalisées soit dans l'espace mais sans profondeur temporelle, soit dans le temps mais sans couverture spatiale satisfaisante. Une écologie spatio-temporelle de la biodiversité est désormais possible à grande échelle spatiale depuis la création de nouvelles bases de données et la mise en oeuvre de nouvelles méthodes. Des prédictions jusqu'ici inexplorées pourraient être testées pour la première fois sur de tels ensembles de données, comme la recherche de liens empiriques entre les pressions majeures d'origine anthropique, le développement du réseau mondial d'aires protégées et la dynamique de la biodiversité.

Part 2:

has been unusually high since the presence of humans compared to fluctuations expected outside of major geological changes. The immediate causes that have been identified are mainly habitat loss, degradation and fragmentation, over-harvesting, pollution, and climate change. Although biodiversity crises are not new, the scale of the current crisis and its particular speed compared to previous ones reflect the extreme impact of human activities (Walters and Scholes, 2017). The previous five major crises, Cambrian, Devonian, Permian, Triassic and Cretaceous, eliminated

This partition of the stability was then used to examine the correlations among the different quantitative metrics and with the velocity by calculating the Pearson's correlation coefficients. A PCA has also been conducted to see if these metrics could be used to partition the populations into different groups and to characterise them.

| Testing potential sources of heterogeneity among population trends and stability

In addition, we investigated the influence of biogeographic and taxonomic patterns, species-level traits and anthropogenic pressures on population trends and stability. To do so we used a generalized linear mixed-effects model (GLMM) framework. We took either the velocity (i.e. population trend), the 𝑀𝑆𝐸 (i.e. population overall variability) or the 𝜂 1 (i.e. the change in variability over time) as the response variables, and geographic coordinates, taxa, realm, species body sizes, conservation status or anthropogenic pressures as the explanatory variables in separate models (fig. 3). Each model included species as a random effect to account for the possible correlation between populations from the same species.

To assess geographical and taxonomic patterns, we modelled trends and stability metrics as a function of latitude, realm and taxa separately, resulting in three models for each metric. To assess the impact of pressures (Red List category, GHF) and species-level traits (body size and mass) on terrestrial vertebrates, we modelled trends and stability metrics by taxa against the three predictors separately, resulting in 12 models for each metric.

The structure is the same for all models with a categorical fixed effect (Realm, Taxa, Red List category, Pressure class), with the exception of the identity of the fixed effects. For example, the taxa model for velocity is structured as followed:

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖,𝑗,𝑘 = 𝛽 0 + 𝛽 𝑘 𝑡𝑎𝑥𝑎 𝑖,𝑗,𝑘 + µ 0,𝑗 + Ɛ 𝑖,𝑗,𝑘 where 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖,𝑗,𝑘 is the estimate of the velocity for the i th population time series from the j th species from the k th taxa, 𝑡𝑎𝑥𝑎 𝑖,𝑗,𝑘 is the taxa of the i th time series from the j th species, 𝛽 0 the global intercept, 𝛽 𝑘 the global slope estimates for the k th taxa (fixed effect), µ 0,𝑗 is the species-level departure from 𝛽 0 (random effect), and Ɛ 𝑖,𝑗,𝑘 the random error (unreliable measurements, random fluctuations). The number of randomly selected points in this area for each case is indicated in brackets. The results for an area of 4km² are taken as reference. We concluded that the sensitivity to the area considered was not very strong for small areas, between 1 and 6 km², the points being close to the x=y regression line with a few exceptions. However, the pattern becomes more and more disparate as the area increases. In order to continue this analysis, it would be interesting to separate the populations whose original record location extends over more than 4km² into several subpopulations. The line of dots at 128 probably corresponds to points on the sea coast. The GHF is only evaluated for land areas and arbitrarily set to 128 for all coordinates referring to sea areas.

SUPPLEMENTARY MATERIALS