Hierarchical modelling of population growth rate from individual capture-recapture data

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leHierarchical modelling of population growth rate from

individual capture-recapture data

Simone Tenan1,2,∗, Roger Pradel3, Giacomo Tavecchia1, Jose Manuel Igual1, Ana Sanz-Aguilar1,4,

Meritxell Genovart1, Daniel Oro1

1 Population Ecology Group, IMEDEA (CSIC-UIB), Miquel Marques 21, 07190 Esporles

(Mallorca), Spain;

2 Sezione Zoologia dei Vertebrati, MUSE - Museo delle Scienze, Corso del Lavoro e della Scienza

3, 38123 Trento, Italy (current address);

3 Biostatistics and Population Biology Group, Centre d’Ecologie Fonctionelle et Evolutive

(Campus CNRS) 1919 Route de Mende, F-34293 Montpellier, France;

4 Estacion Biologica de Donana (CSIC), Americo Vespucio s/n, E-41092 Sevilla, Spain;

∗ E-mail: Corresponding [email protected]

Running title: Hierarchical Bayesian temporal symmetry model

This article has been accepted for publication and undergone full peer review but has not been

through the copyediting, typesetting, pagination and proofreading process, which may lead to

differences between this version and the Version of Record. Please cite this article as doi:

10.1111/2041-210X.12194

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leAbstract

1. Estimating rates of population change is essential to achieving theoretical and applied goals

in population ecology, and the Pradel (1996) temporal symmetry method permits direct esti-

mation and modelling of the growth rate of open populations, using capture-recapture data

from marked animals.

2. We present a Bayesian formulation of the Pradel approach that permits a hierarchical mod-

elling of the biological and sampling processes. Two parametrisations for the temporal sym-

metry likelihood are presented and implemented into a general purpose software in BUGS

language.

3. We first consider a set of simulated scenarios to evaluate performance of a Bayesian variable

selection approach to test the temporal linear trend on survival and seniority probability,

population growth rate and detectability. We then provide an example application on indi-

vidual detection information of three species of burrowing nesting seabirds, whose populations

cannot be directly counted. For each species we assess the strength of evidence for temporal

random variation and the temporal linear trend on survival probability, population growth

rate and detectability.

4. The Bayesian formulation provides more flexibility, by easily allowing the extension of the

original fixed time effects structure to random time effects, an option that is still impractical

in a frequentist framework.

Key-words

Bayesian analysis, Balearic shearwater, Gibbs variable selection, mark-recapture, population dy-

namics, rate of population change, Scopoli’s shearwater, Storm petrel, survival, temporal symmetry

model.

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leIntroduction

The rate of change of population size over time is a process of great interest in population ecology.

Much has been written about the importance of this metric to assess the viability of a population

and the effects of management actions (e.g. Morris & Doak 2002). Different approaches exist to

estimate and model population growth and associated vital rates using capture-recapture data from

open populations, and many are implemented in freely available software such as MARK (White &

Burnham 1999) and POPAN (Arnason & Schwarz 1999). All represent multiple ways of writing the

likelihood first introduced by Jolly (1965) and Seber (1965), and differ in the way that the arrival

of new individuals into the population (i.e., local recruitment along with immigration) is modelled.

The frameworks among which animal ecologists can choose include the superpopulation approach

(Crosbie & Manly 1985; Schwarz & Arnason 1996), the temporal symmetry approach (Pradel 1996),

the parametrisation of Link & Barker (2005) and the restricted occupancy formulation of Royle &

Dorazio (2008).

The Pradel method, i.e., the temporal symmetry approach, has the unique characteristic of

combining in the same likelihood the standard-time and the reverse-time approach, simultaneously

incorporating survival and recruitment parameters and thus allowing inference on population growth

rate. The conceptual basis of the temporal symmetry of capture-recapture data derives from the

observation that the proportion of individuals that are already members of the population in the

previous sampling occasion, is the analogue of the survival rate when capture history data are

considered in reverse time order (Pollock et al. 1974; Nichols et al. 1986). Pradel (1996) termed

that proportion the ‘seniority’ parameter, reflecting the relative contribution of survivors from the

previous occasion to the population growth rate (Nichols et al. 2000).

Among the three parametrisations presented for the temporal symmetry likelihood, one uses

population growth rate ρ as a model parameter and thus allows testing of biological hypotheses

directly on ρ. Population growth rate integrates environmental effects on different vital rates and

the possibility of directly modelling ρ can be useful to investigate overall environmental influences,

especially when an environmental covariate influences multiple vital rates (Nichols & Hines 2002).

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leNichols & Hines (2002) also highlighted that temporal variance of ρ, estimated using a random

effect perspective, is highly relevant to extinction probability and ‘emphasizes the potential utility

of the direct estimation and modelling of ρ for population viability analyses’.

