Limits of large metapopulations with patch-dependent extinction probabilities

Abstract

We propose a model for the presence/absence of a population in a collection of habitat patches. This model assumes that colonisation and extinction of the patches occur as distinct phases. Importantly, the local extinction probabilities are allowed to vary between patches. This permits an investigation of the effect of habitat degradation on the persistence of the population. The limiting behaviour of the model is examined as the number of habitat patches increases to ∞. This is done in the case where the number of patches and the initial number of occupied patches increase at the same rate, and for the case where the initial number of occupied patches remains fixed.

The limiting behaviour of a mainland-island metapopulation

“…This theorem essentially states that the limiting metapopulation will persist if and only if there are sufficiently many islands with whose population has a high probability of survival. A similar result demonstrating the importance of high quality habitat to the survival of the metapopulation was given in McVinish and Pollett (2010).…”

“…Buckley and Pollett (2010a) examine a class of seasonal metapopulation models and derive difference equation approximations to discrete time Markov chain models. Other work of particular relevance to modelling metapopulations include Arrigoni (2003) and McVinish and Pollett (2010). Although many Markov chains models satisfy the conditions to be well approximated by a deterministic path, it would be unreasonable to expect these conditions to hold for the metapopulation model (4)-(5).…”

“…More recently, researchers have tried to bring the modelling assumptions closer to the ecological reality. In particular, we note the work on stochastic landscapes (Cornell and Ovaskainen 2008;Ovaskainen and Cornell 2006), structured metapopulations which incorporate population dynamics within habitat patches (Arrigoni 2003; Pugliese 2004, 2005) and heterogeneity of habitat quality (McVinish and Pollett 2010).…”

Asymptotic distribution of martingale estimators for a class of epidemic models

“…Secondly, there are no deterministic counterpart based on differential equations as it is the case of an epidemic modeled by means of a counting process. Another type of population modeling, which is applied to metapopulations, has been introduced by some authors such as Pollet (2009, 2010), Mcvinish and Pollett (2009) and other references therein. These researchers derive an approximation that preserves the discrete time structure and reduces the complexity of the models.…”

A Central Limit Theorem for a Discrete-Time SIS Model with Individual Variation

“…This model is a generalisation of the metapopulation model presented in McVinish and Pollett (2010). The main result of the paper is a central limit theorem showing that fluctuations in the proportion of infected individuals around the limiting proportion converges to a Gaussian random variable when appropriately rescaled.…”