Using capture-recapture data and hybrid Monte Carlo sampling to estimate an animal population affected by an environmental catastrophe

Abstract

We propose a dynamic model for the evolution of an open animal population that is subject to an environmental catastrophe. The model incorporates a capture-recapture experiment often conducted for studying wildlife population, and enables inferences on the population size and possible effect of the catastrophe. A Bayesian approach is used to model unobserved quantities in the problem as latent variables and Markov chain Monte Carlo (MCMC) is used for posterior computation. Because the particular interrelationship between observed and latent variables negates the feasibility of standard MCMC methods, we propose a hybrid Monte Carlo approach that integrates a Gibbs sampler with the strategies of sequential importance sampling (SIS) and acceptance-rejection (AR) sampling for model estimation. We develop results on how to construct effective proposal densities for the SIS scheme. The approach is illustrated through a simulation study, and is applied to data from a mountain pygmy possum (Burramys Parvus) population that was affected by a bushfire.

Introduction

Open population models are important in ecological and environmental research. Early work on open population inference was largely conducted within the maximum likelihood framework. With advances in simulation techniques, implementation of the integrals required for calculating posterior distribution has become increasingly available. This leads to more studies being conducted via a Bayesian approach. For example, Crome et al. (1996) presented a Bayesian analysis to assess the impacts of logging rain forest on fauna using Before-After-Control-Impact-Pairs experimental data. Brooks et al. (2000) applied Bayesian methods to the Cormack–Jolly–Seber (CJS) alike models to estimate survival and recovery/capture probabilities from band-return and capture-recapture data. Link and Barker (2005) presented a hierarchical extension of the (CJS) models for estimating the correlation between biological birth rate and survival rate. Lee et al. (2006) analyzed data collected from a two-release experiment. A comprehensive review on open population models via a Bayesian approach can be found in Buckland et al. (2007).

Recently Huggins (2007) proposed a martingale estimation equation approach to obtain the size of an open population, where the population size is assumed to be a deterministic function of certain covariates and parameters, and linear models are used to estimate the effects of the covariates on the population size. This approach uses full individual capture histories in its bootstrap estimation of the standard deviations of parameter estimators. Sometimes individuals may be difficult to distinguish from each other in a capture-recapture experiment. For example, in the study of goose population (Macinnes, 1966, Malecki et al., 1981) the data are usually collected by sighting the neck or foot bands of the geese. An individual’s capture history is hence not available in this case. Instead, summary data, i.e., the numbers of marked and unmarked animals captured on each occasion, are often obtainable. In this paper we aim to develop a population model and an estimation approach for this type of data. We assume the population under study is open, and has suffered from an environmental catastrophe during the data collection period. Further, suppose that the population size before the catastrophe has a stationary distribution with mean $N$, the catastrophe has an effect to reduce the mean population size by $N\delta$ for some $0\le \delta \le 1$, and the population size eventually returns to the stationary distribution after the catastrophe. To model a stationary population it is reasonable to assume that the mean number of births and immigrants is the same as the mean number of deaths and emigrants.

In Section 2 we develop a dynamic population model to delineate the evolutions of population sizes and capture-recapture data. The model consists of latent variables, observed capture-recapture variables and the parameters of interest. In Section 3 we develop MCMC methods for computing the posterior distributions. In computing the conditional distribution of the latent variables, because the involved distributions are so convoluted that feasible proposal densities with sizeable acceptance probabilities cannot be constructed, we develop a sequential importance sampling (SIS) scheme (cf. Doucet et al. (2001), Liu (2001) and Chapter 14 of Robert and Casella (2004)). Simulation studies are conducted and reported in Section 4. In Section 5 the model and algorithm are applied to capture-recapture data on mountain pygmy possums (Burramys Parvus) whose habitat was affected by a bushfire during the data collection period. Further discussions are given in Section 6.