Here we present a Bayesian formulation of the Pradel temporal symmetry approach in which

parameter estimates are derived from the sampling of posterior distributions by MCMC methods.

The Bayesian formulation permits the hierarchical modelling of the biological and sampling pro-

cesses, allowing the extension of the original fixed time effects structure to random time effects, an

option that is still impractical in a frequentist framework. We illustrated the modelling approach

with real data on detection histories of individuals belonging to three species of burrowing nesting

seabirds, whose populations cannot be directly censused. In addition to the estimation of unex-

plained temporal variance we tested for temporal trends in model parameters. Our study species

suffer from additive mortality from longline fishing gears and alien predators at sea and at the

colonies, respectively, and it has been suggested that this mortality is causing a decline in their

populations (Barcelona et al. 2010; Cooper et al. 2003). For this reason, population trends for these

seabirds are of particular conservation interest.

Previous evidence suggested that placing constraints on some model parameters may impose

unintended constraints on others, with ρ being defined as a function of survival probability, that

in turn appears in the model (Nichols & Hines 2002). Furthermore, spurious trends in ρ can arise

from changes in the relative number of marked and unmarked animals in the population through

time (Hines & Nichols 2002). We therefore preceded the application with an investigation of the

effectiveness of detecting a temporal trend, and assigned it to the correct parameter, through the

Gibbs variable selection procedure (Dellaportas et al. 2000) applied to simulated data. Finally, we

provided supplementary material to facilitate understanding of likelihood implementation in BUGS

language (Lunn et al. 2009).

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leMethods

The Pradel temporal symmetry approach

Pollock et al. (1974) first pointed out that considering capture histories in reverse time order allows

inference about the recruitment process. By adopting this approach, Pradel (1996) developed a full

likelihood that permits the direct modelling of population dynamics by incorporating survival and

recruitment parameters. We briefly describe the Pradel temporal symmetry approach, to which we

also refer for general notation, by focusing on single-age populations sampled at a single study site.

Consider a sampling scheme of capture-mark-recapture (CMR) data on individuals sampled during

s successive occasions. From the ensemble of capture histories we can derive, for each occasion i,

the number of animals observed (ni), and among them, those observed for the first time (ui), for

the last time (vi), and those removed from (i.e., not released back into) the population (di).

The likelihood of Pradel (1996) includes the following parameters:

φi the survival probability from just after i to just before i+ 1;

pi the probability of being captured at time i when present just before i;

γi the probability that an animal present just before i was already present just after i− 1 (also

called seniority probability);

ri the probability of being captured at time i when present just after i;

µi the probability of being released for an individual captured at i;

ξi the probability of not being seen before i when present just before i;

χi the probability of not being seen after time i when present just after i.

Note that the parameters γi, ri, and ξi are the reverse-time analogue of φi, pi, and χi used

in the Cormack-Jolly-Seber (CJS) model. When neither removals nor introductions derive from

the capture-recapture procedure, as in our study, the statement “just before” and “just after” can

be replaced with “at”, and the forward- and reverse-time capture probabilities are equal (pi =

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leri). The temporal symmetry models use both forward- and reverse-time modelling simultaneously,

conditioning on the number of animals in the population at the beginning of the study (N−1 ). More

generally we can define N−i and N+

i as the abundances just before and just after sampling occasion

i. The expected number of animals in the population at successive occasions is determined by

means of the population growth rate (ρi), which can be expressed by considering the forward- and

reverse-time manner of writing the expected number of animals alive, i.e. N+i φi = N−

i+1γi+1. The

expected rate of population growth can be derived, from the following approximate equality, as:

ρi = E(N−i+1/N

+i ) ≈ φi/γi+1. (1)

Note that eqn 1 does not account for sampling-related mortality, i.e. for animals captured but not

released back into the population. Trap mortality can easily be taken into account and we refer to

Nichols & Hines (2002, eqn 8) and Hines & Nichols (2002, eqn 6) for further details.

The expected number of animals with a specific capture history can be written conditional on

the initial population size N−1 . As an instance, under the temporal symmetry model the expected

number of animals exhibiting capture history ‘011’ can be written as:

E(x011|N−1 ) = N−

1 λ′

1 ξ2 p2 µ2 φ2 p3 µ3, (2)

where N−1 λ

′1 gives the expected number of animals in the population just before sampling occasion 2.

