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Abstract: In this paper, we include stochastic perturbations into SIR and SEIR epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number . The long time behavior of the two stochastic systems is studied. Mainly, we utilize stochastic Lyapunov functions to show under some conditions, the solution has the ergodic property as , while exponential stability as . At last, we make simulations to conform our analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we show there is a stationary distribution of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that a stochastic logistic population under regime switching controlled by a Markov chain is either stochastically permanent or extinctive, and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions. In the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the Markov chain and the parameters of the subsystems of the population model. Finally, we illustrate our conclusions through two examples. [Copyright &y& Elsevier]