Stability in distribution of a stochastic predator–prey system with S-type distributed time delays

Abstract

This paper concerns the dynamics of a stochastic predator–prey Lotka–Volterra system with S-type distributed time delays. Sufficient conditions for the stability in distribution of the solutions (SDS) to the system are obtained. The results show that the dynamic scenarios of the SDS are completely characterized by two parameters δ1>δ2, among which δ1 is just related to the environmental noise, while δ2 is closely related to both time delays and environmental noises: if δ2>0, then the distributions of prey–predator converge weakly to a unique ergodic invariant distribution (UEID); if δ1>0>δ2, then the predator goes to extinction, while the distributions of prey converge weakly to a UEID; if 0>δ1, then both the predator and prey go to extinction.