Note that the initial population size N−1 in Eqn 2 is an unknown random variable. The probability

of each capture history conditional on the total number of animals captured (P (h)), can be derived

by dividing the expected number of animals with a specific capture history (E(xh)) by the expected

total number of animals caught (E(M)). E(M) can be written as the sum of the expected number

of animals seen for the first time at each sampling occasion, i.e. E(M) =∑s

i=1 ξiN−i pi (for i =

1, . . . , s). The unknown initial population sizes N−i cancel in the ratio P (h) = E(xh)/E(M), leaving

the conditional probabilities expressed in terms of estimable model parameters. The likelihood, for

the set of observed animals, is thus the product of the conditional probabilities of all individual

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lecapture histories, i.e. L =

∏h P (h)xh .

Pradel (1996) proposed three different parametrisations of his original likelihood, which are all

implemented in program MARK (White & Burnham 1999). They all maintain detection and sur-

vival probability as model parameters: (i) one parametrisation uses the reverse-time parameter, or

seniority probability (φt,γt,pt, where t denotes time-variant parameters), (ii) another parametrisa-

tion incorporates population growth rate as a parameter (φt,ρt,pt), and (iii) a third one is based on

a measure fi of recruitment rate, defined as the expected number of new individuals at time i + 1

per animal alive and in the population at i (φt,ft,pt). We focused on the first two parametrisations,

denoted (γ, φ) and (ρ, φ) respectively, whose likelihoods differ in the way by which seniority prob-

ability is expressed. In fact, the second parametrisation can be derived from the first original one

(eqn 2 in Pradel 1996) by substituting γi with the ratio φi−1

ρi−1derived from eqn 1 (see Appendix S1

for further details). Note that this model formulation, with seniority probability γi as a derived

parameter, allows values of γ greater than 1.

Data simulation

We used simulated data to investigate the capability of detecting simulated trends in model param-

eters (φi, γi, pi). We also examined whether spurious trends could not be selected in any parameter,

including ρi. For each scenario, we generated 100 data sets of individual capture histories from a

population size N=100, for ten sampling occasions. Population size was assumed constant over

time (i.e. ρi=1), and thus each time we added to the population the number of new individuals

needed to compensate for the number of deaths. A total of six scenarios were considered (see Table

1 in Appendix S2). The first three scenarios had in common a constant survival and seniority prob-

ability (denoted as φ(.)γ(.)) with φi=γi=0.85, and thus a constant population growth rate (ρ(.)),

but differed in the values defined for detectability p. The latter parameter was considered constant

(p(.)=0.5) in the first scenario, and with a temporal logit-linear trend in the other two scenarios

(with trend coefficient βp,time = 0.3 in the second scenario, denoted as p(trend03), and βp,time = 0.5

in the third scenario, p(trend05)) along with a mean value p=0.5. Time was standardized to have

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lemean zero and unit variance. Detectability ranged between 0.39 and 0.61 in the scenario p(trend03),

and between 0.32 and 0.68 in the scenario p(trend05).

A second group of scenarios included an equal temporal logit-linear trend for survival and

seniority probability (φ(trend)γ(trend)) that led again to a constant ρ. We used a mean survival (φ)

and seniority (γ) rate of 0.8, whereas trend coefficients on the logit scale βφ,time and βγ,time were set

to 0.5. Simulated survival and seniority probabilities ranged between 0.66 and 0.89. Detectability

was assumed either constant or with two different temporal trends as in the previous scenarios.

Scenario-specific settings for each parameter are summarized in Table 1 of Appendix S2. Each

simulated data set was analysed using the (γ, φ) and (ρ, φ) parametrisation of the Pradel temporal

symmetry model, under two sets of priors (see section ‘Bayesian analysis’).

Case study

The application we present is based on individual data collected for three seabird species of Procel-

lariiformes during s successive occasions (years), with s equals 12, 8, and 20 for Scopoli’s shearwater

(Calonectris diomedea), Balearic shearwater (Puffinus mauretanicus), and Storm petrel (Hydrobates

pelagicus), respectively. More specifically, data were collected during the breeding period at three

Mediterranean colonies: Pantaleu Islet (Mallorca, Balearic Archipelago, Spain) for Scopoli’s shear-

water (period 2002-2013, n=699), Sa Cella (Mallorca, Balearic Archipelago) for Balearic shearwater

(1997-2004, n=443), and Benidorm Island cave 1 (Alicante, Spain) for European storm petrel (1993-

2012, n=889). The number of individuals recaptured at least once was 475 in Scopoli’s shearwater,

251 in Balearic shearwater, and 590 in Storm petrel. All individuals were caught at nests while

incubating or when feeding chicks, and thus the sample was made up only of breeding adults. Since

all study burrows are marked and monitored over the years, and the goal has been to capture the

two members of the pair of each burrow during the breeding season, field effort has been kept rather

constant. More details of fieldwork procedures can be found in Genovart et al. (2013a), Oro et al.

