Analytical methods for a stochastic mainland-island metapopulation model

The term ‘metapopulation’ is used to describe individuals of a species living as a group of local populations in geographically separate, but connected, habitat patches (Levins 1970, Hanski 1999). Patches may become empty through local extinction and empty patches may be recolonised by migrants from other local populations. A balance between local extinction and colonisation may be reached which allows the metapopulation to persist (Hanski 1999). The relationship between these two processes is therefore an important consideration when formulating mathematical metapopulation models. We suppose that events of the same type occur in seasonal phases, so that extinction events only occur during the extinction phase and colonisation events only occur during the colonisation phase, and that these phases alternate over time. They may correspond to two parts of an annual cycle, for example, where local populations are prone to extinction during winter while new populations establish during spring. We assume that a census takes place either at the end of the colonisation phase (. . . -extinction-colonisationcensus. . . ) or at the end of the extinction phase (. . . -colonisation-extinction-census. . . ), and thus fits naturally within a discrete-time modelling framework. If extinction and colonisation events were to occur in random order, then a continuous-time model would of course be preferred. Here we use a discrete-time Markov chain whose state nt is the observed number of occupied patches at the t-th census. Its transition matrix is the product of two transition matrices that govern the individual extinction and colonisation processes. This approach has been used previously and several models have been proposed (Akçakaya and Ginzburg 1991, Day and Possingham 1995, Hill and Caswell 2001, Klok and De Roos 1998, Tenhumberg et al. 2004, Rout et al. 2007). Each model accounts for local extinction in the same way, but different methods are used to model the colonisation process, reflecting the differing breeding habits and means of propagation of the particular species under investigation. Whilst they account for a range of colonisation behaviour, the models were examined using numerical methods and simulation, and few explicit analytical results were obtained. Furthermore, only the extinction-colonisation-census scenario was considered. Whilst it is certainly true that timing of the census is arbitrary in that it does not affect the dynamics of the metapopulation (Day and Possingham 1995), its timing may affect the efficiency of any statistical procedures used to calibrate the models and successful implementation of management actions. We present a new and quite general approach to modelling the colonisation process, one that permits explicit expressions for a variety of quantities of interest. We concentrate here on a mainland-island configuration: the patches (islands) receive migrants from an external source (the mainland), assumed to be immune from extinction. We evaluate the distribution of nt at any census time t. We then establish a law of large numbers, which identifies a deterministic trajectory that can be used to approximate (nt, t ≥ 0) at any time t when the number of patches is large. We also establish a central limit theorem, which shows that the fluctuations about this trajectory are approximately normally distributed. These results are useful in understanding the patch occupancy process when the parameters of the model are known. For example, the mean and variance of nt, and the expected time to first total extinction, can be exhibited explicitly. We describe briefly much finer results that can be used for model calibration.