## Introduction

Abundance is a parameter of central interest to animal ecologists because it affects virtually all other aspects of biology or conservation of the species of interest. Consequently, estimating abundance is a fundamental goal of many animal sampling problems, and it forms the basis of a vast body of literature on statistical methods in animal ecology as exemplified in the synthetic treatments by Seber (1982), Borchers, Buckland & Zucchini (2002), and Williams, Nichols & Conroy (2002). The main consideration in estimating abundance of most animal populations is that individuals cannot be observed or detected perfectly. That is, the probability of encountering or detecting an animal is less than 1. A myriad of techniques have been devised for dealing with imperfect detection so that estimates of *N* may be obtained from observations of individuals (Williams *et al*. 2002).

Capture–recapture methods in particular are frequently applied to estimate abundance using encounter history data from arrays of traps or passive detection devices. These include classical animal trapping grids, general photography (Goswami, Madhusudan & Karanth 2007), camera traps and DNA collected from dung or ‘hair snares’ (Waits 2004). It is natural to view the resulting capture–recapture data as being generated by a closed population provided the sample occasions are close enough together in time to preclude mortality, recruitment and movement. However, lacking a physical barrier around the sampled area, there is almost always some movement of individuals onto and off the area during the survey – referred to as an absence of geographic closure (e.g. White *et al*. 1982). This phenomenon in the simplest case (‘random temporary emigration’, Kendall, Nichols & Hines 1997) biases *p̂* low, hence *N̂* high, with respect to the nominal area over which traps are physically situated. While we may be able to obtain an estimate of *N* that is relevant to a real population of individuals, we do not know the area from which animals were sampled. In addition to movement of individuals complicating the interpretation of abundance estimates, another problem that arises in capture–recapture studies based on spatially organized trap arrays is that the probabilistic models of the capture process (e.g. Otis *et al*. 1978) used to estimate *N* are no longer strictly appropriate. Specifically, heterogeneity in capture probability can arise as a result of individual locations relative to the array location. That is, some individuals experience more exposure to trapping than others. These problems have received considerable attention in the literature (Dice 1938, 1941; Stickel 1954; Smith *et al*. 1971; Otis *et al*. 1978; Wilson & Anderson 1985) and a large number of *ad hoc* inference procedures for spatial capture–recapture data have been proposed. Conversely, formal model-based inference procedures have only recently been developed (Efford 2004; Borchers & Efford 2008; Royle & Young 2008).

In this study, we consider the problem of estimating tiger density from camera-trap arrays in India. The use of camera traps to estimate tiger abundance was first proposed by Karanth (1995), and received further development by Karanth & Nichols (1998, 2000, 2002). It has since been widely used on several individually identifiable carnivore species (O’Brien, Kinnaird & Wibisono 2003; Trolle & Kéry 2003; Wallace *et al*. 2003; Karanth *et al*. 2004a,b, 2006 Silver *et al*. 2004; Wegge *et al*. 2004; Soisalo & Cavalcanti 2006). Density estimation using photographic capture–recapture methods has so far emphasized *ad hoc* approaches based on estimates of boundary strip width. This is viewed as the weak link in this widely used methodology (Karanth *et al*. 2006; Soisalo & Cavalcanti 2006). Soisalo & Cavalcanti (2006) noted that different estimation methods yielded very different density estimates for jaguars *Panthera onca*. Such differences have important conservation consequences. Because of the relationship between attainable tiger densities and prey densities (e.g. Karanth *et al*. 2004b), tiger management in India is largely a matter of managing prey species. Expected tiger densities can be predicted from estimates of available prey. Both the analysis of these relationships and the subsequent comparisons of predicted and realized tiger numbers depend on reasonable estimates of tiger densities.

We propose a spatial-explicit capture–recapture model for estimating density from trapping arrays that yield encounter histories of individuals in addition to auxiliary spatial information – the traps in which each individual is captured. Motivated by a similar problem (Royle & Young 2008), we formulate a hierarchical model for the observations (the locations of trapping) that is specified conditional on a set of latent variables (random effects). These latent variables are the hypothetical home range or activity centres of every individual in the population. The objective is to estimate the absolute density of home range centres in the region containing the trap array. Classical inference under such models is achieved by removing the latent variables from the conditional likelihood by integration (see Borchers & Efford 2008). Here, we adopt an approach based on Bayesian analysis of the hierarchical model using Markov chain Monte Carlo (MCMC). In particular, we make use of the method of data augmentation (Royle, Dorazio & Link 2007), which yields a simple treatment of this problem as a ‘missing data’ problem, where the home range centres are the missing data (Royle & Young 2008). Formally, data augmentation yields a parameterization of the model that can be implemented in the freely available software WinBUGS (Gilks, Thomas & Spiegelhalter 1994) with little more than a few lines of ‘pseudo-code’ to describe the model.