Theoretical Population Biology (1995) 48 (3) : 333-360. http:dx.doi.org/10.1006/tpbi.1995.1034 A Stochastic Metapopulation Model with Variability in Patch Size and Position Jemery R. Day and Hugh P. Possingham ABSTRACT Analytically tractable metapopulation models usually assume that every patch is identical, which limits their application to real metapopulations. We describe a new single species model of metapopulation dynamics that allows variation in patch size and position. The state of the metapopulation is defined by the presence or absence of the species in each patch. For a system of n patches, this gives 2 n possible states. We show how to construct and analyse a matrix describing transitions between all possible states by first constructing separate extinction and colonisation matrices. We illustrate the model's application to metapopulations by considering an example of malleefowl, Leipoa ocellata, in southern Australia, and calculate extinction probabilities and quasi-stationary distributions. We investigate the relative importance of modelling the particular arrangement of patches and the variation in patch sizes for this metapopulation and we use the model to examine the effects of further habitat loss on extinction probabilities. 1. INTRODUCTION There has been much recent interest in incorporating spatial structure into models of population dynamics (Pulliam, 1988; Howe et al., 1991; Verboom et al., 1991a; Verboom et al., 1991b; Mangel and Tier, 1993; Adler and Nuernberger, 1994). Traditionally, population models make the restrictive assumption that populations are well mixed and have no spatial structure. These assumptions imply that interaction between any two individuals in the population is equally likely. One method for incorporating spatial structure is to use the concept of a metapopulation (Andrewartha and Birch, 1954; den Boer, 1968), which allows the chance of interaction between individuals to vary according to their relative location. A metapopulation refers to a population inhabiting a collection of discrete patches (Levins, 1969; Hanski, 1991). Each patch is homogeneous and contains a local population in which individuals mix freely. The extinction of local populations and the recolonisation of empty patches are key features of metapopulation dynamics. Many metapopulation models, including both stochastic and deterministic models, rely on the assumption that the system comprises either an infinite or a very large number of identical patches (MacArthur and Wilson, 1967; Levins, 1969; Richter-Dyn and Goel, 1972; Nisbet and Gurney, 1982; Chesson, 1984; Woolhouse, 1988; Hanski, 1991; Hastings, 1991; Gotelli and Kelley, 1993; Hanski and Gyllenberg, 1993). This assumption simplifies mathematical analyses, but such models may miss properties of species restricted to small numbers of patches. It is important to be able to explore the effect of the relative size and spatial position of patches. Variation in the size and quality of patches ( Pulliam, 1988; Howe et al., 1991) and variation due to different spatial arrangements of patches will have different effects on population dynamics (Doak et al., 1992; Holt, 1992; Adler and Nuernberger, 1994). Previous attempts to model these aspects of metapopulation dynamics have relied largely on Monte Carlo simulations (Boyce, 1992; Burgman et al., 1993; Hanski and Thomas, 1994; Possingham et a!., 1994). Other approaches incorporating some form of heterogeneous spatial structure use an array of interlocking cells ( Pulliam eta!., 1992; Perry and Gonzalez-Andujar, 1993). These detailed models also involve simulation, especially if they include stochastic effects. Hanski (1994a; 1994b) looks at presence-absence data for metapopulations and uses this to estimate probabilities of patch extinction and recolonisation. From these probabilities, he uses simulation to predict the future state of a metapopulation. Other authors develop and analyse dynamic metapopulation models which incorporate variation in patch sizes without resorting to simulation. Anderson (1991) considers a stochastic continuous time metapopulation model which explicitly models the size of each local population in a collection of variable sized patches. To ensure analytical tractability, Anderson's model does not include density dependence. Hanski and Gyllenberg (1993) allow continuous variation in