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The authors have declared that no competing interests exist.

Current address: U.S. Geological Survey, North Carolina Cooperative Fish and Wildlife Research Unit, Department of Applied Ecology, North Carolina State University, Raleigh, North Carolina, United States of America

Understanding the influence of individual attributes on demographic processes is a key objective of wildlife population studies. Capture-recapture and age data are commonly collected to investigate hypotheses about survival, reproduction, and viability. We present a novel age-structured Jolly-Seber model that incorporates age and capture-recapture data to provide comprehensive information on population dynamics, including abundance, age-dependent survival, recruitment, age structure, and population growth rates. We applied our model to a multi-year capture-recapture study of polar bears (

Age structure affects population dynamics and how populations respond to environmental change [

Capture-recapture methods provide a robust framework to estimate demographic processes and incorporate auxiliary information to improve inferences [

Here, we develop a novel and generalizable age-structured JS open population model to jointly estimate age-specific demographic rates, abundance, population age structure, annual changes in age structure, and recruitment from capture-recapture and age data. Our approach integrates six processes into one state-space JS model [

We begin by describing a state-space JS model using the Schwarz and Arnason superpopulation formulation [

We applied the age-structured JS model to a case study investigating polar bear (

We follow the methods of [_{i}, where _{i} = 1 if the individual is part of the superpopulation and 0 otherwise. We assume
_{i} = 1 is known for any observed individual. Individuals recruit into the superpopulation during one of _{k},

For purposes of working with an augmented data set, entry probabilities are re-expressed as conditional probabilities, _{k}, the probability of entry at

The sequential state process (i.e., recruitment and survival) for individual _{ik}), is now described as,
_{i1} = 1 if individual _{i}), the product _{ik} _{i} = 1 if individual _{ik} denotes the detection or non-detection of individual _{k}) and the superpopulation (

For notation simplicity, we did not include individual (

Age data (_{ik}; the numeric age of individual _{ik}). For example, we know that an individual first captured at occasion _{i1}, as a random variable described by an initial age distribution (_{1}) denotes age 0 individuals that have not yet entered. We use (_{i1} + 1) so that age-0 (not yet entered) references the first category (_{1}). We define _{1} is equivalent to 1 − _{1} in the JS model (i.e., not entered at occasion 1; _{1} = (1 − _{1}) is the probability that an individual has not yet entered by occasion 1 (_{i1} = 0), and _{ik}) is estimated for all individuals in the population (observed and unobserved) and thus reflects the population-level age-structure, which can vary from annually observed ages for a variety of reasons (e.g., small sample sizes, variation in detection by age; see Case Study Results). Our approach makes no assumptions about stable age distributions or asymptotic properties but instead allows age structure to reflect data collected across the entirety of the study.

Directly linking the state and aging processes (Eqs _{0}, _{1}, _{2} describe the survival intercept at age 0 (or some centered value) and relationships of survival with age and age^{2}, respectively. Incorporating additional covariates, fixed and random effects, or individual heterogeneity on survival, recruitment, and detection parameters follow the same approaches as in JS models [

We developed two simulation studies to evaluate model performance. We generated and analyzed 200 datasets with _{ik} − 5) in the regression model to improve convergence (hereafter ‘age-specific survival simulation’). Although not necessary, we found that centering ages aided convergence similar to centering or scaling covariates [

We analyzed data from the constant-survival simulation using both the JS and age-structured JS models to assess effects of including age data on the precision of parameters (_{k}), and annual population growth rates (_{k+1}/_{k}), which are often a primary interest in JS studies. We calculated the percent reduction in coefficient of variation for survival probabilities, annual abundances, and annual population growth rates to evaluate changes in precision between the JS and age-structured JS models for these key parameters. For the age-specific survival simulation, we used the age-structured JS model with a quadratic survival model. Here, our primary objective was to assess the ability of the age-structured JS model to return unbiased parameter estimates, particularly for age-specific survival.

We repeated analyses of both the constant and age-specific survival simulations using different values of maximum age in year 1 (

We used a 7-year dataset (2012–2018) of individually marked polar bears in WHB to investigate multiple components of polar bear demography. The data are from a long-term study on polar bear ecology in the Hudson Bay region [

We fit three models to the data set: JS model without age data, age-structured JS model with constant survival, and age-structured JS model where survival was a quadratic function of age [

Models were fit in a Bayesian framework using Markov chain Monte Carlo (MCMC) methods. Both the JS and age-structured JS models are easily fit in common MCMC software packages such as WinBUGS, JAGS, or NIMBLE. In our study, models were fit using NIMBLE v0.8.0 [^{th} iteration to reduce file size. For the case study, we increased the number of chains to six and the number of iterations to 220,000 to increase the number of effective samples. We assessed convergence using diagnostic plots and the Gelman-Rubin statistic (_{0}) ~Beta(1,1) and independent Normal(0, sd = 10) for _{1} and _{2}. Results are reported as posterior medians and 2.5 and 97.5 percentiles (95% CRI) of retained posterior samples.

