Abstract
In this paper, we analyze two stochastic predator-prey models with distributed delay and stage structure for prey. For the nonautonomous periodic case of the model, by using Khasminskii's theory of periodic solution, we show that the system has at least one positive T-periodic solution. For the model which is disturbed by both white and telegraph noises, we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. The positive recurrence implies that both prey and predator populations will be persistent in the long term.