Complex dynamics of a stochastic uni-directional consumer-resource mutualism system

Like predation and competition, mutualism is recognized as a consumer-resource interaction, which includes bi-directional and uni-directional mutualisms. In this paper, we firstly propose a stochastic uni-directional consumer-resource system of two species in which the consumer has both positive and negative effects on the resource, while the resource has only a positive effect on the consumer. We then mathematically analyze the system, to demonstrate the existence, uniqueness, asymptotic pathwise behavior and stochastically ultimately boundedness of the global positive solution, and to establish sufficient conditions for the global attractivity and the existence of ergodic stationary distribution of the system. We also establish sufficient conditions for the extinction and persistence in mean of the resource, the consumer or the entire system. Numerical simulations are carried out to demonstrate the analytical results.