The stability, persistence and extinction in a stochastic model of the population growth

TL;DR

This paper analyzes a stochastic logistic population model with white noise perturbations, revealing conditions for stability, oscillation, and extinction, including explicit thresholds for noise-induced stabilization and population extinction.

Contribution

It provides a rigorous analysis of the global qualitative behavior of a stochastic population model, including explicit criteria for stabilization and extinction due to noise.

Findings

Solutions oscillate around a specific interval

Critical noise level induces stabilization at zero

Population extinction occurs almost surely in finite time

Abstract

In this paper we consider the global qualitative properties of a stochastically perturbed logistic model of population growth. In this model, the stochastic perturbations are assumed to be of the white noise type and are proportional to the current population size. Using the direct Lyapunov method, we established the global properties of this stochastic differential equation. In particular, we found that solutions of the equation oscillate around an interval, and explicitly found the end points of this interval. Moreover, we found that, if the magnitude of the noise exceeds a certain critical level (which is also explicitly found), then the stochastic stabilisation ("stabilisation by noise") of the zero solution occurs. In this case, (i) the origin is the lower boundary of the interval, and (ii) the extinction of the population due to stochasticity occurs almost sure (a.s.) for a finite time.