Global asymptotic properties of a stochastic model of population growth

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 with positive initial conditions converge to a certain bounded region in the model phase space and oscillate around this region thereafter. Moreover, we found that, if the magnitude of the noise exceeds a certain critical level (which is also explicitly found), then the stochastic stabilization (“stabilization 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.