Rémi Choquet
email: remi.choquet@cefe.cnrs.fr

Jean-Dominique Lebreton

Olivier Gimenez

Anne-Marie Reboulet

U-CARE: Utilities for performing goodness of fit tests and manipulating CApture-REcapture data

Keywords: Arnason-Schwarz model, Contingency tables, Directional tests, Memory model, Overdispersion coefficient, Test of multinomial mixture, Test WBWA, Test 3G.SR, Test M.ITEC, Transient, Trap-dependence

), (Pradel et al. 2005). It is a free and stand-alone application for Windows.

The estimation and comparison of demographic rates in animal and plant populations [START_REF] Lebreton | Modeling survival and testing biological hypotheses using marked animals: A unified approach with case studies[END_REF], [START_REF] Gregg | Comparison of size vs. life-state classification demographic models for the terrestrial orchid Cleistes bifaria[END_REF] is currently based on following marked individuals over time and possibly space and analyzing the resulting data with adequate statistical models. Such models, in which individuals may move among states (such as geographical sites or breeding statuses) over discrete time periods, must account for the non-exhaustive detection of individuals. These so-called capture-recapture models are still evolving [START_REF] Pradel | Principles and interest of GOF tests for multistate capture-recapture models[END_REF]. In their current form, they are based on two key assumptions.

1) The individuals are assumed to be independent, 2) Within the history of each individual, the successive pairs of each release associated with its subsequent reencounter (if any) are assumed to be independent [START_REF] Burnham | On a unified theory for release-resampling of animal populations[END_REF].

While fitting multi-state capture-recapture models can now be carried out without much difficulty with a variety of computer software applications (M-SURGE (Choquet et al. 2004 #340), E-SURGE [START_REF] Choquet | Program {E-SURGE}: A software application for fitting multievent models[END_REF] or MARK [START_REF] White | Program MARK: Survival estimation from populations of marked animals[END_REF]) for a frequentist approach, MARK for Bayesian inference), the examination of the above mentioned assumptions can be done in an optimal way [START_REF] Pradel | A Proposal for a Goodness-of-Fit Test to the Arnason-Schwarz Multisite Capture-Recapture Model[END_REF] only with program U-CARE. And yet this step is critical. For instance, if a major structural effect such as the presence of transient individuals on the study site is overlooked, survival will be severely underestimated. More generally, spurious effects will often be detected if no model fits the data. This is best seen when considering how model selection proceeds.

In the frequentist paradigm, model selection is generally based on the Akaike information criterion (AIC, see [START_REF] Burnham | Model selection and Inference: A practical Informationtheoretic approach[END_REF]). The AIC is equal to the deviance plus two times the number of estimable parameters. In presence of lack of fit, the deviance tends to be inflated, thus leading to the selection of over-parameterized models and potentially to erroneous biological conclusions. Moreover, the estimate of precision of the Maximum Likelihood Estimators (MLE) will be overly optimistic. The consequences of lack-of-fit are thus too deleterious to be ignored.

A preliminary assessment of Goodness-Of-Fit (GOF) is thus a crucial prerequisite, just as in any statistical analysis [START_REF] D'agostino | Seasonal survival of Greater Snow Geese and effect of hunting under dependence in sighting probability[END_REF]. Unlike in regression, residuals are not easily available in CR to check the validity of a model. Furthermore, the omnibus approach to goodness-offit testing that consists of comparing expected vs. observed sufficient statistics is impractical due to the sparseness of the data. A specific approach was developed (reviewed in [START_REF] Pradel | A Proposal for a Goodness-of-Fit Test to the Arnason-Schwarz Multisite Capture-Recapture Model[END_REF]. By making this approach available in an easy-to-use software application, we aim at encouraging practitioners to assess the validity of their assumptions. Sound model selection, i.e. preceded by appropriate GOF assessment, is indeed becoming more and more common in the literature (e.g. (Henaux et al. 2007), [START_REF] Jenouvrier | Recruitment processes in long-lived species with delayed maturity: estimating key demographic parameters[END_REF])) but it is still far from being systematic. We describe here the main features of program U-CARE. More details on tests and tools can be found in the user's manual [START_REF] Choquet | U-CARE 2.2 User's Manual[END_REF]).

U-CARE (<http://purl.oclc.org/NET/U-CARE>) is a free, easy-to-use, stand-alone, menu-driven computer application for Windows, with a set of options for managing data and GOF tests capabilities for both single-state and multi-state capture-recapture models (see Figure 1). The two kinds of test are presented as the sum of components examining different aspects of the data through a range of contingency tables. This structure guides the choice of an appropriate model. For instance, if the only significant subcomponent of the multistate goodness-of-fit test is TEST 3G.SR, a two age-class structure for survival should be used [START_REF] Pradel | Transient animals in a resident population of snow geese: local emigration or heterogeneity[END_REF], while, if this is TEST WhereBeforeWhereAfter (WBWA), a memory model is recommended [START_REF] Pradel | A Proposal for a Goodness-of-Fit Test to the Arnason-Schwarz Multisite Capture-Recapture Model[END_REF]. Due to historical reasons, biological relevance is particularly furthered for single-state data through the computation of directional tests. We briefly examine hereafter the following four aspects of data analysis for which U-CARE can be very useful:

1) How to prepare the data?

2) How to detect overdispersion?

3) How to correct for overdispersion? 4) How to conduct specific tests? 1) How to prepare the data? First, the menu TRANSFORM DATA offers some tools for selecting a data subset, such as a particular range of years or some groups (e.g. males or females, 'ringed as young' or 'ringed as adults'), and for recoding data, such as pooling states, groups or years. Among other features, the menu FILE makes it possible to convert data between two main formats (Biomeco [START_REF] Lebreton | BIOMECO (Biometrie-Ecologie)[END_REF] and Mark [START_REF] White | Program MARK: Survival estimation from populations of marked animals[END_REF]).

2) How to detect overdispersion?

When the data are ready, the next step should be the assessment of the fit of a general model to the data at hand. For single-state data, the classical model is the model with time dependent only parameters (Cormack-Jolly-Seber) (CJS); for multistate data, this is a time and state dependent model (Arnason-Schwarz) (AS). For both of them, the probability of the encounter history of each individual is calculated conditional on its first capture. Optimal goodness of fit tests of the assumptions inherent in the CJS or the JMV model, this last being a slight generalization of the AS model, are then derived based on the classical partitioning, according to sufficient statistics T, of the likelihood P(data/parameters): P(data/parameters)=P(data/T) x P(T/parameters).

The compatibility of the data with the hypergeometric distributions in P(data/T) are tested asymptotically by contingency table chi-squared tests. These tests are organized into several interpretable components by further partitioning P(data/T) [START_REF] Pradel | Principles and interest of GOF tests for multistate capture-recapture models[END_REF]. If systematic departures from the general model can be ruled out, lack of fit may result from the lack of independence among individuals (the two members of a pair, social groups, a contagious disease…). Then, the value of the Pearson statistic is much larger than the residual degrees of freedom. One simple measure for overdispersion is defined as the ratio of the Pearson statistic X 2 by its number of degrees of freedom df This ratio can be calculated for each component individually or overall. Although there is no clear cut decision rule to decide that an observed lack of fit results solely from overdispersion, a reasonable rule of thumb is: the ratio is greater than 1 for all components, and there is no component for which the ratio greatly exceeds the others. This procedure, although not perfect, works generally well as long as the overall ratio does not exceed 3, sometimes 5 (Burnham and Anderson 2002numéro de page?); if it does, an important factor has likely been left out of the model and should be identified. On the other hand, small structural effects that result in an overall between 1 and 3 can without much damage be assimilated to overdispersion and treated as noise (see below for how to do this in practice).

To illustrate these general ideas, we will now treat the example of the study of movements of Canada geese Branta canadensis between three wintering regions, Mid-Atlantic, Chesapeake and Carolinas, between 1984[START_REF] Lebreton | BIOMECO (Biometrie-Ecologie)[END_REF][START_REF] Hestbeck | Estimates of movement and site fidelity using mark-resight data of wintering Canada Geese[END_REF]. The option 'GOODNESS-OF-FIT for Multi-state' of the main menu of U-CARE opens onto the goodness-of-fit test of the JollyMove (JMV) model [START_REF] Brownie | Capture-Recapture studies for multiple strata including non-Markovian transitions[END_REF]). Multi-state models [START_REF] Hestbeck | Estimates of movement and site fidelity using mark-resight data of wintering Canada Geese[END_REF]) allow for transitions between states (the wintering sites for the geese), survival probabilities and encounter probabilities. In the JMV model, transitions vary by state of departure, state of arrival and time interval. Survival probabilities vary by state of departure and time interval. Encounter probabilities vary by previous state, current state and date. This model, in contrast with the better-known Arnason-Schwarz (AS) model, allows encounter probabilities to vary by previous state. However, there is currently no optimal GOF test available for the AS model. Program U-CARE contains specific tests for transience (Test 3G.SR, null hypothesis H0: 'There is no difference in the probability of being later reencountered between "new" and "old" individuals encountered simultaneously.'), trap-dependence (Test M.ITEC, H0(i): 'There is no difference in the probabilities of being reencountered in the different states at i+1 between the animals in the same state at occasion i whether encountered or not encountered at this date, conditional on presence at both occasions.'), and memory (Test WBWA H0: 'There is no difference in the expected state of next reencounter among individuals previously encountered in the different states'). The ratio for the JMV model is computed from these three main components plus two complementary tests 3G.SM and M.LTEC [START_REF] Pradel | Principles and interest of GOF tests for multistate capture-recapture models[END_REF]).

If we were to consider that the lack of fit is due solely to overdispersion, the overall ratio for the geese would be: However, the ratio relative to the WBWA component is huge: 23.6 (see Figure 2)! 3) How to correct for overdispersion?

