Extinction in a self-regulating population with demographic and environmental noise

TL;DR

This paper develops a comprehensive stochastic model for population fluctuations, capturing extinction dynamics under demographic and environmental noise, and provides analytical and simulation results for extinction times.

Contribution

It introduces an exact unified stochastic framework combining demographic and environmental stochasticity, with analytical solutions for extinction times and equivalence of different modeling approaches.

Findings

Exact analytical expressions for extinction times.

Demonstration of equivalence between master equation, Fokker-Planck, and stochastic equations.

Approximate extinction times using steepest descent method.

Abstract

We present an explicit unified stochastic model of fluctuations in population size due to random birth, death, density-dependent competition and environmental fluctuations. Stochastic dynamics provide insight into small populations, including processes such as extinction, that cannot be correctly treated by deterministic methods. We present exact analytical and simulation-based results for extinction times of our stochastic model and compare the different effects of environmental stochasticity and intrinsic demographic stochasticity. We use both the discrete master equation approach and an exact mapping to a Fokker-Planck equation (the Poisson method) and stochastic equation, showing they are precisely equivalent. We also calculate approximate extinction times using a steepest descent method. This model can readily be extended to accommodate metapopulation structure and genetic variation in the population and thus represents a step towards a microscopically explicit synthesis of population dynamics and population genetics.