Modeling and dynamic analysis of a stochastic mutualism model with distributed delays

Abstract

A stochastic two-species mutualism model that incorporates nonlinear perturbations and distributed delays is proposed and analyzed in this paper, in where saturation effects and distributed delays with weak kernel are contained in the interspecies mutualism term. We first investigate that a nonlinear stochastic mutualism model possesses a unique global positive solution regarding arbitrarily given initial value. Therewith the sufficient conditions for two-species extinction and the existence of a unique stationary distribution (USD) are obtained. Whereafter, it is derived that the stationary solution near the quasi-positive equilibrium adheres a unique probability density function with the aid of tackling Fokker–Planck (FP) equation corresponding to the linearized system and applying the theory of algebraic equations we have developed. At last, several numerical examples are presented to validate the validity of our theoretical findings.