(2004), and Sanz-Aguilar et al. (2011). The two shearwaters are mainly threatened by mortality

at sea from fishing gears, a factor that does not affect storm petrels, which are affected by alien

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leterrestrial predators such as rats and carnivores (e.g. Barcelona et al. 2010; Cooper et al. 2003).

Scopoli’s shearwater and Storm petrel include Mediterranean sub-species with vulnerable popula-

tions, whereas the Balearic shearwater has the highest IUCN category of threat for a taxon in the

wild, Critically Endangered, and is an endemic species of the Balearic archipelago (IUCN 2013).

The study colonies were all free of alien terrestrial predators and only the storm petrel colony was

affected by adult predation from yellow-legged gulls (Sanz-Aguilar et al. 2009a).

Bayesian analysis

Temporal symmetry models were fitted using a Bayesian formulation and the Markov chain Monte

Carlo (MCMC) framework (Robert & Casella 2004). Since the Pradel model involves likelihoods

not directly available in BUGS, we used the “zeros trick” (Spiegelhalter et al. 2007) that allows the

creation of an arbitrary likelihood. We refer to Appendix S1 for a detailed description of likelihood

specification.

Simulated data were analysed using both the (γ, φ) and (ρ, φ) parametrisation of the Pradel

model, under two sets of priors (see below). Given the conditions assumed for each scenario, we

calculated posterior probability of detecting trends in model parameters or, in other words, the

strength of evidence for temporal trends. This degree of support for a trend in a specific parameter

can be assessed by calculating the posterior variable inclusion probability, i.e. the probability

that a variable (‘TIME’ in this case) is “in” the model, using a Bayesian variable selection approach

(O’Hara & Sillanpaa 2009). Following Dellaportas et al. (2000) and Ntzoufras (2002), each temporal

slope βη,time (present in each linear predictor) was multiplied by an “inclusion parameter” ωη (for

η ∈ {φ, γ, p} in the (γ, φ) parametrisation, and η ∈ {φ, ρ, p} in the (ρ, φ) parametrisation), a latent

binary variable with a Bernoulli prior distribution with parameter 0.5, indicating no constraints on

the probability that a given covariate is included in the model. Taking the example of the (ρ, φ)

parametrisation, testing for a temporal trend in each parameter by means of the Gibbs variable

selection approach involved the calculation of the probability that the covariate ‘TIME’ was included

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lein the model. The corresponding ‘full’ linear predictors, and related constraints, are as follows:

logit(φi) = αφ + ωφ βφ,timeTIMEi

log(ρi) = αρ + ωρ βρ,timeTIMEi

logit(pi) = αp + ωp βp,timeTIMEi

(3)

where αφ, αρ, and αp are the overall means of survival probability (on logit scale), population

growth rate (on log scale) and capture probability (on logit scale), respectively. ωφ, ωρ, and ωp are

the inclusion parameters (in each linear predictor) for covariate TIMEi, which was standardized to

have mean zero and unit variance (for sampling occasion i from 1 to 10). If ωη=1 the linear predictor

includes trend variable ‘TIME’, whereas if ωη=0 the linear predictor does not include ‘TIME’. For

each scenario and model parametrisation, we derived the mean across 100 replicates of the Bayesian

point estimate of the mean inclusion probability, i.e the probability that ωη=1, associated with a

trend in each linear predictor.

In the application with real data, we extended the model applied on simulations to a hierarchical

formulation in order to estimate the temporal random variance, unexplained by the deterministic

trend term, for each parameter in the (ρ, φ) parametrisation (φ,ρ,p). Consequently, the extended

‘full’ linear predictors used in the Gibbs variable selection were the following:

logit(φi) = αφ + εφ,i + ωφ βφ,timeTIMEi, εφ,i ∼ N(0, σ2φ)

log(ρi) = αρ + ερ,i + ωρ βρ,timeTIMEi, ερ,i ∼ N(0, σ2ρ)

logit(pi) = αp + εp,i + ωp βp,timeTIMEi, εi ∼ N(0, σ2p)

(4)

where εφ,i, ερ,i, and εp,i is the deviation from the overall mean of each parameter, whereas σ2φ, σ

2ρ,

and σ2p is the temporal variance of each parameter (on logit scale for φ and p, on log scale for ρ).

From this mixture model, we derived not only the posterior inclusion probabilities ωη, but also

model-averaged estimates (averaged across the different models included in the posterior sample)

for φi, ρi, pi, the related overall means (αφ, αρ, αp) and standard deviations (σφ, σρ, σp). These

estimates take into account both parameter and model uncertainty, with respect to trend covariate

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ledependence. Model-averaged estimates were also obtained for covariate coefficients (βφ,time, βρ,time,

βp,time) by averaging across posterior samples where the corresponding ωη=1 ( i.e., conditional on

the covariate ‘TIME’ being present in the model) (Royle & Dorazio 2008).