Given the large value of the overall ratio obtained for the Canada geese, we must try and find a more general model that takes into account the effects unveiled by the GOF test: the strong memory effect and a transient effect. The ratio of Test 3G.SR is indeed 9.8. A model studied in (Rouan et al.) combines these two effects. After discarding the components corresponding to the effects incorporated in this model (memory: component WBWA; transience: component 3G.SR), the new ratio is This value is typical of large data sets with several thousands of individuals where individual differences inherent in any animal population are inevitably detected. This ratio can be used as a variance inflation factor for the model of Rouan and derived models. There is an option in M-SURGE, E-SURGE and MARK to introduce this factor. The impact of the correction factor is null on parameter estimates, but estimated (co)variances are multiplied by , widths of the Wald-CIintervals are times larger and the deviance is divided by . This in turn affects model selection where the AIC is replaced with the QAIC [START_REF] Burnham | Model selection and Inference: A practical Informationtheoretic approach[END_REF]. This procedure of identifying a starting model is valid more generally and should be pursued until an acceptable value for the ratio is reached. 4) How to conduct more specific tests? U-CARE provides details that may give some additional information, notably the tables of observed and expected numbers. For example, Figure 3 shows that newly marked individuals are less recaptured than previously marked individuals at occasion 5 on sites 1 and 3 but not 2. This is a general pattern over all occasions which suggests that transients may be absent from the central site 2. A more specific model could be considered. For an application to seasonal data, see [START_REF] D'agostino | Seasonal survival of Greater Snow Geese and effect of hunting under dependence in sighting probability[END_REF]). U-CARE also gives directional tests (REF) focusing more closely on the detection of transience and trap-dependence. Eventually, the option 'TOOLS' contains some specific tools, not easily found, like the ability to test for mixtures of multinomial in a contingency table [START_REF] Pradel | A Proposal for a Goodness-of-Fit Test to the Arnason-Schwarz Multisite Capture-Recapture Model[END_REF], with an improved algorithm.

To cite U-CARE or acknowledge its use, cite this Software Note as follows, substituting the version of the application that you used for ''Version 2.3'': CHOQUET R., LEBRETON J.-D., GIMENEZ O., . U-CARE: Utilities for performing goodness of fit tests and manipulating Capture-REcapture data. Ecography ?: ?-?. (Version 2.3)


Aurelien Besnard
email: besnard@cefe.cnrs-mop.fr

Olivier Gimenez

Jean-Dominique Lebreton

Aurélien Besnard

A model for the evolution of crèching behaviour in gulls

Keywords: Crèching behaviour, ESS, Gulls, predation, habitat instability

Crèche is the aggregation of chicks outside the nesting territories in which chicks continue to be fed only by their own parents. Several adaptive functions of crèching have been proposed, the most frequent one being a reduction in predator pressure. Using an evolutionary stable strategy approach based on the computation of individuals' fecundity, we examined which regime of aerial and terrestrial predation is likely to favour the evolution and stability of the crèching strategy in gulls. Our results confirm the hypothesis that habitat instability associated with high levels of terrestrial predation favours the evolution and maintenance of the crèching behaviour. Moreover, our results suggest that a low aggressiveness against predators by offspring may be a preadaptation to the crèching strategy. In contrast, the high synchronisation often observed in crèching species does not favour the evolution of a crèching behaviour and is thus probably under selection pressures different from those modelled here.

INTRODUCTION

Coloniality, defined as aggregation on a breeding site that is dissociated from foraging sites [START_REF] Wittenberger | The adaptive significance of coloniality in birds[END_REF][START_REF] Siegel-Causey | The evolution of coloniality[END_REF]) is a complex reproductive strategy that brings many individuals simultaneously to a relatively small breeding site. Although the evolution of coloniality has been intensively studied, the factors responsible for its evolution and maintenance are still subject of much discussion (see [START_REF] Danchin | The evolution of coloniality: the emergence of new perspectives[END_REF] for a review). Coloniality is usually associated with several life history traits such as collective anti-predator behaviour [START_REF] Wittenberger | The adaptive significance of coloniality in birds[END_REF][START_REF] Siegel-Causey | The evolution of coloniality[END_REF], synchronisation of the breeding activities [START_REF] Gochfeld | Mechanism and adaptive value of reproductive synchrony in colonial seabirds[END_REF]) and a high frequency of aggressive interactions [START_REF] Tinbergen | On the Functions of territory in gulls[END_REF][START_REF] Tinbergen | Comparative studies on the behaviour of Gulls (Laridae): a progress report[END_REF].

The formation of large groups of chicks in or near the colony site is one of the most intriguing traits associated with coloniality. These groups are referred to as a crèche, the crèching strategy being defined as an aggregation of chicks outside the nesting territories in which the chicks continue to be fed only by their own parents.

Crèching has been reported in seven of the 14 colonial waterbirds families (A. Besnard and C. Tourenq, unpubl. results). However, the proportion of crèching species within these families varies greatly, ranging from a single crèching species (e.g., the banded stilt, Cladorhynchus leucocephalus) in the Recurvirostridae [START_REF] Del Hoyo | Handbook of the birds of the world[END_REF] to all members of the family (e.g. flamingos) in the Phoenicopteridae [START_REF] Del Hoyo | Handbook of the birds of the world[END_REF].

Several functions of the crèching behaviour have been proposed, the most frequently mentioned being the reduction of predation [START_REF] Pettingill | Crèche behavior and individual recognition in colony of Rockhopper Penguins[END_REF][START_REF] Schaller | Breeding behavior of the White Pelican at Yellowstone Lake, Wyoming[END_REF][START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Buckley | The breeding ecology of royal Terns Sterna (Thalasseus) maxima maxima[END_REF][START_REF] Spurr | Behavior of the Adélie Penguin[END_REF][START_REF] Evans | Some causal and functional correlates of creching in young White Pelicans[END_REF][START_REF] Carter | Creching behavior of Brandt's Cormorant chicks[END_REF]. In a previous study of the crèching behaviour in gulls (Laridae, Larini following [START_REF] Sibley | Phylogeny and classification of birds[END_REF], we demonstrated that crèching species all occupy unstable habitats such as lagoon or river islets (Besnard et al. submitted). Such unstable habitats are characterised by a high probability that colony sites become unsuitable following a drying of the lagoon or a flooding of the breeding site. This instability of the habitat could exert an important selective pressure on crèching behaviour and thus its evolution.

In non-crèching larids, the chicks usually remain on the nesting territories where they are fed and protected against conspecifics by their parents until they fledge [START_REF] Del Hoyo | Handbook of the birds of the world[END_REF]. In crèching species, the chicks leave the nesting territories only few days after hatching and roam in the colony for a short period before they congregate in a crèche at the periphery of the colony and possibly definitively leave the colony site [START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Isenmann | Contribution à l'étude de la biologie de la reproduction et de l'étho-écologie du goéland railleur, Larus genei[END_REF][START_REF] Del Hoyo | Handbook of the birds of the world[END_REF]. To explain this behaviour we propose that crèching may allow for chicks to leave the colony site when it becomes unsuitable for rearing chicks (flooded or accessible to terrestrial predators following the drying of the water surrounding the colony) while maintaining the colony structure and its ability to defend against predation (Besnard et al. submitted).

In order to test this hypothesis, in this paper we study the effect of a high predation rate associated with a high probability for the colony site to become accessible to terrestrial predation over the course of a single breeding season. An Evolutionary Stable Strategy (ESS) approach is used. Individual fecundity was computed using a reproductive output model based on cumulative predation risks over time. We thus examined which characteristics of predation favour the evolution and the maintenance of the crèching strategy. Since all crèching larids demonstrate an extreme synchronisation in laying date [START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Isenmann | Contribution à l'étude de la biologie de la reproduction et de l'étho-écologie du goéland railleur, Larus genei[END_REF][START_REF] Zubakin | Types of coloniality in the family Laridae[END_REF][START_REF] Mierauska | Competitive Interactions between the Herring Gull L. a cachinnans and the Great Blac-headed Gull L. ichthyaetus at Sivash Lake (South Ukraine)[END_REF], and are described as being poorly aggressive against the predators of their chicks [START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Isenmann | Contribution à l'étude de la biologie de la reproduction et de l'étho-écologie du goéland railleur, Larus genei[END_REF][START_REF] Veen | Functional and Causal aspects of nest distribution in colonies of the Sandwich Tern[END_REF][START_REF] Zubakin | Types of coloniality in the family Laridae[END_REF], we focus on the effects of synchronisation of the laying and anti-predator behaviour on the evolutionary stability of the crèching strategy.

THE MODELS

All parameters used in the models are summarised in Table 1. We considered a colony of approximately constant size (i.e. N pairs) over the course of the breeding season.

We were only interested in the predators of chicks. We distinguished two different phases of the breeding period, the incubation and the rearing period. Let   t F be the cumulative distribution of laying dates, i.e. the fraction of the N pairs which started to breed before time t and m the individual reproductive output, assumed to be constant.

The number of eggs  and chicks  on the colony site at time t are given by:

        i t t F t F mN t     (1)         b i t t F t t F mN t      (2)
where i t denote the duration of the incubation, r t is the rearing period and b t stands for the total time for which young were potential preys (from laying to fledging). The total number of items (eggs or chicks) in the colony at time t , denoted ) (t n is simply obtained as the sum of (1) and (2):

        b t t F t F mN t n    (3)
Assuming that eggs and chicks have the same probability of being taken by a predator and let k be the instantaneous rate of predator attacks, the probability p for an egg or a chick to be preyed between time t and dt t  is given by:

  dt t n k p  (4)
which reflects the dilution effect of the predation effort over   t n items present at time t (cf. selfish herd effect Hamilton, 1971).

