Abstract: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 cl… Show more
“…In this model, these events are separated into distinct phases as in [12] and [3]. First, a colonisation phase occurs during which unoccupied patches are colonised by individuals from the occupied patches.…”
“…In this model, these events are separated into distinct phases as in [12] and [3]. First, a colonisation phase occurs during which unoccupied patches are colonised by individuals from the occupied patches.…”
“…There has been a high level of interest in the field since the late 60s [1,2], and this has continued to the present (see [3,4,5,6] and references therein). Of significant concern to ecologists is the survival of the population and under what conditions the population might become extinct.…”
We develop a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics. Using a deterministic approximation derived from a functional law of large numbers, we develop conditions for extinction and persistence of the metapopulation in terms of the birth, death and migration parameters. Interestingly, we observe the Allee effect in a metapopulation comprising two patches of greatly different sizes, despite there being decreasing patch specific per-capita birth rates. We show that the Allee effect is due to way the migration rates depend on the population density of the patches.
“…A number of researchers ( [1,2,3,4] among many others) have used branching processes to approximate the initial spread of an epidemic in a large population. The branching process approximation is less frequently applied in the context of metapopulation modelling, but [5] provides one example.…”
Hanski's incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.
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