Posterior distributions, for both parameter estimates and variable inclusion probabilities, were

assessed for prior sensitivity (e.g. King 2009) by using two sets of priors for the intercepts (αs)

and slopes (βs) in the linear predictors (prior set 1: Normal(0, 100) distributions; prior set 2:

Uniform(−5, 5) distributions). The hierarchical priors were specified by assuming each σ in eqn 4

drawn from a Uniform(0, 2) distribution.

Summaries of the posterior distribution were calculated from one Markov chain (in the simulation

study) and from three independent chains (in the application) initialized with random starting

values, run 500,000 times after a 50,000 burn-in and re-sampling every 20 draws. Chain length

was increased to 1,000,000 (after burn-in) iterations in the application with real data, in order to

improve sampling of the parameters associated with low inclusion probabilities. The models were

implemented in program JAGS (Plummer 2003), that we executed from R (R Core Team 2012)

with the package R2jags (Su & Yajima 2012). An R script with the model specification in BUGS

language is provided in Appendix S1.

Results

Simulation study

In the presence of constant survival and seniority probability, and thus constant population growth

rate, the probability of detecting a spurious trend was generally very low (average posterior inclusion

probability ω < 0.17 for φ, < 0.09 for γ, and < 0.01 for ρ, across 100 replicate data sets; Appendix

S2, Table 2). Under the same conditions, the probability of including in model a real trend on

detectability ranged, on average, from 0.40 to 0.56 for a logit-linear trend coefficient βp,time = 0.3,

and from 0.84 to 0.95 for βp,time = 0.5.

When a temporal trend was imposed on survival and seniority probability, but not on the

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lepopulation growth rate, the probability of detecting a real trend in φ and γ was always > 0.25

(Appendix S2, Table 3). The related posterior inclusion probabilities ranged, on average, as follows:

for φ: 0.40 – 0.77 (no trend in p), 0.39 – 0.74 (βp,time = 0.3), 0.33 – 0.68 (βp,time = 0.5); for γ:

0.40 – 0.52 (no trend in p), 0.38 – 0.45 (βp,time = 0.3), 0.26 – 0.42 (βp,time = 0.5). Under the last

two scenarios, the average probability of including in the model a real trend in detectability was

0.40 – 0.63 (βp,time = 0.3) and 0.75 – 0.81 (βp,time = 0.5). In addition, the probability of detecting

a spurious trend in ρ was always low (< 0.02) and not affected by the simulated conditions on p.

Interestingly, (ρ, φ) parametrisation models performed better than (γ, φ) parametrisation models in

the presence of trend in φ and γ (last three scenarios; Table 3 in Appendix S2). In general, posterior

inclusion probabilities were slightly higher under the Uniform(−5, 5) priors, even if estimates did

not substantially differ under the two sets of priors.

Case study

Posterior probabilities for the inclusion of temporal trends in model parameters (ωη with η ∈

{φ, ρ, p}) showed good agreement for the two sets of priors, even though ωη estimates were slightly

higher with the Uniform(−5, 5) priors (Table 1). The degree of support for trends in model param-

eters was very low (i.e. < 0.2, Ntzoufras 2009) in all species, except for the probability of a trend in

Scopoli’s shearwater detectability (ωp=0.25, under prior set 2). The two sets of priors yielded sim-

ilar posterior distributions for the parameters (Appendix S3). We thus discussed model-averaged

posterior estimates obtained under the set of Normal(0, 100) priors.

Mean survival probability (φ = expit(αφ)) was the lowest for Balearic shearwater (0.773, 0.672 –

0.863, 95%CRI), the highest for Storm petrel (0.824, 0.785 – 0.861), and intermediate for Scopoli’s

shearwater 0.810 (0.784 – 0.838; Table 2, Fig. 1). In all three species, 95% credible interval for

the estimate of the survival trend coefficient (βφ,time) encompassed zero, in agreement with the low

inclusion probabilities for temporal trends in φ. However, Balearic shearwater and Storm petrel

survival probabilities were characterized by substantial unexplained temporal random variation (σφ;

0.526, 0.055 – 1.368, and 0.494, 0.281 – 0.808 respectively) compared to Scopoli’s shearwater (0.193,

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le0.019 – 0.445).

On average, Scopoli’s shearwater and Storm petrel populations are both estimated to slightly

increase (i.e. ρ > 1, where ρ = exp (αρ)) over the whole study period, by 0.2 and 0.4% per year,

respectively. In contrast, Balearic shearwater showed an average decreasing trend of 4.9% per year.