Our model considers two kinds of predation: aerial and terrestrial. The aerial predation rate, a k , is assumed to be constant over the course of the breeding season. Gulls, usually breeding on sites surrounded by water, are protected against terrestrial predators. Terrestrial predation can only occur after the drying of colony periphery. The occurrence of such a drying during the breeding season is more or less probable according to the habitat characteristics occupied by a species (from highly probable in temporary marsh or lagoon to highly improbable in permanent marsh or lake). Let a be the date at which the colony becomes accessible to terrestrial predation. The terrestrial predation rate, denoted g k , is null before a and constant after a until the end of the breeding season (Model 1).

Gulls often actively defend the colony site against predators using "mobbing" behaviour [START_REF] Patterson | Timing and spacing of broods in the Black-headed gull Larus ridibundus[END_REF][START_REF] Kruuk | Predator and anti-predator behaviour of the Black-headed Gull (Larus ridibundus L.)[END_REF][START_REF] Del Hoyo | Handbook of the birds of the world[END_REF] which consists in collective flights directed towards the predator. While mobbing has been demonstrated

to be an effective deterrent against aerial predators (reviewed in [START_REF] Wittenberger | The adaptive significance of coloniality in birds[END_REF], its impact on terrestrial predator is often described as null (Veen 1977, Wittenberger and[START_REF] Wittenberger | The adaptive significance of coloniality in birds[END_REF]. In order to take the mobbing behaviour into account, a k is assumed to be a decreasing function of the number of breeding pairs still present at time t on the colony site (Model 2).

In order to retain the simplicity of the models, mortality is assumed not to affect the number of eggs or chicks present on the breeding sites. We do not expected qualitative results of the models to be changed by this assumption.

Crèching versus Non-crèching strategy

Chicks of non-crèching species stay on the colony site until they fledge [START_REF] Tinbergen | On the Functions of territory in gulls[END_REF][START_REF] Del Hoyo | Handbook of the birds of the world[END_REF] and are preyed on by aerial and terrestrial predators during the entire breeding season. In contrast, chicks of crèching species usually leave the site when it becomes accessible to terrestrial predators and move to a "safe" site [START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Isenmann | Contribution à l'étude de la biologie de la reproduction et de l'étho-écologie du goéland railleur, Larus genei[END_REF][START_REF] Zubakin | Okologie und Verhalten der Reliktmöwe (Larus relictus Lönnb.)[END_REF]Besnard et al., submitted). We thus considered that the chicks of the crèching species remain at their birth site as long as it remains protected against terrestrial predators. Contrary to the non-crèching chicks, they move to a site protected against the terrestrial predators in the immediate proximity of their birth site as soon as it becomes accessible. After having left their birth site, chicks escape terrestrial predators but remain exposed to aerial predators.

We examined the ecological conditions under which the Non-Crèching Strategy The fecundity of the four classes of individuals were derived for two cases: one with a k constant (no active defence against predators: Model 1) and another one with a k a decreasing function of the breeding pairs on the colony site (active defence against predators: Model 2). When there is no active defence against predators, the noncrèching and crèching chicks suffer from the same aerial predation while only the noncrèching chicks suffer from terrestrial predation. Under these circumstances, CS always outperforms NCS. The predictions from Model 1 are that crèching is an ESS as soon as terrestrial predation is not null. However, we chose to present the computation of this first model in order to progressively detail the different steps in our modelling approach.

When adults present an active defence against predators, the pay-off of each strategy is affected by the number of chicks in the crèche or in the colony at all dates. Under these circumstances, the modelling becomes crucial. Fecundity for both models are respectively summarised in Tables 2 and3 and more details are given in appendices A and B. To sum up, CS is evolutionarily stable if the actual terrestrial predation on the chicks is not zero. On the contrary, NCS is always invasible by CS. As a consequence, the CS is convergent as soon as the realised terrestrial predation on the chicks is not zero. In contrast, if terrestrial predation is zero or if the colony site never dries up, none of the strategies are evolutionarily stable. The evolutionary stability of the strategy does not depend on the rate of aerial predation, the length of incubation or the laying date distribution.

Model 2: with active defence

      0 ) ( 1      dt t n k N k k b t l a a a g (7a)
and relation (7a

) holds if       0 1    a a g k N k k (7b)
In order to qualitatively explore conditions in which NCS is an ESS, a k is assumed to be N-dependent with the form:

               1 max N k N k a (8)
where  is the curvature of the function describing the protection efficiency with the number of adult defending the site and max k is the avian predation rate when only one pair breeds on the site. Hence, the overall predation rate decreased when  increased.

Under these assumptions, the inequality (8) reduces to :

        1 1 max N k k g  (9)
Figure 1 shows the maximum rate of terrestrial predation g k permitting the evolutionary stability of NCS for different levels of  and max k . Qualitatively, an increase in active defence ( ) or an increase in maximum rate of aerial predation ( max k )

favours the evolutionary stability of NCS since it requires an increase in the rate of terrestrial predation for CS to be invasive.

Fig. 1 approximately here

The CS is evolutionarily stable if the following inequality is satisfied (Table 3

right part).           0 ) ( ) ( (              b t l a b t l a g i a b i a dt t w k dt t n t t F t F k t t F t t F k (10)
To qualitatively study the evolutionary stability of CS, we suppose the cumulative distribution of the laying dates to be of the exponential form:

t t f      exp 1 ) ( (11)
where  describes the synchronisation of the laying. The active defence and its efficiency are modelled as above.

The CS is thus evolutionarily stable if the following inequality is satisfied.

                  0 1 exp exp exp 1 exp 2 exp exp 1 exp exp 2 1 exp exp exp 2 2 2 2 2 max             i b i b b b i i b t t t t a t l t t t l t l a g m N k Nm k                 ( 12
)
Figure 2 shows in which conditions CS is evolutionary stable for some usual population parameters of larids. The main results are that the stability of CS depends on the relative levels of terrestrial ( g k ) and aerial predation (

 max k
) and in a more complex way on the synchronisation of laying dates ( ). In the range of biologically realistic parameters, CS is frequently evolutionary stable. CS is invasible by NCS only if the population is submitted to a high rate of realised aerial predation relatively to the terrestrial one and shows at the same time a highly synchronised laying (figure 2).

Fig. 2 approximately here

We only developed here the equations of fecundity computed for individuals whose chicks hatched before the site became accessible to the terrestrial predation.

When the chicks hatch after the site became accessible, they progressively reach the "safe" site. Thus until they hatch, the late laid eggs experience exactly the same predation as the non-crèching eggs. Thus, if the inequality satisfying the evolutionary stability are slightly different, the qualitative results are of the same form and our conclusions hold for the two different situations.

DISCUSSION

Three main conclusions can be derived from our results: first the crèching strategy can simply evolve from a non-crèching strategy if the probability for the site to become accessible to terrestrial predation is high. Second, in a species with no anti-predator behaviour, the crèching strategy always invades and is stable in a large range of ecological situations. Third, when individuals actively defend their chicks against terrestrial predation, crèching and non-crèching strategy may both be ESS but remain invasible depending on the predation rates, the efficiency of the defence against predators and the synchronisation of laying.

In all cases, evolutionary stability is strongly influenced by the relative rates of aerial and terrestrial predation. The non-crèching strategy is favoured and generally the ESS if the aerial predation is high compared to terrestrial predation while crèching strategy is favoured and is the ESS when terrestrial predation is high relative to aerial predation. Since gulls generally occupy sites surrounded by water [START_REF] Del Hoyo | Handbook of the birds of the world[END_REF], they are protected against the terrestrial predators and crèching strategy cannot evolve from a non-crèching strategy. However, in lagoons, where drying is frequent during the breeding season, terrestrial predation remains possible and is even highly probable over the course of the breeding season. This result confirms our original hypothesis that habitat instability associated with terrestrial predation could exert a strong selective pressure in favour of the crèching behaviour.

It has been shown that crèching species show little aggression against the predators of their eggs and chicks [START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Isenmann | Contribution à l'étude de la biologie de la reproduction et de l'étho-écologie du goéland railleur, Larus genei[END_REF][START_REF] Veen | Functional and Causal aspects of nest distribution in colonies of the Sandwich Tern[END_REF][START_REF] Zubakin | Types of coloniality in the family Laridae[END_REF]. We demonstrated in this paper that when no anti-predator behaviours exists, the crèching strategy eventually converges. When a species has a reduced but not null aggressiveness against predators, the crèching strategy is no more convergent but is still invasive and evolutionary stable in many ecological contexts (in particular when terrestrial predation is not null). A little aggressive species could maintain the noncrèching strategy only under a very low rate of terrestrial predation. The non-crèching strategy could thus be maintained while being weakly aggressive only in highly stable habitats protected against terrestrial predation or species living in habitats where no terrestrial predation occurs. To our knowledge, this is the first demonstration that weak aggressiveness observed in crèching species could be a pre-adaptation to this strategy since it favours its evolution and its stability. Since a little aggressive non-crèching species is extremely likely to evolve to crèching and unlikely to then evolve back to the non-crèching strategy, it can explain why no intermediate stage seems to exist between non-crèching and crèching and why no reversion (evolution from crèching to noncrèching strategy) seems to have occurred in larids (A. Besnard, unpublished results).

Bearing in mind that the evolutionary stability of the non-crèching strategy is not conditional to the synchronisation of the laying date, we can reject the hypothesis that the high synchronisation of laying, observed in many crèching species [START_REF] Beer | Adaptation to nesting habitat in the reproductive behaviour of the Black-billed Gull (Larus bulleri)[END_REF][START_REF] Isenmann | Contribution à l'étude de la biologie de la reproduction et de l'étho-écologie du goéland railleur, Larus genei[END_REF][START_REF] Zubakin | Types of coloniality in the family Laridae[END_REF][START_REF] Mierauska | Competitive Interactions between the Herring Gull L. a cachinnans and the Great Blac-headed Gull L. ichthyaetus at Sivash Lake (South Ukraine)[END_REF]Besnard et al., submitted) is a pre-adaptation to the crèching strategy. An extreme synchronisation of laying associated with a high rate of aerial predation favours the invasion of the crèching strategy and thus exerts a selective pressure against the maintenance of crèching behaviour. It can thus be suggested that the high synchronisation observed in several crèching species is a response to a selective pressure other than those studied here. We modelled the crèching strategy as a massive departure from the colony site when it becomes accessible to terrestrial predation and by a progressive aggregation at the newly colonised site by late-hatched chicks. In the field, it seems that crèche formation is punctual and that late breeders who do not join the crèche often abandon their eggs (J.-D. Lebreton, pers. obs.). Such a constraint against a late laying should exert a strong selective pressure for the synchronisation of laying.

