Extinction in population dynamics

TL;DR

This paper investigates how stochastic effects influence extinction in a population within a refuge, showing that fluctuations can cause extinction even when deterministic models predict survival, especially under strong competition.

Contribution

It extends traditional reaction-diffusion models by incorporating stochastic population fluctuations, revealing critical conditions for extinction likelihood beyond mean field predictions.

Findings

Stochastic fluctuations can lead to extinction even in large refuges.

Weak competition allows refuge size to reduce extinction risk.

High competition rates make extinction likely regardless of refuge size.

Abstract

We study a generic reaction-diffusion model for single-species population dynamics that includes reproduction, death, and competition. The population is assumed to be confined in a refuge beyond which conditions are so harsh that they lead to certain extinction. Standard continuum mean field models in one dimension yield a critical refuge length L_c such that a population in a refuge larger than this is assured survival. Herein we extend the model to take into account the discreteness and finiteness of the population, which leads us to a stochastic description. We present a particular critical criterion for likely extinction, namely, that the standard deviation of the population be equal to the mean. According to this criterion, we find that while survival can no longer be guaranteed for any refuge size, for sufficiently weak competition one can make the refuge large enough (certainly larger than L_c) to cause extinction to be unlikely. However, beyond a certain value of the competition rate parameter it is no longer possible to escape a likelihood of extinction even in an infinite refuge. These unavoidable fluctuations therefore have a severe impact on refuge design issues.