Local approximation of a metapopulation's equilibrium

TL;DR

This paper develops a method to approximate the equilibrium state of a metapopulation model with randomly distributed patches, showing the approximation's accuracy and providing bounds on occupation probabilities.

Contribution

It introduces explicit bounds and conditions under which the equilibrium of a complex metapopulation model can be approximated by a simpler Levins's model.

Findings

Approximation is accurate with many contributing patches.

Explicit bounds are provided for occupation probabilities.

The model accounts for randomness in patch distribution.

Abstract

We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset $\Omega$ of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at $z$ being occupied is shown to be close to $q_1(z)$, the equilibrium occupation probability in Levins's model, at any point $z \in \Omega$ not too close to the boundary, if the local colonization pressure and extinction rates appropriate to $z$ are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.