In summary, our results thus lead to three main conclusions concerning the evolution of the crèching behaviour in Larids. First they confirm our original hypothesis stipulating that crèching behaviour evolved in unstable habitats associated with a high terrestrial predation. Second, they demonstrate for the first time that a reduced aggressiveness against predators is a pre-adaptation to the evolution of crèching behaviour. Finally, high synchronisation of laying dates does not appear to represent a pre-adaptation to the evolution of the crèching and actually favours its invasibility in species actively defending their chicks.

The evolution of the crèching strategy under the selective pressure of two different predation rates (terrestrial and aerial) could have been modelled using simple models by comparing, for instance, the instantaneous rate of death for crèching or noncrèching chicks. However, our model allow for one to more easily incorporate parameters such as the synchronisation of laying and to later build up more complex behaviour of the crèching species than those examined here without changing the philosophy of the modelling. Such a flexibility in our way of modelling should favour its utilisation to examine the evolution of crèching strategy in other bird families submitted to different selective pressures.

Figure 1 :

Maximum rate of terrestrial predation g k permitting the evolutionary stability of NCS for different levels of  and max k . NCS is evolutionarily stable when

        1 1 max N k k g 
. An increase in active defence ( ) or an increase in maximum rate of aerial predation ( max k ) favours the evolutionary stability of NCS since it requires an increase in the rate of terrestrial predation for CS to be invasive. N was fixed to 400 pairs. CS is stable when the surface defined by the fecundity ratio is upper than the horizontal plan. The fecundity ratio is given by eq. ( 12). Parameters in eq. ( 12) where fixed in order to be biological realistic for gulls : N=400, m=2, 25

 i t , 30  b t , l=32, a=43, 01 . 0  g K , 5 . 0 max  K , =1.


   ) ( ) ( ) ( dt t k t n k dt t n k a l b t l a g a a               ) ( ) ( ) (  Crèching    b t l l a dt t n k ) ( dt t n k b t l l a    ) ( Condition satisfying the ESS 0 ) (    dt t n k b t l a g 0 ) (    b t l a g dt t k 
      dt t n N k k dt t n N k b t l a a g a l a       ) ( ) (   dt t k t n t t F t F k dt t n k a l b t l a g i a a               ) ( ) ( ) ( ) ( ) (  Crèching     dt t n k dt t n N k t l a a a l a      ) ( 1 ) (         dt t n t t F t t F k dt t n N k a l b t tl a b i a a         ) ( ) ( Condition satisfying ESS       0 ) ( 1      dt t n k N k k b t l a a a g     0 ) ( ) ( ) ( ) ( 2            b t l a b t l a g b i a dt t w k dt t n t F t t F t t F k dt t n k NC NC a l a e l f a f    ) ( / ) ( ) ( (A4)
where NC means Non-Crèching. After the site became accessible to terrestrial predators, the offspring is submitted to the terrestrial predation g k and the aerial predation a k . Doing in the same way as above, we obtain : Here offspring are not submitted to terrestrial predation after a because they leave the site when it became accessible to terrestrial predation. The procedure described above remains valid.

dt t n k k NC NC NC NC b b t l a g a e a f t l f       ) ( / / ) ( ) ( (A5) using (A4) in (A5), we obtain dt t n k k dt t n k NC NC b a l b t l a g a a e l f t l f               ) ( ) ( / ) ( ) ( (
The fecundity is thus given by Since many CS birds leave the site when it becomes accessible to terrestrial predators, the number of items on the colony sites submitted to terrestrial predation is given by the number of eggs still unhatched. The number of items submitted to aerial predation is still given by the total number of items on both sites. We thus obtain : All the parents of the crèching chicks defend the new colonised site simultaneously protected against the terrestrial predation, the fecundity is thus given by:

dt t n k NC C b b t l l a e l f t l f      ) ( / ) ( ) ( (
dt a l b t l a t g k t n a k dt t n a k C NC b e l f t l f                  ) ( ) ( ) ( / ) ( ) (  (A10) (
        dt t n t t F t t F k dt t n N k C C b a l b t tl a b i a a e l f t l f           ) ( ) ( / ) ( ) ( (B5)
Then the fecundity ratio satisfying the ESS of CS is given by:

    0 ) ( ) ( ) ( ) ( 2            b t

(

  hereafter NCS) or the Crèching Strategy (CS) are ESS. If adult survival is equal for two strategies A and B, strategy A is non-invasible by the strategy B when, in a population fixed for A, the fecundity of an A individual is greater than that of a B individual (Maynard Smith 1982). If only two strategies, A and B, are possible, the study of invasibility of A is equivalent to the study of the evolutionary stability of A. Strategy A is ESS if, in a population fixed for the strategy A, the ratio of the fecundity of an individual exhibiting strategy A to that of an individual with strategy B is greater than one. The number of surviving offspring at fledging, later called fecundity, for a pair laying at date l according to the biological rules stated above was derived. The four equations of fecundity were computed: (a) a NCS individual in a NCS population, (b) a CS individual in a NCS population, (c) a NCS individual in a CS population and (d) a CS individual in a CS population.

Figure 2 :

 2 Figure 2 : Conditions in which CS is evolutionarily stable for different levels of

  A6) (b) Derivation of the fecundity of an individual demonstrating CS in a population fixed for NCS.

  A7)where C means Crèching and NC means Non-Crèching.The ratio of the fecundity satisfying the ESS of NCS is given by 1 Derivation of the fecundity of an individual demonstrating NCS in a population fixed for CS.

  d) Derivation of the fecundity of an individual demonstrating CS in a population fixed for CS.Chicks leave the site when it becomes accessible to terrestrial predators. Thus, they only experience the aerial predation diluted in the whole population of eggs and chicks.Taking into account what it precedes, Derivation of the fecundity of an individual demonstrating CS in a population fixed for NCS.The chicks of the CS invader are defended only by their own parents on the site they reached after their birth site became accessible to terrestrial predators. The fecundity of an CS in a NCS is thus given by: Derivation of the fecundity of an individual demonstrating NCS in a population fixed for CS.When the site becomes accessible to terrestrial predators, chicks already hatched leave the colony site. It remains on the colony sites all the eggs still unhatched and protected by their parents. The fecundity is thus given by: Derivation of the fecundity of an individual demonstrating CS in a population fixed for CS.

Table 1

 1

	approximately here
	RESULTS

t n is strictly positive whatever t . Hence, NCS is always invasible by CS.

CS is evolutionarily stable if the following inequality holds (Table

2 right part):

Table 3

 3

approximately here NCS is evolutionarily stable if the following inequality is fulfilled (table 3 left part).

Table 1 :

 1 Definition of the different parameters used in the ESS modelling approach to model the individual fecundity of crèching or non-crèching birds in population fixed for crèching or noncrèching strategies.Total number of egg and chicks, i.e. number of items submitted to predation at date t

	F	  t		Cumulative distribution function of laying date at date t
	N		Total number of breeding pairs in the area
	m		Clutch size
	mNF	  t	Cumulative number of eggs laid at date t
	t	i				Length of incubation period
	t	r			Length of the rearing period (from hatching to fledging)
	t	b			Total time of rearing from laying to fledging, hence	t	b		t	i		t	r
		  t		Number of incubated eggs at date t
		  t		Number of chicks (i.e. hatched but not yet fledged) at date t
	n	  t
	k	a		Rate of aerial predation
	k	g		Rate of ground (terrestrial) predation
	l					Laying date of an individual
	a				Date at which the colony site becomes accessible to terrestrial predation

Table 2 :

 2 Fecundity of the two possible strategies in function of the strategy fixed in the population when no active defence against the predators exists; the fecundity ratio gives the condition of ESS.

										Population fixed for the strategy
						Non-Crèching				Crèching
	Rare Strategy	Non-Crèching	a l 	n k	t a	dt	l a 		b t	k	n g	t	k	a	dt

Table 3 :

 3 Fecundity of the two possible strategies in function of the strategy fixed in the population when an active defence against the predators exists; the fecundity ratio gives the condition of ESS

			Population fixed for the strategy
		Non-Crèching	Crèching
	Rare	Non-
	Strategy	Crèching

AKNOWLEDGMENTS

We thank Kate Lessells for her precious commentaries in our modelling approach and John Thompson for his help on improving english of the manuscript.

Appendix A: Derivation of individual fecundity in a system where adults do not defence the colony site against the predators (a) Derivation of the fecundity of an individual demonstrating CS in a population fixed for NCS From (4), the number of offspring at date dt t  is given by : )

Integrating between a and l both sides of equation (A3) and taking exponential, we obtain the number of offspring at date a of an individual laying at date l :

Appendix B: Derivation of individual fecundity in a system where adults actively defence the colony site against the predators (a) Derivation of the fecundity of an individual demonstrating CS in a population fixed for NCS.