Nevertheless, credible intervals of mean population growth rates encompassed one (i.e. population

stability) in all three species (Fig. 2), and the probability of a mean growth > 1 was 0.55 for both

Scopoli’s shearwater and Storm petrel, whereas the probability of a mean growth < 1 was 0.86

in Balearic shearwater. Temporal random variation of population growth rate (σρ) was generally

quite small, with the largest in Storm petrel (0.115, 0.068 – 0.182), intermediate values in Balearic

shearwater (0.095, 0.003 – 0.324), and the lowest in Scopoli’s shearwater (0.043, 0.003 – 0.105).

Detectability (p) of Scopoli’s shearwater showed a clear positive relationship with time (βp,time

= 0.233, 0.024 – 0.441), while a sizeable temporal random variability characterized p in Balearic

shearwater and Storm petrel (Table 2).

Recruitment, in the form of seniority probability (γ), was estimated by deriving it from survival

probability and population growth rate. In Scopoli’s shearwater γ was almost constant, varying

from 0.785 (0.732 – 0.832, γ7) to 0.846 (0.795 – 0.901, γ9), whereas in Balearic shearwater values

ranged from a minimum of 0.701 (0.469 – 0.932, γ2) to a maximum of 0.874 (0.800 – 0.954, γ6), and

in Storm petrel from 0.696 (0.582 – 0.810, γ12) to 0.935 (0.814 – 1.055, γ11).

Discussion

Simulation study

The interest in the temporal symmetry method, compared to other open population modelling

approaches for CMR data, lies in the possibility of directly testing covariate effects on population

growth rates (e.g. Hunter et al. 2010; Ozgul et al. 2010; Tenan et al. 2012). However, a legitimate

concern is that a trend in capture probability may be mistaken for a trend in some other parameter.

Similarly, wrongly modelling parameters as functions of some time-variant covariates may induce

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lecompensatory patterns in other parameters. In this paper, we investigated the efficiency of detecting

the correct linear trends under the (γ, φ) and (ρ, φ) parametrisations in a Bayesian implementation

of the temporal symmetry model. More specifically, we evaluated the performance of a well-known

Bayesian variable selection procedure, the Gibbs variable selection (Dellaportas et al. 2000), in

terms of probability of detecting real trends and avoiding spurious trends under scenarios with

three degrees of trend in detection probability p (no trend, a moderate trend βp,time = 0.3, and a

strong trend βp,time = 0.5). Results showed very low probabilities of detecting spurious trends in

model parameters (< 0.02 for ρ) under all simulated conditions for p. In addition, trend strength in

p (i.e. the βp,time value) only weakly affected the probability of correctly assigning a trend to survival

φ and seniority γ probability. Directly testing for a trend in ρi, by means of a variable selection

procedure, showed good performance in terms of a low probability of selecting spurious trend in

population growth rates, at least under the limited range of scenarios considered. On the other

hand, simulation results showed probabilities of including a real trend (ωη, with η ∈ {φ, ρ, γ, p})

were often < 0.5. Thus, regarding population growth rate, the approach appears conservative:

trends may not be detected, but if they are, they are highly likely to be real. If the approach is

to select the best model, i.e. the more probable combination of covariates in the model, then one

has to define a threshold inclusion probability (values of 0.2 and 0.5 have been proposed in the

literature; Ntzoufras 2009; Barbieri & Berger 2004). As an alternative, model-averaged estimates

can be derived, as done here in the application, taking into account both parameter and model

uncertainty and avoiding the awkward definition of threshold inclusion probabilities. Although a

certain amount of trend will always pass in the average estimates with the latter alternative, this

effect, being weighted by small inclusion probabilities, should be small if no actual trend was present.

This is what we observed in the range of scenarios considered (see Tables 4 to 7 in Appendix S2).

In particular, robust estimates of population growth rate parameters were obtained even in the

presence of trends in other demographic parameters.

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leCase study

Our application focused on testing for linear trends in survival probability, population growth rate,

and detection probability, as well as estimating unexplained temporal random variation in these

parameters, for three species of burrowing nesting seabirds. Testing temporal trends in survival

was initially motivated by evidence for additive causes of anthropogenic mortality (e.g. Laneri

et al. 2010) that affect Scopoli’s and Balearic shearwater (longline by-catch) potentially inducing

a decreasing trend in survival. Furthermore, inbreeding depression (Hogg et al. 2006) and density

dependence (both positive and negative; Ehrlen & Groenendael 1998; Stephens & Sutherland 1999)

can generate temporal trends in population growth rate, which can be even larger in social species,

like our study species. Finally, a trend in detection probability can be induced by an array of

factors, such as an increase in skills in fieldwork procedures or a variation of resource availability

over time (e.g. Williams et al. 2002).

