## Introduction

Survival is a key demographic process for the dynamics of populations. Knowledge about the magnitude of survival, how it varies temporally and spatially and by which factors temporal and spatial variation is induced are essential topics in demographic studies. Survival is very often estimated from capture–recapture data: individuals are marked and released into the study population and are then followed through time by re-encountering them. Since detection is not perfect, the resulting capture–recapture data are typically analysed with the Cormack–Jolly–Seber (CJS) model (see Lebreton *et al*. 1992; Williams, Nichols & Conroy 2002), which allows the separate estimation of the probabilities of survival and recapture. However, if individuals emigrate from the study population, these parameters may be biased and the bias depends on the type of emigration. If emigration is temporary and random, only recapture probabilities are biased, while temporary non-random emigration will bias both parameters (Schaub *et al*. 2004). Finally, if emigration is permanent, the estimated survival probability is biased low as mortality and emigration cannot be distinguished with CJS models. The estimated parameter is referred to as the apparent survival probability which is the product of the probabilities of true survival and of study area fidelity (Lebreton *et al*. 1992). Consequently, apparent survival is lower than true survival unless study area fidelity equals one. The underestimation of true survival from capture–recapture data is a major limitation of the method.

The degree of underestimation of true survival by apparent survival can be substantial (Cilimburg *et al*. 2002; Marshall *et al*. 2004) and depends on the size of the study area and on the dispersal behaviour of the species under study (Zimmerman, Gutierrez & LaHaye 2007). If the study area is large relative to dispersal, a majority of dispersing individuals will settle within the study area and apparent survival approximates true survival well. Thus, the degree of underestimation depends on a combination of the study design (specifically, study area) and the dispersal behaviour. This is worrying for several reasons. First, different groups or age classes of individuals in the study population may have different dispersal behaviour resulting in different estimates of apparent survival, even if true survival does not vary by group or age class. Therefore, it is risky to test whether true survival differs among groups or age classes. Secondly, if the temporal variation of survival is of interest, it is unclear how much the observed pattern reflects temporal variation in true survival, how much of it reflects variation in site fidelity or of a combination of both. Thirdly, if spatial variation of survival is studied by comparing apparent survival from different study areas, the estimated variation could stem from variation in true survival or from variation in site fidelity or a combination of both, which might be due to differences in the study area size and shape. For these reasons, it would be desirable to estimate true instead of apparent survival also from capture–recapture data.

Some attempts have been made to reduce the negative bias of survival and all of them need additional data. The most promising ones are those where the capture–recapture data are analysed jointly with data that allow the estimation of true survival, such as band-recovery data (Burnham 1993), observations of marked individuals outside the study area (Barker 1997) or telemetry data (Powell *et al*. 2000). If dispersal takes place among just a few well-defined sites (e.g. colonies in seabirds), sampling at all of these sites and the use of multistate capture–recapture models allow the estimation of true survival (Lebreton *et al*. 2009). Captures/resightings outside the core study area can also help to reduce the bias (Marshall *et al*. 2004). Gilroy *et al*. (2012) developed recently an interesting model that explicitly used information about the locations of encounters within the study area. Using a two-step approach, they generated a spatial projection of dispersal probability around each capture location which is used in a second step to estimate permanent emigration probability and finally survival. Finally, Ergon & Garnder (2014) developed a spatial robust design model that can be applied to a sampling design with fixed trapping locations. The model uses the spatial information about the trapping locations and allows the estimation of dispersal and true survival.

Here, we develop a spatially explicit capture–recapture model for open populations that allow the estimation of true survival, that is, a spatial generalization of the CJS model. Besides the information about *whether* a specific individual was encountered at a given occasion, it is often recorded *where* the encounter occurred. A fraction of dispersal, namely dispersal that occurs within the study area, can be observed. This information can be used to gauge emigration from the study area and thus estimate true survival. Our model is based on a similar idea as is the model by Gilroy *et al*. (2012). However, in contrast to their model, we estimate and model dispersal and survival jointly within a single unified hierarchical model. This has the advantages that estimation errors are fully accounted for and of much increased flexibility in modelling. Our model shows similarities with the model of Ergon & Garnder (2014). A main difference is that they considered a different sampling design with fixed trapping locations while we assumed that recapture is possible anywhere within a defined study area. We introduce our model, present a simulation study to assess its performance and finally apply it to a real data set on a passerine bird.