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In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

This paper investigates a deterministic and stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence. Firstly, for the corresponding deterministic system, the global asymptotic stability of disease-free equilibrium and the endemic equilibrium is proved through the stability theory. Secondly, for the stochastic system, the threshold conditions which decide the extinction or permanence of the disease are derived. By constructing suitable Lyapunov functions, we investigate the oscillation behavior of the stochastic system solution near the endemic equilibrium. The results of this paper show that there exists a great difference between the deterministic and stochastic systems, which implies that the large stochastic noise contributes to inhibiting the spread of disease. Finally, in order to validate the theoretical results, a series of numerical simulations are presented. [ABSTRACT FROM AUTHOR]

In this paper, the stochastic SIR epidemic model with Beddington-DeAngelis incidence rate is investigated. We classify the model by introducing a threshold value λ. To be more specific, we show that if λ<0 then the disease-free is globally asymptotic stable i.e., the disease will eventually disappear while the epidemic is persistence provided that λ>0. In this case, we derive that the model under consideration has a unique invariant probability measure. We also depict the support of invariant probability measure and prove the convergence in total variation norm of transition probabilities to the invariant measure. Some of numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions for global asymptotic stability of a stochastic non-autonomous Lotka-Volterra competitive system with infinite delays are established. Some recent results are improved and generalized. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate a stochastic two-prey one-predator model. Firstly, under some simple assumptions, we show that for each species xi, i=1,2,3, there is a πi which is represented by the coefficients of the model. If πi<1, then xi goes to extinction (i.e., lim t→+∞xi( t)=0); if πi>1, then xi is stable in the mean (i.e., $\lim_{t\rightarrow+\infty}t^{-1} \int_{0}^{t}x_{i}(s)\,\mathrm {d}s=\mbox{a positive constant}$ ). Secondly, we prove that there is a stationary distribution to this model and it has the ergodic property. Thirdly, we establish the sufficient conditions for global asymptotic stability of the positive solution. Finally, we introduce some numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]