Survival analysis of an impulsive stochastic facultative mutualism system with saturation effect

A system of impulsive stochastic differential equations is proposed as a two-species facultative mutualism model subject to impulsive and two coupling noise source perturbations, in which the saturation effect is taken into account. A set of sufficient criteria for extinction (exponential extinction and extinction) and permanence (permanence in time average and stochastic permanence) of the system are established. Extensive simulation figures are demonstrated to support the theoretical findings. Meanwhile, we look at the effects of coupling white noises, impulses, intrinsic growth rates, intra-specific competition rates and inter-specific mutualism rates on the survival of populations.

The stationary distribution and ergodicity of a stochastic mutualism model

This paper is concerned with a stochastic mutualism system with toxicant substances and saturation terms. We obtain the sufficient conditions for the existence of a unique stationary distribution to the equation and it has an ergodic property. It is interesting and surprising that toxicant substances have no effect on the stationary distribution of the stochastic model. Simulations are also carried out to confirm our analytical results.