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The authors consider the problem of the existence, with probability 1, of Green’s function of a stochastic m-point Cauchy problem for a higher-order parabolic equation with white noise perturbations taken with negative values only. Estimates are obtained in spaces of functions whose norm contains the mathematical expectation. [ABSTRACT FROM AUTHOR]

In this paper we analyse a stochastic model driven by fractional Brownian motion representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the Dirichlet boundary value problem for a multi-dimensional nonlocal conservation law with a multiplicative stochastic perturbation in a bounded domain. Using the concept of measure-valued solutions and Kruzhkov's semi-entropy formulations, a result of existence and uniqueness of entropy solution is proved. [ABSTRACT FROM AUTHOR]

This article discusses a stochastic SIQS epidemic model with saturated incidence. We assume that random perturbations always fluctuate at the endemic equilibrium. The existence of a global positive solution is obtained by constructing a suitable Lyapunov function. Under some suitable conditions, we derive the stochastic boundedness and stochastic permanence of the solutions of a stochastic SIQS model. Some numerical simulations are carried out to check our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mutualism ecological model with stochastic perturbations. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded, uniformly continuous and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistent are also established. At last, some figures are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R 0 is obtained. Furthermore, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable and if R 0 > 1 , then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R 0 > 1 , we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R 0 ≤ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R 0 > 1 , then there is a unique stationary distribution to stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic nonautonomous predator-prey model with modified Leslie-Gower and Holling II schemes in the presence of environmental forcing. The deterministic model is the modified Holling-Tanner model which is an extension of the classical Leslie-Gower model. We show that there is a unique positive solution to the stochastic system for any positive initial value. Sufficient conditions for strong persistence in mean and extinction to the stochastic system are established. [ABSTRACT FROM AUTHOR]

The deterministic and stochastic SIQS models with non-linear incidence are introduced. For deterministic model, the basic reproductive rate R0 is derived. Moreover, if R0≥1, the disease-free equilibrium is globally asymptotically stable and if R0>1, there exists a unique endemic equilibrium which is globally asymptotically stable. For stochastic model, sufficient condition for extinction of the disease that is regardless of the value of R0 is presented. In addition, if the intensities of the white noises are sufficiently small and R0>1, then there exists a unique stationary distribution to stochastic model. Numerical simulations are also carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a stochastic multi-group Lotka–Volterra mutualistic system under regime switching. It is well known that the population is forced to expire when the perturbation is sufficiently large. The main aim here is to investigate its ergodic property and positive recurrence by stochastic Lyapunov functions under small perturbation, which can be used to explain some recurring phenomena in practice and thus provide a good description of permanence. The mean of the stationary distribution is estimated. Simulations are also carried out to confirm our analytical results. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain sufficient criteria for the existence of periodic solutions to deterministic SIR and SEIR epidemic models with modified saturation incidence rates by means of using the continuation theorem based on coincidence degree theory, and we show that the solution is unique and globally stable. Second, we discuss their corresponding stochastic epidemic models with random perturbation have a unique global positive solution respectively, and we utilize stochastic Lyapunov functions to investigate the asymptotic behavior of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the problem of suppression explosive solutions by noise for nonlinear deterministic differential system. Given a deterministic differential system with coefficients satisfying a more general one-sided polynomial growth condition, we introduce Brownian noise feedback and therefore stochastically perturb this system into the nonlinear stochastic differential system . We show that appropriate guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic systems may explode in a finite time. Under some weaker conditions, we reveal that the single noise can also make almost every path of the solution of corresponding stochastically perturbed system grow at most polynomially. [Copyright &y& Elsevier]

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: We study a crystalline version of the modified Stefan problem in the plane. The feature of our approach is that, we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon. The existence of solution to our stochastic system is established. Galerkin’s method is one of the main tools used in the proof of our assertion. [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: In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. Numerical simulations conclude the paper. [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]