Anabela Da Silva
email: anabela.dasilva@fresnel.fr

Charles Handschin

Christophe Riedinger

Julien Piasecki

Serge Mensah

Amélie Litman

Hassan Akhouayri

Anabela Da

Taking advantage of acoustic inhomogeneities in photoacoustic measurements

Keywords: Photoacoustic imaging, Medical and biological imaging, Inverse problems

Photoacoustic offers promising perspectives in probing and imaging subsurface optically absorbing structures in biological tissues. The optical fluence absorbed is partly dissipated into heat accompanied with microdilatations that generate acoustic pressure waves, the intensity which is related to the amount of fluence absorbed. Hence the photoacoustic signal measured offers access, at least potentially, to a local monitoring of the absorption coefficient, in 3D if tomographic measurements are considered. However, due to both the diffusing and absorbing nature of the surrounding tissues, the major part of the fluence is deposited locally at the periphery of the tissue, generating an intense acoustic pressure wave that may hide relevant photoacoustic signals. Experimental strategies have been developed in order to measure exclusively the photoacoustic waves generated by the structure of interest (orthogonal illumination and detection). Temporal or more sophisticated filters (wavelets) can also be applied. However, the measurement of this primary acoustic wave carries a lot of information about the acoustically inhomogeneous nature of the medium. We propose a protocol that includes the processing of this primary intense acoustic wave, leading to the quantification of the surrounding medium sound speed, and, if appropriate to an acoustical parametric image of the heterogeneities. This information is then included as prior knowledge in the photoacoustic reconstruction scheme to improve the localization and quantification.

INTRODUCTION

The first goal of Photoacoustic tomography is to provide indirectly, by the resolution of the inverse problem, a measurement of the optical absorption coefficient of heterogeneously absorbing media. Quantification of this absorption coefficient can be obtained if the physics of the optical diffusion process is also considered in the model, and a discrimination between absorption and diffusion is achieved. Interrogating the medium with multiple sources illumination 1 is a method that allows this. Local illumination produce a primary intense photoacoustic wave, localized in a volume close to the interface. Usually, one seeks at positioning sources and tranducers in a geometric way such that this signal doesn't affect much the measures of the photoacoustic signal coming from the inhomogeneity of interest. However, this primary intense photoacoustic wave, propagating through the medium, can serve as an acoutic probe sensitive to the acoustic inhomogeneities present in the medium, like tumors, heterogeneous both optically (neo-angiogenesis) and acoustically (detected with conventional ultrasound echography). With local sources illumination, the interesting point is that the detection of this acoustically sensitive signal and the more conventional photoacoustic signal is detected within two separate times intervals. A classical time filter can be applied to separate in time the two species of signals, that can afterwards be exploited separatly: the acoustic echoes bring spatial information that can be introduced as prior knowledge in the photoacoustic signal processing to retrieve the optical properties. In the present work, through a demonstrating experimental photoacoustic measurement performed on a phantom presenting an optical/acoutic inhomogeneity, the information content of the acquired signals is presented and dicussed, and the filtering protocol to exploit the acoustic information extracted from the measured photoacoustic signal is illustrated through simulations performed on a synthetic phantom.

EXPERIMENTAL DEMONSTRATION

In photoacoustics, the medium is heated with a time varying optical source. The thermal expansion of the tissues generates acoustic pressure waves that can be measured with transducers. Sound and light velocities being so different, the optical fluence is going to be deposited instantaneously within the whole medium according to the spatial distribution of the absorption and the diffusion coefficients of the probed tissue. The deposited energy is dissipated into heat, microdilatations are produced that give birth to a pressure wave that can propagate through the medium. If the medium is homogeneous in terms of both optical and acoustic properties, a single acoustic wave pressure echo, is going to be measured due to the deposition of energy, mostly near the illuminated surface because of the short scattering mean free path (few tens µm in biological tissues). Hence its time-offlight corresponds roughly to the distance between the illuminated surface and the detector position. However, the fact is that this initial pressure wave is going to propagate and potentially being able probe the medium as a conventional acoustic wave. Hence, if acoustic heterogeneities are present in the medium, for example because of a different density as in tumors (mammary tumors are visible with US echography), then this primary photoacoustic wave is going to be perturbed, and echoes related to the presence of acoustic heterogenities will be sensed at longer times-of-flight. If optically absorbing heterogeneities are present within the medium, more energy will be absorbed locally giving birth to the conventional photoacoustic echoes. In order to eyelight this phenomenon, hereafter is shown a measurement performed on an Agar-Agar based phantom. Our setup (Fig. 1, Left) comprises a pulsed laser (Quantel, France, Nd:Yag, 1064 nm, 330 mJ, 5 ns, 10 Hz, frequency doubled by a KTP crystal at 532 nm) and with immersion acoustic transducers (diameter 10 mm, focal length 1, central frequency 3.5 MHz, Olympus, France) are used. The illumination beam was enlarged in the transverse direction (size of the beam 40 mm). The collected signal is then preamplified (Olympus, France, gain 40 dB, cutting frequency 10 MHz) and drived to an oscilloscope (Tektronix, USA, 4 channels, bandpass width 200 MHz) for signal analyzis. The phantom is a cylinder (∅=5cm) made of a aqueous gels of a mixture of agar-agar (volume fraction 2%, Jeulin, France) and titanium dioxyde (TiO 2 ) powder (0.05%, Sigma-Aldrich, France) to mimick the level of optical scattering of the tissues (reduced scattering coefficient estimated 6 cm -1 , and absorption coefficient 0.02cm -1 at 532 nm). The inclusion is a black hair placed as in Fig. 1, Left. The measurement, collected with a transducer placed as in Fig. 1, Left, perpendicular to the illumination path, is plotted on (Fig. 1, Right).

The photoacoustic echo arrives approximately at t i ∼ 18 µs, corresponding to the distance inclusiontransducer. As the phantom contains an acoustic heterogeneity, it can be orbserved the presence of a second echo, arriving at a time-of-flight corresponding to the path source-inclusion-tranducer: t si + t id ∼ 28µs. This secondary acoustic echo, due to the presence of a burried structure acoustically inhomogeneous, allows retrieving spatial information on its location. This information can afterwards included in the reconstruction of the optical parameters as prior information to improve the inversion process.

Two echoes of different nature are present in the measurements: the photoacoustic ones (Fig. 1,Right,(b)) and the acoustic ones (Fig. 1, Right, (c)). The main difficulty lies in sorting temporally these signals of different nature. One can note that the two species of echoes belong to two different time spaces, delimited by the position of the interface (Fig. 1, Right, (a)): (i) photoacoustic echo (b) is produced by a perturbating object T belonging to a circle C of radius T D = t (b) /v s , v s being the sound velocity; (ii) acoustic echo (c) senses the presence of an object T belonging to an ellipse E of foci the position of the detector D and the projection of the laser beam at the surface of the phantom such that ST + T D = t (c) /v s . The perturbation T belongs to the intersection of C and E within the phantom area. For multiple objects localization, one has to multiply the measurements and plot all possible occurences, as illustrated in the next section.

ILLUSTRATION WITH SYNTHETIC DATA

A 2D example of simulations explains how this measure can be exploited and introduced in a reconstruction algorithm to retrieve the optical parameters. A homogeneous disk phantom is considered containing two inhomogeneous regions, one disk (diameter 5 mm) and one square (2.5× 2.5 mm 2 ). The geometry is shown on Fig. 2. The following table lists the values of the physical parameters. Synthetic pressure measurements were simulated by, first, computing for each source the initial pressure distribution p os (r) = Γ(r)H s (r), Γ [-] being the Grueneisen coefficient, H s (r) = µ a (r)φ s (r) [W.cm -3 ] the absorbed power density transformed into heat at position r, µ a (r) [cm -1 ] the optical absorption coefficient, φ s (r) [W.cm -2 ] the fluence deposited by source s, calculated by solving numerically (Finite Element Method) the Diffusion Equation in the Continuous Wave regime. The k-wave toolbox were used to solve the acoustic propagation. [START_REF] Cox | k-space propagation models for acoustically heterogeneous media: Application to biomedical photoacoustics[END_REF] In the present example, 360 transducers were evenly distributed at the periphery of the object and a set of 8 different point sources experiments (also evenly distributed) were run. The final sinogram (Fig. 2, 2 nd image) was composed of 8×360 time-resolved mesurements, the contribution of the homogeneous background was subtracted, an example of these measurements is plotted on Fig. 2 (3 rd image), showing the complexity of the signals. From the sinograms, an algorithm of peak detections is applied, [START_REF] Yoder | Matlab function 'peakfinder.m'[END_REF] and the spatial histogram of probabilities of intersections between circle C and E is calculated. The gradient of the resulting image is taken to obtained enhanced contours and a final thresholding allows rejecting artifacts due to sources positions (Fig.

(4 th image)).

A minimization based approach has been adopted for the resolution of the inverse problem, in which it is sought to find (µ a , D) which minimizes the error functional F = 1 2 s Ω (H m s (r)-H s (r, µ a , D)) 2 dr, where H m s is the measured initial pressure distribution map due to source s, that can be obtained from the measured pressures, after resolution of the acoustic propagation inverse problem. The reconstructions of the optical parameters have been performed with an adjoint-assisted gradient-based method. [START_REF] Paul | Quantitative photoacoustic imaging: Measurement of absolute chromophore concentrations for physiological and molecular imaging[END_REF] The reconstructions obtained with and without use of the localization map are shown on Fig. 3.

CONCLUSION

A demonstration experiment performed on a phantom with physical properties close to those of biological tissues shows the effect of perturbation of acoustic nature on the photoacoustic measurements. The specific measurement of this perturbation can be used to retrieve morphological information on the sample that can be used afterwards to constrain the reconstruction of the optical parameters. This is illustrated through an example of reconstruction performed on synthetic data.

Figure 1 .

 1 Figure 1. Left:Schema of the experimental setup and phantom; Right: Phantom measurement with hair inclusion: (a) photoacoustic response of the surface, (b) photoacoustic response of the inclusion, (c) acoustic response of the inclusion.

Figure 2 . 1

 21 Figure 2. 1 st image: geometry, 2 nd : differential sinogram, 3 rd : measurement number 145 with peak detection (blue circles: photoacoustic peaks, read circles: acoustic peaks), 4 th : resulting map of probability of presence of inhomogeneities.

Figure 3 .

