In this article,we discuss an SIR model with stochastic perturbation. We show that there is a nonnegative solution that belongs to a positively invariant set. Then,by stochastic Lyapunov functional methods,we deduce the globally asymptotical stability and exponential meansquare stability of the disease-free equilibrium under some conditions,which means the disease will die out. Comparing with the deterministic model,there is no endemic equilibrium. To show when the disease will prevail,we investigate the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model. Last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.