In our case, only a trend in Scopoli’s shearwater detectability was supported by the data, and

temporal random variation was generally small in the population growth rates of all species. The

estimated mean survival probability for Scopoli’s shearwater (0.810) was generally low compared to

other estimates from the same colony but referred to a shorter time series (newly marked individuals

0.83 (0.76 – 0.88, 95%CI) and 0.78 (0.71 – 0.83), residents 0.87 (0.83 – 0.91) and 0.87 (0.82 – 0.90)

in Genovart et al. 2013a and Sanz-Aguilar et al. 2011, respectively) or from other Mediterranean

islands (ranging from 0.74 to 0.96; Igual et al. 2009; Jenouvrier et al. 2008, 2009). This low level

of adult survival has been strongly associated to the large-scale Southern Oscillation index (SOI)

influencing the frequency of strong hurricanes (Genovart et al. 2013a), and to other non-climatic

factors such as long-line incidental bycatch (Barcelona et al. 2010; Laneri et al. 2010; Ramos et al.

2012). Furthermore, the presence of transients in our sample (e.g. Jenouvrier et al. 2009) and the

impossibility of discerning them in the analyses, is known to negatively bias survival estimates

for residents (Pradel et al. 1997). The annual percentage in the number of transients at the two

colonies considered by Genovart et al. (2013a) varied from 0% to 21%. Low survival rates in

this species might be compensated for by recruitment, which was here estimated in the form of

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leseniority probability and encompasses not only local recruitment but also immigration. Previous

studies already suggested that even if the species is mostly philopatric (97%), there is non-negligible

dispersal (particularly natal dispersal) between colonies (Genovart et al. 2013b).

It is known that adult survival in Balearic shearwater is unusually lower than the typical values

observed in Procellariform seabirds (Oro et al. 2004). Our results confirmed a low mean survival

probability (0.773), very close to the estimate of 0.780 (0.02, SE) reported by Oro et al. (2004),

higher than 0.662 (0.04, SE) estimated by Tavecchia et al. (2008), and characterized by a remarkable

temporal random variation. The population is estimated to decline on average by 4.9% per year,

matching the 4.8% value reported by Oro et al. (2004) using population age-class matrix models

that incorporated both demographic and environmental stochasticity. Note that even if we used

two more years of data than Oro et al. (2004), inference of trend might be limited by the relatively

short study period.

Inference for the vulnerable Storm petrel was based on data from the same colony studied by

Sanz-Aguilar et al. (2009a) but using a time-series that is five years longer. Sanz-Aguilar et al.

(2009a) showed that predation by yellow-legged gulls negatively affected survival probability of

syntopic petrels, whose values soared from 0.75 (0.71–0.78, 95%CI) to 0.89 (0.82–0.94) after the

removal of specialist predatory gulls. The average survival probability estimated here (0.824) was

consistent with these values, but lower than estimates from predator-free colonies (0.90–0.95, Biar-

ritz, Atlantic France, Hemery 1980; 0.88 (0.85–0.91 95%CI), Marettimo, Italy, Sanz-Aguilar et al.

2009b). As a very interesting result, clearly reflected in Fig. 2, population growth rates were es-

timated to be positive after the beginning, in 2004, of a program of culling specialist gulls. The

unexplained temporal variation in survival we found here may be due to gull predation as numbers

of predated petrels were highly variable during the study period (Sanz-Aguilar et al. 2009a).

The modelling approach

We implemented in a Bayesian framework the temporal symmetry approach of Pradel (1996) that

permits direct modelling of ρi, the rate of change in population size between sampling occasion i

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leand i + 1. We showed how a Bayesian variable selection approach (O’Hara & Sillanpaa 2009) can

be used to test hypotheses directly on different model parameters, even though more investigation

is needed to explore potential biases in the estimation of model parameters (and related effects)

under other specific sampling situations. Moreover, by adopting a Bayesian approach we were able

to test for temporal random variation in the population growth rate, an option that is not available

in other computer software in which this modelling approach is implemented (e.g. MARK). A

Bayesian implementation of the temporal symmetry approach can help to make this underused

method more accessible, a method that has already been applied to other ecological themes, such

as characterizing the realized growth of host populations in relation to disease effects (Lachish et al.

2007), or estimating stopover duration (Chernetsov 2012).

Acknowledgements

We thanks Robert B. O’Hara and the anonymous referees for constructive comments on previous

versions of this manuscript. We also thanks Aaron Lemma for IT assistance. Funds were partially

provided by FEDER – Balearic Government and Fundacion Biodiversidad. We are grateful to

the Ward and Environmental Monitoring Service of Benidorm Island (Serra Gelada Natural Park,

Generalitat Valenciana), Ciudadanos por la Ciencia, Dr. E. Minguez, Dr. A. Martınez-Abraın and

B. Sarzo. Roger Pradel received a research stay grant from Universitat de les Illes Balears.

Data accessability

• The R and BUGS script for running the model is included in supplementary material (Ap-

pendix S1).

