Survival and stationary distribution of a stochastic facultative mutualism model with distributed delays and strong...

To investigate the roles of both coupling noises and distributed delays with strong kernels, a novel delayed stochastic two-species facultative mutualism model is established, in where the strong kernels indicate that the maximum influence on the growth rate response at some time is due to population densities at the previous time, and the saturation effect is also incorporated because the facultative capacity of each species is finite and their interspecific mutualism should be upper bounded in real life. We first transfer the two-species stochastic model with strong kernels into an equivalent six-dimensional model through a linear chain technique. Later, sufficient conditions for the extinction exponentially, persistence in the mean, permanent in time average and stationary distribution are respectively obtained. Finally, numerical simulations are supplied to support our theoretical results. Our analytical results show that the coupling noise intensities play an important role in the long-time behaviors while the strong kernels are independent of the above asymptotic properties.