 3 Figure 3. Optical parameters reconstructions. Top: absorption coefficient, target values, reconstruction of µa alone without prior, reconstruction of µa by simultaneously reconstruction D, without prior, horizontal crossplot (Y=0). Bottom: diffusion coefficient, target values, reconstruction of D simulatneously with µa, without prior, reconstruction of D simulatneously with µa with prior, horizontal crossplot (Y=0).

Table 1 .

 1 Physical properties of the synthetic phantom.

		Backgound Disk inclusion Square inclusion
	Absorption coefficient [cm -1 ]	0.1	0.11	0.1
	Diffusion coefficient [cm]	0.0333	0.0303	0.0303
	Density [kg.m -3 ]	1000	1500	1700
	Sound velocity [m.s -1 ]	1485	1485	1485

ACKNOWLEDGMENTS

Agence National pour la Recherche (ANR) (AVENTURES-ANR-12-BLANBS01-0001-04); SATT-SE programme de maturation Mammoscan.


Roger Pradel

Lory Maurin-Bernier

Olivier Gimenez

Meritxell Genovart

R Émi Choquet

Daniel Oro

Estimation of sex-specific survival with imperfect clues about sex

Keywords: capture-recapture, parameter redundancy, multievent models capture-recapture, parameter redundancy, multievent models

In monomorphic species, determination of sex from behavior is prone to errors.

Ignoring this fact renders problematic the estimation of sex-specific parameters.

We develop capture-recapture survival models that account for uncertainty in the assessment of sex. There is a legitimate concern that some parameters may not be identifiable in these models. We examine parameter redundancy for four basic models with survival and encounter probabilities constant or time-dependent. We further develop a more refined and more appropriate model for an Audouin's gull data set where four distinct behavioral clues have been used. We examine how useful it is to incorporate the least reliable of the clues and the genetic determination of sex available for only a handful of individuals. We finally discuss the implications of our findings for the design of field studies as well as further directions of research.

Introduction

Sex differences in survival, dispersal or movement may have important consequences in demography, mating and parental investment patterns [START_REF] Breitwisch | Mortality patterns, sex ratios, and parental investment in monogamous birds[END_REF][START_REF] Gowaty | Differential dispersal, local resource competition, and sex ratio variation in birds[END_REF], even for species where the two sexes have very similar body size and appearance. Thus, a modern study in ecology cannot ignore a priori this factor. However, naming the sex of an individual may be particularly tricky for monomorphic species observed in the field. Often the field biologist will rely on behavior to distinguish males from females. Some behavioral clues like position during copulation are thought to be quite reliable but other clues like the relative body size in a pair not so much and yet the latter type are easier to gather. With this kind of approach, it is thus not rare that an animal which has once been recorded as a male is later referred to as a female. Confidence in the correct determination of sex will progressively increase with repeated and consistent observations and, eventually, it may be decided that there is no reasonable doubt about the true sex of a particular individual. However, this does not hold for animals seen only a few times and for those with a contradictory record of given sex. A natural tendency for studying nonetheless sex-specific parameters is then to consider the sole individuals for which the sex has been determined with reasonable certainty. However, this approach has shortcomings, especially when the aim of the study is survival. First, it may represent a huge waste of data. For instance, in the field study that has motivated this paper, approximately 80% of the individuals had never been sexed and would thus be discarded.

Second, to be sexed with reasonable certainty an individual must often have gone through several observations and hence have survived meanwhile. Then, estimation of survival based on the subsample of known-sex individuals will inevitably be biased high. Another possibility is to segregate the animals into three groups: males, females and unknownsex individuals. The survival of the unknown-sex individuals can then be constrained to be a weighted average of the survivals of males and females [START_REF] Oro | Determinants of local recruitment in a growing colony of audouin's gull[END_REF] with the weights reflecting the proportions of males and females among the unknown.

However, these weights are not easy to determine as the relative proportions will depend on the sex-ratio but also on the relative catchability of males and females and on the ability to identify the sex, which may differ between males and females. Thus, while this second method is preferable, it is still imperfect and anyway does not make full use of the information available. We know of one example where sex uncertainty was directly incorporated in a survival analysis [START_REF] Conroy | Development and application of a mark-recapture model incorporating predicted sex and transitory behaviour[END_REF]. This study dealt with serins Serinus serinus, a species which is monomorphic only at the juvenile stage. A biometrical measure (wing length) was used to predict the sex of the captured juveniles and an initial state 'predicted male' or 'predicted female' was then assigned to each bird on this basis. If they were ever seen again, their true sex was then determined and they were assigned the new state 'known male' or 'known female'. This procedure allowed the estimation of transition probabilities between the 'predicted sexes' and the 'known sexes' and thus the incorporation of the never recaptured juveniles in the analysis. In this way, survival could be estimated without bias. This model was cast within the framework of multistate capture-recapture models [START_REF] Arnason | Parameter estimates from mark-recapture experiments on two populations subject to migration and death[END_REF][START_REF] Arnason | The estimation of population size, migration rates and survival in a stratified population[END_REF][START_REF] Schwarz | Estimating migration rates using tag-recovery data[END_REF]. Unfortunately, this approach does not work for species which remain monomorphic all their life and this is why we had to develop specific models. We present here a general solution to the problem of sex uncertainty by introducing explicitly probabilities of errors. In essence, as compared to previous approaches, instead of summarizing the data previous to the analysis, we model the very process of sex identification. This work was motivated by the demographic study of an Audouin's gull Larus audouinii colony at the Ebro Delta (Spain) (e.g. [START_REF] Oro | Determinants of local recruitment in a growing colony of audouin's gull[END_REF][START_REF] Cam | Assessment of hypotheses about dispersal in a long-lived seabird using multistate capture-recapture models[END_REF] where the birds are sexed upon observation according to four criteria (see [START_REF] Oro | Food availability and nest predation influence life history traits in audouin's gull, larus audouinii[END_REF], for details): position during copulation, begging food during courtship, courtship feeding and relative body size (the males being on average slightly heavier than the females, see [START_REF] Oro | Audouin's gull account[END_REF]Genovart et al., 2003a); or not sexed at all. The information can be coded as three basic events: "the animal is judged to be a male", "the animal is judged to be a female", "no judgment is issued". As this scheme should be relevant to many studies, we start (section 2) by developing a set of four general models that extend the Cormack-Jolly-Seber capture-recapture model and constant-parameter versions of it [START_REF] Cormack | Estimates of survival from the sighting of marked animals[END_REF][START_REF] Jolly | Explicit estimates from capture-recapture data with both death and immigration-stochastic model[END_REF][START_REF] Jolly | Mark-recapture models with parameters constant in time[END_REF][START_REF] Seber | A note on the multiple-recapture census[END_REF] to uncertainty in sex assignment. These models incorporate probabilities of judgement and probabilities of error and can be further developed if needed. They are described in subsection 2.1. Because of the additional parameters involved, there is a legitimate concern that not all parameters may be identifiable. In subsection 2.2, we set off to study redundancy by the method of Catchpole, Morgan and Freeman (1997;[START_REF] Oro | Audouin's gull account[END_REF]. The models of subsection 2.1 do not account for likely differences in probability of error with each of the four criteria used in the Audouin's gull study, nor do they incorporate such peculiarity of this study as a trend over time in the attempt to identify the sex of the birds. In the next section 3, we construct a more refined and more specialized model, suited to our data. At the same time, we assess the effect of dealing or not dealing with different pieces of information: is it useful to gather the least reliable clue? Should genetic sexing be developed? The last section is a discussion of the interest of this kind of models and of the implications of our findings in terms of field work.

2 Four basic models

Description

The typical data are presented in Fig. 1. There, the true sex of every single individual is unknown. To write the probability of a given encounter history h, we start by conditioning on the first capture event-as is usual in capture-recapture (CR)-and then apply the law of total probabilities: P (h) = P (f )P (h|f ) + P (m)P (h|m). Thus, P (f ) and P (m) are the sex proportions among the unmarked captured, and P (h|f ) and P (h|m) are the conditional probabilities of h for a female and a male respectively. We need the following parameters: • p m , p f sex-specific encounter rates 2. New parameters

1. Traditional CR parameters • φ m , φ f sex-specific survival rates
• µ, proportion of males in the population

• e m , e f sex-specific probabilities to judge the sex of an individual

• x m , x f sex-specific probabilities to issue the correct judgement.

It is now possible to write out P (m) as

pmµ pmµ+p f (1-µ) (P (f ) = 1 -P (m)).
As for the conditional probabilities, let us write them for h = 1 3 2. If this relates to a male, it has been sexed the first time (probability e m ) correctly (probability x m ). It has then survived to occasion 2 (probability φ m ), when it was encountered (probability p m ), but not sexed (probability 1e m ). It survived again (φ m ), was encountered at occasion 3 (p m ), and sexed (e m ) incorrectly (1x m ). All put together, we obtain

P (h|m) = e m x m φ m p m (1 -e m )φ m p m e m (1 -x m ).
In a similar way, we would get

P (h|f ) = e f (1 - x f )φ f p f (1 -e f )φ f p f e f x f
. The complete probability for this encounter history is thus

P (h) = p m µ p m µ + p f (1 -µ) e m x m φ m p m (1 -e m )φ m p m e m (1 -x m ) + p f (1 -µ) p m µ + p f (1 -µ) e f (1 -x f )φ f p f (1 -e f )φ f p f e f x f . (1)
The probability of each individual encounter history will be obtained in this way and hence the likelihood of the entire data set which, assuming independence of fates, is the product of them. With #h denoting the number of animals with encounter history h, the likelihood can be written in compact form:

L = h P (h) #{h} .
Maximum Likelihood Estimators (MLE) are derived by maximizing L as a function of the different parameters. These estimators are known to have excellent statistical properties such as being asymptotically unbiased and this is the approach to parameter estimation retained in leading CR software. However, because the current models do not fall in the category of existing models, we could not take advantage of this feature and wrote our own maximizing programs in MATLAB [START_REF] Hanselman | Mastering MATLAB 6[END_REF].

