Stability of a stochastic logistic model with distributed delay

Abstract

This paper is concerned with the stability of the solutions to the stochastic logistic model with distributed delay, which is represented by the equation dx(t)=x(t)(1-ax(t)-b∫-τ0x(t+θ)dμ(θ))[rdt+σdBt], where Bt is a standard Brownian motion. This study shows that the above stochastic system has a global positive solution with probability 1 and establishes the sufficient conditions for stability of the zero solution and the positive equilibrium. Several numerical examples are introduced to illustrate the results. Some recent results are improved and generalized. © 2012 Elsevier Ltd.