Abstract:We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.AMS 2000 subject classifications: Primary 60J10, 92B05; secondary 60J80.Received January 2010. MetapopulationsA metapopulation is a population confined to a network of geographically separated habitat patches that may suffer extinction locally and be recolonized through dispersal of individuals from other patches. The term was coined by Levins [41] Levins [40] was the first to provide a succinct mathematical description of a metapopulation, proposing that the number n t of occupied patches at time t in a group of N patches should follow the law of motionwith c being the colonization rate and e being the local extinction rate. This is Verhulst's model [63] for population growth and Levins used Pearl's rationale [51,52,53] to derive it. Furthermore, Levins was able to divine an explicit solution to (1) in the case where both c and e are time dependent, and he derived a diffusion approximation for n t (surprisingly, the time-inhomogeneous * This is an original survey paper. 53

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