Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure

TL;DR

This paper analyzes the stochastic dynamics of population colonization under a strong Allee effect and low immigration, deriving exact solutions to predict colonization times and probabilities beyond previous approximation methods.

Contribution

It introduces an exact zero-flux solution to the master equation for low immigration rates, improving predictions of colonization times under strong Allee effects.

Findings

Accurate evaluation of pre-colonization population distribution

Derivation of mean time to colonization across various immigration rates

Validation of results against previous WKB approximations at high immigration

Abstract

We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average pre-colonization population size is small thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate non-zero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasi-stationary probability distribution of population sizes in the pre-colonization state, and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.