Extinction rates of established spatial populations

TL;DR

This paper develops a mathematical framework using WKB approximation to analyze the extinction dynamics of spatial populations under intrinsic noise, classifying regimes and predicting optimal refuge sizes for population persistence.

Contribution

It introduces a novel application of WKB approximation to spatial population extinction, including regimes with Allee effects and optimal refuge size prediction.

Findings

Classified extinction regimes with and without Allee effects.

Derived Hamilton's equations for most probable extinction paths.

Predicted an optimal refuge size maximizing population persistence.

Abstract

This paper deals with extinction of an isolated population caused by intrinsic noise. We model the population dynamics in a "refuge" as a Markov process which involves births and deaths on discrete lattice sites and random migrations between neighboring sites. In extinction scenario I the zero population size is a repelling fixed point of the on-site deterministic dynamics. In extinction scenario II the zero population size is an attracting fixed point, corresponding to what is known in ecology as Allee effect. Assuming a large population size, we develop WKB (Wentzel-Kramers-Brillouin) approximation to the master equation. The resulting Hamilton's equations encode the most probable path of the population toward extinction and the mean time to extinction. In the fast-migration limit these equations coincide, up to a canonical transformation, with those obtained, in a different way, by Elgart and Kamenev (2004). We classify possible regimes of population extinction with and without an Allee effect and for different types of refuge and solve several examples analytically and numerically. For a very strong Allee effect the extinction problem can be mapped into the over-damped limit of theory of homogeneous nucleation due to Langer (1969). In this regime, and for very long systems, we predict an optimal refuge size that maximizes the mean time to extinction.