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Abstract: In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings. [Copyright &y& Elsevier]

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: This paper discusses a randomized non-autonomous logistic equation , where is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172], the authors show that has a unique positive T-periodic solution provided , and are continuous T-periodic functions, , and . We show that this equation is stochastically permanent and the solution is globally attractive provided , and are continuous T-periodic functions, , and . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized nonautonomous logistic equation where is 1-dimensional standard Brownian motion. We show that has a unique positive T-periodic solution provided , , and are continuous T-periodic functions, , and . [Copyright &y& Elsevier]