Abstract
In this article, we study two stochastic predator-prey models with distributed delay and general functional response. For the nonautonomous periodic case of the system, by utilizing Khasminskii's theory of periodic solution, we prove that the system admits a positive T-periodic solution. For the system which is perturbed by both white and colored noises, we establish sufficient conditions for the existence of positive recurrence of the solutions to the system. The existence of positive recurrence implies stochastic weak stability.