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In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this letter, a stochastic mutualism model with Lévy jumps and harvesting is considered. Under some simple assumptions, sufficient and necessary criteria for the existence of optimal harvesting policy are established. The optimal harvesting effort and the maximum of sustainable yield are also obtained. The effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we focus on the stability problem for a class of stochastic delayed recurrent neural networks. Different from the traditional stability criteria, we introduce and study a new stability criterion: the mean-square exponential input-to-state stability. To the best of our knowledge, this new stability criterion has never been discussed in the field of stochastic recurrent neural networks. The main objective of the paper is to fill the gap. With the help of the Lyapunov–Krasovskii functional, stochastic analysis theory and Itô's formula, we prove that the addressed system is mean-square exponentially input-to-state stable. Moreover, two numerical examples and their simulations are presented to verify the theoretical results well. [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]