Impact of Colored Environmental Noise on the Extinction of a Long-Lived Stochastic Population: Role of the Allee Effect

TL;DR

This paper analyzes how colored environmental noise combined with demographic noise influences the extinction risk of populations with the Allee effect, providing new analytical insights into mean time to extinction under various noise conditions.

Contribution

It introduces a novel analytical framework using WKB approximation and Hamiltonian mechanics to study extinction dynamics under correlated environmental noise and the Allee effect.

Findings

Extinction risk increases significantly with stronger environmental noise.

The mean time to extinction shifts from exponential to negligible dependence as noise strength grows.

New analytic results for different noise correlation times and strengths are derived.

Abstract

We study the combined impact of a colored environmental noise and demographic noise on the extinction risk of a long-lived and well-mixed isolated stochastic population which exhibits the Allee effect. The environmental noise modulates the population birth and death rates. Assuming that the Allee effect is strong, and the environmental noise is positively correlated and Gaussian, we derive a Fokker-Planck equation for the joint probability distribution of the population sizes and environmental fluctuations. In WKB approximation this equation reduces to an effective two-dimensional Hamiltonian mechanics, where the most likely path to extinction and the most likely environmental fluctuation are encoded in an instanton-like trajectory in the phase space. The mean time to extinction (MTE) is related to the mechanical action along this trajectory. We obtain new analytic results for short-correlated, long-correlated and relatively weak environmental noise. The population-size dependence of the MTE changes from exponential for weak environmental noise to no dependence for strong noise, implying a greatly increased extinction risk. The theory is readily extendable to population switches between different metastable states, and to stochastic population explosion, due to a combined action of demographic and environmental noise.