Analytical methods for stochastic discrete-time metapopulation models

The term ‘metapopulation’ is used to describe a population of individuals that live as a group of local populations in geographically separate, but connected, habitat patches. The patches are situated within an otherwise uninhabitable landscape which dispersing individuals traverse in search of suitable habitats. The central concepts associated with metapopulation dynamics are that of local extinction, the extinction of a local population, and recolonisation, where migrants establish new populations in unoccupied (empty) patches. The relationship between the processes of local extinction and colonisation is therefore an important consideration when formulating mathematical metapopulation models. We shall consider a particular type of metapopulation dynamic where extinction events and colonisation events are assumed to occur during separate time periods, or phases, that alternate over time. This seasonal dynamic may be thought of as an annual cycle, say, where local populations are prone to extinction during winter and new populations establish during spring. In particular, we model the number nt of occupied patches at time t as a discrete-time Markov chain (nt : t = 0, 1, 2, . . . ) with transition probabilities that alternate according to the seasonal phases. The models are naturally constructed in a discrete-time setting due to the assumed dynamic, however it will be made clear that whether the Markov chains are timehomogeneous (where the population census is taken after every cycle) or time-inhomogeneous (where the census is taken after each seasonal phase) depends on the monitoring scheme under investigation. We present a number of metapopulation models with the assumed seasonal dynamic where, in particular, the local extinction process is modelled in the same way in each case whilst the colonisation process is modelled according to various means of propagation. We assume that each local population goes extinct with the same, constant, probability, and that all events are independent. Hence, the number of extinction events that occur during the extinction phase is binomial. For metapopulation networks with a finite number N of patches, we also assume that the number of colonisation events that occur during the colonisation phase is v binomial. We investigate both state-independent and state-dependent colonisation probabilities, where the former is defined with a constant probability and the latter depends on the current number of occupied patches. Metapopulation models defined with a state-independent colonisation process are referred to as mainland models because empty patches are thought to be colonised by migrants from an outside source population (the ‘mainland’) in this case. For models defined with state-dependent colonisation processes, we refer to these as island models when colonists originate from occupied patches (islands) or mainland-island models when both types of colonising behaviour are assumed. The overall two-phase model is called a chain binomial metapopulation model since the extinction and colonisation processes together define a sequence of binomial random variables. We also investigate similar models but for networks with infinitely-many patches (N = ∞); the number of colonisation events is modelled as a Poisson random variable in such cases. The discrete-time Markov chain approach is well established in the applied metapopulation literature, however models of this type are usually examined via numerical methods and simulation. The models presented here are accompanied with extensive analytical treatments. For most of our finite-patch models, we evaluate conditional state distributions explicitly and use these distributions to establish convergence results (in the sense of convergence in distribution). These results include a law of large numbers, which identifies an approximating deterministic trajectory, and a central limit law, which establishes that the scaled fluctuations about the deterministic trajectory have an approximate normal (Gaussian) distribution. We show that the infinite-patch models are equivalent to branching processes. This body of work culminates by presenting limit theorems for discrete-time metapopulation models. First, we prove limit theorems for a general class of inhomogeneous Markov chains that exhibit the particular property of density dependence. These theorems include a law of large numbers and a central limit law (in the sense of convergence in finite-dimensional distribution), which establishes that the scaled fluctuations about this deterministic trajectory have an approximating autoregressive structure. Second, we apply these results to our Markov chain metapopulation models (both finite-patch and infinite-patch) where we demonstrate that the limiting behaviour of any of our metapopulation models with density-dependent phases can be evaluated explicitly, even in situations where a conditional state distribution could not be determined. vi