Parameter redundancy

The aim of this subsection is to study the impact of the introduction of sex uncertainty on parameter identifiability. As one can easily realize by browsing through the extensive capture-recapture literature, the number of traditional CR models in use is virtually unlimited as new models can be created by changing the kind of effects considered on each type of parameters. For instance, parameters may be constant over Catchpole, Morgan and Freeman (1997;[START_REF] Oro | Audouin's gull account[END_REF] have developed a formal method (hereafter the CMF method) for studying parameter redundancy in models belonging to the exponential family of probability distributions. This method can be applied to the multinomial distribution of animals over the observable encounter histories. It indicates which parameters are directly identifiable and provides identifiable functions of the redundant parameters (see [START_REF] Catchpole | Detecting parameter redundancy[END_REF] and [START_REF] Catchpole | Estimation in parameter-redundant models[END_REF] for details).

All the calculations being formal, we carried them out with MAPLE. Because the number of capture histories 4 k -1 increases rapidly with the number of time steps k, we could apply the procedure only for k = 3, but this is sufficient as the results obtained are easily demonstrated to hold in general.

A first conclusion (see Table 1) is that improving the realism of the models by adding new constant nuisance parameters to account for sex uncertainty does not restrict the ability to estimate the parameters of interest, i.e. the survival probabilities. Indeed, the only restriction we observe in the estimation of survival relates to the fully time-dependent model and this restriction is the same that was already present in the corresponding fully time-dependent CJS model. Actually, the new model improves over the CJS model applied to each sex separately in allowing the estimation of the ratio of survivals of males and females at the last occasion; we believe that this result is due to the assumption of a constant sex-ratio in the population. The other case of parameter redundancy concerns i.e. the relative catchability of males and females at the initial occasion is estimable.

p 1 m , p 1 f p 1 f p 1 m (φ t * s , p t * s ) 15 13 p 1 m , p 3 m , p 1 f , p 3 f , φ 2 m , φ 2 f p 1 f p 1 m , φ 2 f φ 2 m , φ 2 m p 3 m , φ 2 f p 3 f the initial
This is easily understood from the likelihood. The initial encounter probabilities appear in it only in the terms P (m) and P (f ), and there only through their ratio. For instance, P (m) can be rewritten

P (m) = p m µ p m µ + p f (1 -µ) = µ µ + p f pm (1 -µ) .

Dual solutions

A redundant parameter is also one that can take all of a continuous range of values while the likelihood remains at its maximum. For a given data set, this can be seen by drawing the profile likelihood curve, i.e. the curve of the maximum value that the likelihood can assume for each value of the parameter under scrutiny: for a redundant parameter, the profile likelihood presents a characteristically flat area at its top. The shape of the profile likelihood is interesting more generally. For instance, when the profile likelihood decreases rapidly away from the optimal value of the parameter, this parameter is estimable with great precision. Away from its maximum, the profile likelihood may exhibit local maxima. If one of these happens to be exactly on the same level as the 'absolute' maximum, then this is an instance of non-identifiability without redundancy: two entirely different values of the parameter maximize the likelihood equally well but not the values in-between. This situation cannot be detected by the CMF method. Thus, we examined the profile likelihood curves of the model with traditional parameters hold constant (φ s , p s ) applied to the Audouin's gull data already mentioned (Fig. 1). The parameters were maintained within range by logit transformations. Instead of drawing the profile likelihoods, we actually drew the profile deviances (D = -2 log L) and hence observed the minima rather than the maxima (Fig. 2).

A striking feature of model (φ s , p s ) is the existence for each scalar parameter of two distinct values where the deviance is minimized. These values organize into two sets which represent two alternative solutions. This result is in fact data-independent and holds as well for the three other models of this section: (φ t * s , p t * s ), (φ t * s , p s ) and (φ s , p t * s ) (see the appendix for a theoretical demonstration). The deep reason for this fundamental duality comes down to this: given that the true sex of any individual is never known, it is equally acceptable, from the model point of view, to consider that an animal estimated many times to be, say, a male is indeed a male and that most judgments were correct, or that it is a female and most judgments were wrong. Yet, one solution produces probabilities of error above 50%, which is probably not acceptable from the practitioner's point of view. More precisely, one solution is derived from the other by exchanging the survival, encounter and judgment probabilities of males and females, replacing x m with 1x f and x f with 1x m and finally reversing the sex-ratio (see Appendix). As a consequence, the profile deviance curves of sex-specific parameters are the same for males and females. In the case of the Audouin's gull data, they present two close minima (Fig. 2). The profile deviance curve of the proportion of males is symmetrical around 0.5 with two minima, which happen to be very distant with our data, at 0.14 and 0.86, and, incidentally, utterly unrealistic form a biological point of view. 3 A more realistic model for the Audouin's gull data

The four models introduced in the previous section, fitted to the Audouin's gull data, produce very unreasonable if similar estimates: very high proportions of males that are almost never judged and probabilities of error close to 50'% (Table 2). This is not entirely surprising given that they do not incorporate several known important features of the study. In this section, we exemplify the flexibility of our approach by building a more realistic model incorporating our knowledge of the biology and of the way the field work had been conducted.

To start, there was no good reason to believe that the error attached to each of the 4 criteria used (copulation, begging food, courtship feeding and relative body size) was the same. Copulation for instance was suspected to be the most reliable and relative body size, the most error-prone. We thus decided to distinguish the different criteria and recoded the data accordingly (see Table 3). All along the study, sex determination has been a secondary activity but it has been conducted on stringent criteria. This is reflected in the very few obvious mistakes (sexing is consistent over time for a given individual), but the roughly 80% of never-sexed individuals in the data set. Yet, during the course of

the following characteristics:

• p t , encounter probability: time but not sex-dependent.

• φ s , survival rate: possibly sex-dependent but constant over time.

• µ, proportion of males: held constant.

• e T : logit-linear trend over time (T ) in the attempt to judge the sex (e). Note that because birds are always judged by pair, we put no sex-dependency on e.

• m4 T : logit-linear trend over time (T ) in the frequency of use of the criterion 'relative body size' (m4) among the instances when a judgment was issued.

• m1: frequency of use of the criterion 'copulation' among the behavioural criteria (i.e. excluding 'body size'): held constant.

• m2 and m3: same as m1 for 'begging food' and 'courtship feeding' respectively (one of m1, m2 or m3 is redundant).

• xi: probability to be right when using criterion i (i = 1, . . . 4): held constant. m2, x1, x2, x3, x4) applied to the Audouin's gull data set and relative to the proportion of males µ. The genetic sexing of 24 individuals renders the profile deviance steeper at its bottom where it retains just one minimum (Analysis A; continuous line) as compared to when the genetic information is not used (Analysis B; dotted line). There is a narrow ridge between the 2 dual solutions in Analysis B.

15

A main concern when we started building these models was the risk of parameter redundancy. The models would have been useless if no parameter of interest was estimable.

This fear had been voiced in particular by [START_REF] Nichols | Estimation of sexspecific survival from capture-recapture data when sex is not always known[END_REF] when they examined the restricted situation where the sex is not always assessed but, if so, without error.

Our results show on the contrary that it is quite possible to run models incorporating probabilities of error and nonetheless gain knowledge about such important biological parameters as survival rates. Although these models have in general two mathematical solutions, one is so unreasonable that it should not be difficult to sort out which one is to be retained.

A more serious difficulty is numerical. These models, and more generally the multievent models [START_REF] Pradel | Multievent: an extension of multistate capture-recapture models to uncertain states[END_REF] to which they belong-but also the now classical multistate models-present local minima to which the optimization algorithm occasionally converges. The profile deviance curves may help diagnose the problem and improved algorithms may be sought (see, in particular, section 6.2 in [START_REF] Choquet | M-SURGE 1.8 User's manual[END_REF], but it is also possible to guard against local minima by augmenting the information available.

Often, such information already exists, but could not be exploited in the traditional CR models. One of the major aims of the newly developed models was indeed to alleviate and valorize the field work by comprehending and exploiting a wider range of data.

In this paper, we have examined two ways of gathering additional information: using less efficient clues and sexing genetically some animals. Genetic sexing is of course paramount for the determination of sex, but for practical reasons it cannot usually be performed on every single individual. In our example, genetically sexing very few birds proved sufficient to greatly improve the shape of the deviance and hence the efficiency of the optimization algorithm. We also observed that it eliminated the wrong one of the two dual solutions. Finally, genetic sexing occasionally rendered identifiable new parameters in parameter-redundant models. It should also be noticed that genetic sexing is only one approach to specifying the a priori probability to be a male, P (m), for some individuals. Other, not necessarily 0 -1 predictors of sex frequently available in old studies, like biometrical measures, can be incorporated in the model in the same way (P (m) can be set to some intermediate value between 0 and 1). This approach of incorporating external information has been used by [START_REF] Fujiwara | Estimating population projection matrices from multi-stage mark-recapture data[END_REF] although in a way that, past the first encounter, may not be correct: the conditioning in their stage-assignment matrix, P (stage|measure), is the reverse of what is needed, P (measure|stage), in the calculation of the likelihood (see [START_REF] Pradel | Multievent: an extension of multistate capture-recapture models to uncertain states[END_REF], for a formal demonstration). As for the use of less efficient clues, our example proved that they may be valuable beyond what seems at first sight attainable based on their frequency of use and efficiency (percentage of error). We thus encourage people to gather such clues especially if they are not time consuming (e.g. [START_REF] Redman | Sexing northern gannets from DNA, morphology and behavior[END_REF]. After three or four visits, it will always be possible to assess the quality of the prospective clues as well as the improvement brought about by such and such identification method in terms of gained precision. If different methods for sex identification are in competition, it is then possible to decide objectively, from the informed assessment of their relative costs and benefits, which ones to favor and which ones to discard.