We evaluated goodness-of-fit using a posterior-predictive check to evaluate the ability of the model to predict the number of observed individuals each year (_{k}), which is a shared metric across modelling approaches. For each iteration, we generated the expected number of observed individuals (

Both the JS and age-structured JS model produced unbiased estimates of survival, recruitment, and abundance in the constant-survival simulations (_{k}, resulting in improved precision of annual growth rates (

Boxplots include medians (black line), interquartile range (box), and range of values (whiskers). Red horizontal lines denote data generating values. Data generation fixed the maximum age in year 1 (

The age-structured JS model also performed well in age-specific survival simulations, providing minimally biased to unbiased estimates of age-specific survival, abundance, recruitment, and initial age structure (

Boxplots include medians (black line), interquartile range (box), and range of values (whiskers). Red horizontal lines denote data generating values. Data generation fixed the maximum age in year 1 (

We analyzed encounter histories from

Points and error bars are posterior medians and 95% credible intervals, respectively.

All models provided similar estimates of superpopulation abundance (~ 500–600 individuals) but differed in the survival and recruitment processes leading to these superpopulation abundances (

Points and error bars are posterior medians and 95% credible intervals, respectively. Analyses used a subset of the larger, long-term WHB polar bear study and, therefore, do not reflect the status of the entire subpopulation ([

Posterior medians and 95% credible intervals are shown. Cumulative survival and life expectancy estimates are conditional on surviving to two years of age (i.e., independent bears). Upper credible bound for life expectancy from the age structure model with constant survival is > 100 years and not shown (see

Individuals 6–8 years of age were disproportionately represented in the 2012 age structure, suggesting strong recruitment from the 2004–2006 birth years. Cascading effects of this large cohort led to significant changes in population age structure and recruitment across years (

Red points are year-specific proportions from observed data. Grey polygon denotes prime breeding ages (10–15 years of age). The annual proportion of the population in prime breeding age is summarized in lower right panel (median and 95% credible intervals).

We developed an age-structured JS model to improve estimation of demographic parameters and, thus, inference about population dynamics from capture-recapture and age data. The novelty of our approach arises from integrating model components describing age structure, aging, survival, recruitment, and abundance into a single hierarchical model that overcomes the challenges of unknown ages in JS models (for observed and unobserved individuals) and individuals born prior to the study [

Evaluating age-specific demography, particularly demographic senescence (i.e., degradation of survival or breeding probabilities associated with aging), is essential to understanding the impacts of age structure on population dynamics [

Individuals 6–8 years of age were disproportionately represented in the 2012 age structure, suggesting large recruitment classes in 2004–2006. WHB female body mass was above average when these large age classes were dependent young (2005–2009; Lunn unpublished data), supporting previous findings that maternal body mass in polar bears is positively correlated with increased reproduction and survival of their young [

The generality of our approach provides a promising tool for future investigations into the effects of aging on population dynamics. Thus far, we have assumed age can be identified when an individual is observed. Although not fully explored herein, our approach allows for both missing age data and the incorporation of age proxies (e.g., morphometric data such as dental cementum or size) when annual age data are missing. Missing age data for some portion of observed individuals are addressed using the same process as unknown ages for augmented individuals, whereby age in year one is considered a random variable (Eqs

Simulation studies demonstrated that both the JS and age-structured JS models provided unbiased estimates of demographic rates and abundances; however, incorporating age data improved precision of demographic rates and population growth rates, increased the power to detect trends in abundance, and allowed unbiased estimation of age-dependent survival and changes in annual age structure. The age-structured JS model was generally robust to uncertainty in the selection of maximum age in year 1 (

We made several simplifying assumptions in our case study by not allowing for individual or temporal variation in survival (except by age), detection, reproduction, or movement, although there is capacity within our framework to generalize the model to these factors. Our case study consisted of a subset of a broader, long-term WHB polar bear study, thus our results may not represent the status of the entire subpopulation and are not intended to be used for management purposes [

Age structure of free-ranging populations may fluctuate in response to environmental stressors, especially factors that disproportionately affect reproduction or age-dependent survival (e.g., weather, competition; [

Jointly modeling abundance, survival, recruitment, age structure, and the aging process within the JS framework provides an important advance in our ability to evaluate population dynamics and provides crucial information for species management and conservation. Integration of age and capture-recapture data within the JS framework allows exploration of a wider range of demographic processes, including evolutionary and life history analyses (e.g., senescence, life expectancy, reproductive success) and the effects of age structure on population persistence, while also improving our ability to explore interacting hypotheses in evolutionary, behavioral, and population ecology. Recognizing how demographic rates, abundance, and age structure interact within the JS framework in turn can help improve the explanatory power of JS models and more accurately forecast future population dynamics.

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We thank D. Andriashek, A. McCall, D. McGeachy, L. Sciullo, and the many helicopter pilots and graduate students for help in the field. We thank J.A. Royle and three anonymous reviewers for helpful comments on earlier drafts of this manuscript. Any use of trade, firm or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

^{®}in the Canadian Arctic