## Introduction

A great limitation when it comes to studying senescence in wildlife populations results from the unavailability of information on the ages of the individuals under study. With many species of mammals and birds, it is not possible to identify the ages of the individuals unless these are caught as juveniles. As a result, one can assume a constant survival probability for all ages of adults and therefore only distinguish two age classes, namely juveniles and adults (Brownie *et al*. 1978). However, this does not enable the study of senescence and is, in many cases, unrealistic. Another approach would be to include only the individuals for which the ages are known with the disadvantage of discarding large parts of the data set or to use time since first capture as a proxy for age (Reed *et al*. 2008) which relies on the assumption that all individuals are of the same (unknown) age when first caught.

Two models have been proposed recently that aim to tackle the issue of estimating age-dependent survival probabilities for individuals of unknown age: Colchero & Clark (2012) developed a Bayesian hierarchical model, available in the R package BaSTA (Colchero *et al*. 2012), while McCrea *et al*. (2013) developed a mixture model for recovery data which requires that the ages of some of the individuals in the sample are known.

The same obstacles are presentedõ when modelling the transient presence of migratory birds at stop-over sites. In the stop-over setting, the probability of leaving the site is thought to depend, amongst other factors, on the time the individuals have spent at the site. However, this information is generally unavailable. Pledger *et al*. (2009) developed a model that builds on the unknown time of arrival of the individuals and allows the estimation of the probability of remaining until the next sampling occasion to be linked to the unknown time already spent at the site (see also Pradel (2009) for a different approach). They established that the age-dependent model, where age is the time since arrival in the stop-over setting, is identifiable even when all individuals are of unknown age (see Data S1 in the Supporting Information for the verification of this result using symbolic algebra methods in Maple) and applied their model to a stop-over data set of semipalmated sandpipers (*Calidris pusilla*).

The Pledger *et al*. (2009) model assumes that the time the individuals have spent at the site before the start of the study is negligible. This assumption is realistic for data sets collected at stop-over sites where it is common for the duration of the study to exceed the stop-over duration of the individuals. However, this is not the case when studying survival by sampling resident populations at permanent sites. Additionally, the model assumes homogeneous individuals that share common population parameters. However, it has been repeatedly shown that wildlife populations can be substantially heterogeneous, both in terms of survival (Cam *et al*. 2002) and capture probabilities (Gimenez & Choquet 2010; Pledger *et al*. 2010). In the first case, individuals with an overall higher probability of survival will prevail at older ages, which can result in the average probability of survival to increase for older ages, potentially masking the effect of senescence (Vaupel & Yashin 1985; Peron *et al*. 2010). In the latter case, ignoring any differences in capture probabilities between individuals can lead to ‘leakage’ between parameters as an over-simplistic model for capture probabilities can lead to the selection of a more complicated model for survival probabilities than necessary (Catchpole *et al*. 2004; Pledger *et al*. 2009) and/or to biased estimates of survival probabilities (Fletcher *et al*. 2012).

In this paper, we demonstrate how the Pledger *et al*. (2009) model can be applied to data sets collected at permanent sites where individuals are born and die, and we show a possible way to model the birth times of the individuals that may have been born prior to the start of the study. Using this model allows the estimation of age-dependent survival probabilities, enabling the study of senescence, without any information on the ages of the individuals being required. Furthermore, we extend the model to allow for heterogeneous probabilities of survival and capture using finite mixtures, following the work of Pledger *et al*. (2010). Therefore, the two major innovations in the present paper are as follows: (i) combining heterogeneity with age structure when age is unknown in the same model and (ii) modelling entry terms before the first sample that is required for populations already well established before the start of the study.

The models and methods are presented in 'Materials and methods' section. The 'Simulations' section demonstrates the importance of extending the possible birth times of the individuals beyond the study time limits, especially when the study length is not considerably greater than the lifespan of the species, as well as the importance of accounting for heterogeneous capture probabilities when the focus is on obtaining unbiased estimates of age-dependent survival probabilities. The application of the methods to a long-term data set of brushtail possums is given in the 'Application to real data: possums' section, and a discussion of the results is found in 'Discussion' section.