φ f , p f , e f , x f . This function depends on the particular encounter history h. Let write

P (h|f ) = g h (φ f , p f , e f , x f ). (2)
The probability of the same history conditional on the animal being this time a male is obtained by using the male parameters instead of the female ones. However, when a judgment was right for a female, it is wrong for a male (and vice-versa) so that x f should be replaced with 1x m (see the example of section 2.1 leading to equation 1 if this is not immediately clear). Thus,

P (h|m) = g h (φ m , p m , e m , 1 -x m ). (3)
The unconditional probabilities P (m) and P (f ) are respectively

p m µ p m µ + p f (1 -µ) and p f (1 -µ) p m µ + p f (1 -µ) . (4)
The transformation Θ : [0, 1] 9 -→ [0, 1] 9

(θ 1 , . . . θ 9 ) → (θ 2 , θ 1 , θ 4 , θ 3 , 1θ 5 , θ 7 , θ 6 , 1θ 9 , 1θ 8 ) plays a key role. We first establish Lemma 1. The probability of any encounter history, seen as a function of the parameter vector (φ m , φ f , p m , p f , µ, e m , e f , x m , x f ), is invariant under Θ.

Figure 1 :

 1 Figure 1: Excerpt from the Audouin's gull data coded according to three criteria: 1=judged male; 2=judged female; 3=no judgement made. #{h} is the number of individuals having encounter history h. The data set has 4025 individuals spread among 917 encounter histories over 10 years of study.

Figure 2 :

 2 Figure 2: Profile-deviance curves of the parameters of model (φ s , p s ) applied to the Audouin's gull data set. For each parameter, the deviance (y-axis) presents two distinct minima. The curves for p, φ and e are the same for males and females.

D

  ∀h, P (h) • Θ = P (h)Proof. When the parameters are changed in the following way:(φ m , φ f , p m , p f , µ, e m , e f , x m , x f ) Θ -→ (φ f , φ m , p f , p m , 1µ, e f , e m , 1x f , 1x m ), P (f ) becomes p m µ p m µ + p f (1µ) = P (m); [from (4)] P (m) becomes p f (1µ) p m µ + p f (1µ) = P (f ); [from (4)] P (h|f ) becomes g h (φ m , p m , e m , 1x m ) = P (h|m); [from (2) and (3)] P (h|m) becomes g h (φ f , p f , e f , x f ) = P (h|f ); [from (2) and (3)] so that, eventually, P (h) itself is unchanged.20A consequence of this lemma is that the likelihood and the deviance are equally unchanged under Θ. Consequently, if the parameter vector ( φm , φf , pm , pf , μ, êm , êf , xm , xf ) maximizes the likelihood, its transform by Θ, ( φf , φm , pf , pm , 1 -μ, êf , êm , 1 -xf , 1 -xm ), maximizes it too. Hence, the theorem:Theorem 1. The likelihood of model (φ s , p s ) is maximized at two generally distinct points one of which is the transform of the other by the unipotent mapping Θ.We now examine some ensuing properties of the profile deviance curves. Let D be the deviance of model (φ s , p s ). D : [0, 1] 9 -→ [0, +∞], is a function of the 9 probability parameters: φ m , φ f , p m , p f , µ, e m , e f , x m , x f , which has the property that D • Θ = D. IfE a i = θ ∈ [0, 1] 9 , θ i = a ,the profile deviance function for parameter i is P i (a) = P 9 (1a).

  last interval is critical. This problem disappears if survival or encounter parameters are hold constant. The introduction of sex uncertainty may well add new general identifiability problems. The minimal generalization of the CJS model consists in having the survival and encounter parameters, the 'traditional' parameters of subsection 2.1, still time-dependent and in considering that all of the 'new' parameters are constant over time. In this subsection, we examine whether this model is parameter-redundant. We also examine the three models derived by holding survival or encounter probabilities or both constant over time. In the following, we denote the four models considered by their sole variable part, i.e. (φ t * s , p t * s ), (φ t * s , p s ), (φ s , p t * s ), (φ s , p s ) where s stands for sex and t for time.

time or timedependent; they may vary by sex, age... It is thus impossible to tell in advance which parameters are or are not identifiable in each possible model. However, it is useful to have some general guidance as to where problems are likely to occur. In traditional CR, the fully time-dependent Cormack-Jolly-Seber model (CJS) plays this role. In this model, it is known that survival over the last time interval is confounded with encounter probability at the last occasion and it is a general feature of the fully time-dependent CR models that the

Table 1 :

 1 Identification of estimable quantities in 4 models incorporating probabilities of error. Superscripts are for time steps. Only 3 occasions were considered. φ 2 and p 3 are thus the last survival and last encounter parameters respectively.

	Model	#par Model rank non estimable parameters estimable functions
	(φ s , p s )	9	9	none	-
	(φ t * s , p s )	11	11	none	-
	(φ s , p t * s )	13	12

  encounter probabilities, p 1 m and p 1 f (absent from the CJS model). When encounter probabilities are time-dependent (models (φ t * s , p t * s ) and (φ s , p t * s )), only the ratio

	p 1 f
	p 1 m

Table 2 :

 2 Maximum-likelihood estimates for each of the four models. For time-dependent parameters, the median is given. Each model has also a dual solution with probabilities of errors >50%.

	Model	encounter probability (p)	survival (φ)	probability of judgment (e)	probability to be right (x)	proportion of males (µ)
		m	f	m	f	m	f	m	f	(m)
	(φ s , p s )	0.61 0.72 0.88 0.95 0.02 0.79 0.53 0.56	0.86
	(φ t * s , p s ) 0.60 0.72 0.89 0.97 0.02 0.79 0.53 0.56	0.87
	(φ s , p t * s ) 0.61 0.74 0.87 0.95 0.02 0.79 0.53 0.56	0.88
	(φ t * s , p t * s ) 0.61 0.76 0.86 0.97 0.02 0.79 0.53 0.56	0.88

Table 3 :

 3 Codes used to take into account the unequal reliability of each of the 4 criteria used to assess the sex of Audouin's gulls upon observation at the Ebro delta colony sexing has gained ground; especially the criterion 'relative body size', initially used very sparingly, has become more common by the end of the study. These features were incorporated in a model denoted (φ s , p t , µ, e T , m4 T ,

	(Spain).
	judged from	to be not judged
		m f
	copulation	1 5	9
	begging food	2 6
	courtship feeding 3 7
	body size	4 8
	the study,

Table 4 :

 4 Results of model (φ s , p t , µ, e T , m4 T , m1, m2, x1, x2, x3, x4) applied to the Audouin's gull data set with different amount of information. Analysis A uses all 4 criteria and the genetic determination of the sex of 24 birds; Analyses B and C do not use the genetic determination of sex; Analysis C does not use the judgments based on body size. Standard errors are given between parentheses. With less information (from left to right), precision decreases. For analyses B and C, we have retained the dual solution that yields proportions of error <50%. Improvement in the shape of the deviance brought about by a limited amount of external information shown on the profile deviance of model (φ s , p t , µ, e T , m4 T , m1,

	Analysis	A	B	C
	proportion of males	0.53 (0.027) 0.53 (0.029) 0.55 (0.034)
	female survival	0.91 (0.013) 0.91 (0.014) 0.93 (0.024)
	male survival	0.86 (0.013) 0.86 (0.014) 0.84 (0.025)
	error copulation	0.06 (0.041) 0.06 (0.041) 0.08 (0.055)
	error begging food	0.05 (0.029) 0.06 (0.031) 0.09 (0.077)
	error courtship feeding 0.00 (0.161) 0.00 (0.155) 0.00 (0.286)
	error body size	0.11 (0.064) 0.09 (0.074)	NA

This model has 23 parameters including 1 intercept and 1 slope for e and 1 intercept and 1 slope for m4 and 10 capture probabilities.

Additionally, a very few birds (24) had been sexed genetically (Genovart et al., 2003a,b). We fitted the previous model using (analysis A) and not using (analysis B) this limited information. When we used it, P (m) in the capture history of the genetically sexed individuals was set to 1 or 0 as appropriate. On the other hand, we considered the effects of not using the presumably less reliable criterion, 'body size'. To do that, the corresponding observations were recoded as a 'no judgment' observation. When doing this latter analysis (analysis C), the genetic determination of sex was not used. Thus, we have a gradient of decreasing amount of information from analysis A to analysis C (Table 4).

All three analyses yield estimates in agreement with what is known of the biology of the species. For instance, survival estimates are very close to those estimated previously by CR in the study site (e.g. [START_REF] Oro | Food availability and nest predation influence life history traits in audouin's gull, larus audouinii[END_REF][START_REF] Cam | Assessment of hypotheses about dispersal in a long-lived seabird using multistate capture-recapture models[END_REF]. We also note that, as anticipated, body size is the least reliable clue but the copulation criterion does not come out as the obvious best method. The results with and without genetical information are very similar (first 2 columns of Table 4). However, the known sex of only 24 birds suffices to break the tie between the dual solutions: while analyses B and C still have two solutions (only the reasonable one is presented in Table 4), analysis A has just one, as can be seen on the profile deviance curves relative to the proportion of males (Fig. 3).

In other, not shown models, where sex-ratio is initially not identifiable, the additional information brought about by the 24 birds renders this and other parameters identifiable.

As for the criterion 'body size', even though mistakes are made on average once in every 10 judgments and only 15.70% of the judgments made use this criterion, dropping it results in a disproportionate loss of precision (compare the standard errors in the last two columns of Table 4).

Appendix

The aim of this appendix is to establish the existence of two distinct dual solutions to the maximization of the likelihood of model (φ s , p s ) presented in subsection 2.1, as well as some related properties of its profile deviance curves. Why this happens is rooted in the law of total probabilities applied to an arbitrary encounter history h. P (h) = P (f )P (h|f ) + P (m)P (h|m)