• Input data for the BUGS model: http://dx.doi.org/10.6084/m9.figshare.977874.

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leTables

Table 1: Posterior probabilities (ωη with η ∈ {φ, ρ, p}) of including a linear trend for the modelparameters of the (ρ, φ) parametrisation. Estimates are reported for two prior sets for the intercepts(αs) and slopes (βs) in the linear predictors in eqn 4 (prior set 1: Normal(0, 100) distributions; priorset 2: Uniform(−5, 5) distributions).

Inclusion probabilitySpecies prior set ωφ ωρ ωp

Scopoli’s shearwater 1 0.019 0.005 0.1162 0.054 0.012 0.253

Balearic shearwater 1 0.037 0.012 0.0392 0.086 0.028 0.093

Storm petrel 1 0.074 0.018 0.0302 0.166 0.047 0.074

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leTable 2: Model-averaged parameter estimates obtained under the set of Normal(0, 100) priors forthe intercepts (αs) and slopes (βs) in the linear predictors of eqn 4. Mean survival (φ) and detection(p) probabilities are given on probability scale, i.e., φ = expit(αφ) and p = expit(αp) where expit isthe inverse-logit function, and the related βs and σs on logit scale. Population growth rate-relatedparameters (βρ and σρ) are on log scale, whereas mean population growth rate ρ = exp(αρ). Median,2.5 and 97.5 percentiles are also reported.

Species Parameter mean SD 2.5% 50% 97.5%

Scopoli’s shearwater φ 0.810 0.013 0.784 0.810 0.838ρ 1.002 0.018 0.965 1.002 1.038p 0.792 0.017 0.758 0.792 0.826σφ 0.193 0.110 0.019 0.181 0.445

βφ,time -0.111 0.104 -0.306 -0.114 0.100σρ 0.043 0.027 0.003 0.039 0.105

βρ,time -0.021 0.024 -0.067 -0.021 0.027σp 0.245 0.122 0.033 0.234 0.520

βp,time 0.233 0.105 0.024 0.233 0.441

Balearic shearwater φ 0.773 0.046 0.672 0.774 0.863ρ 0.951 0.059 0.839 0.950 1.068p 0.468 0.079 0.311 0.467 0.629σφ 0.526 0.327 0.055 0.465 1.368

βφ,time 0.179 0.440 -0.713 0.162 1.128σρ 0.095 0.091 0.003 0.071 0.324

βρ,time -0.069 0.136 -0.367 -0.050 0.155σp 0.845 0.287 0.443 0.789 1.582

βp,time 0.144 0.399 -0.700 0.153 0.892

Storm petrel φ 0.824 0.019 0.785 0.824 0.861ρ 1.004 0.029 0.947 1.003 1.062p 0.709 0.038 0.632 0.710 0.780σφ 0.494 0.136 0.281 0.477 0.808

βφ,time 0.247 0.139 -0.025 0.245 0.532σρ 0.115 0.029 0.068 0.111 0.182

βρ,time 0.054 0.030 -0.006 0.055 0.113σp 0.763 0.151 0.524 0.743 1.113

βp,time 0.179 0.188 -0.193 0.181 0.556

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leFigures

Year

Sur

viva

l pro

babi

lity

2003 2005 2007 2009 2011

0.7

0.8

0.9

(a)

Year

1998 2000 2002

0.5

0.6

0.7

0.8

0.9

(b)

Year

Sur

viva

l pro

babi

lity

1993 1995 1997 1999 2001 2003 2005 2007 2009 2011

0.6

0.7

0.8

0.9

1.0

(c)

Figure 1: Model-averaged annual estimates (with 95%CRI) of survival probability (φi) for Scopoli’sshearwater (a), Balearic shearwater (b), and Storm petrel (c).

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Year

Pop

ulat

ion

grow

th r

ate

2003 2005 2007 2009 2011

0.85

0.95

1.05

1.15

(a)

Year

1998 2000 2002

0.7

0.9

1.1

1.3

(b)

Year

Pop

ulat

ion

grow

th r

ate

1993 1995 1997 1999 2001 2003 2005 2007 2009 2011

0.8

1.0

1.2

1.4

(c)

Figure 2: Model-averaged estimates (with 95%CRI) of population growth rate (ρi) for Scopoli’sshearwater (a), Balearic shearwater (b), and Storm petrel (c). The dotted line indicates populationstability.

Supporting Information

Additional Supporting Information may be found in the online version of this article.

Appendix S1. R and BUGS script.

Appendix S2. Simulation study.

Appendix S3. Bayesian learning plots.

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