Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay

Dynamics of a stochastic Logistic model with distributed delay are considered. We first transfer a scalar stochastic Logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique. Then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one. Moreover, in the case of persistence, we prove that there exists a unique stationary distribution by the Markov semigroups theory. The results show that, the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution, though for the deterministic model with strong kernel or weak kernel, the average delay may induce the existence of a group of small amplitude periodic solutions.