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In this paper, we consider a stochastic food chain chemostat model with variable yields. First, we prove the stochastic model has a unique global positive solution. Second, by employing suitable Lyapunov functions, Itô ′ s formula and some other important inequalities, the existence of a unique ergodic stationary distribution of a stochastic food chain chemostat model is researched, which can help us better understand the statistical characteristics of stochastic food chain chemostat models. Second, we investigate the extinction of the microorganism and the bacteria. Moreover, the case of extinction for bacteria but persistence for microbial species is considered. Finally, some numerical simulations are carried out to illustrate our theoretical results and the influence of the variable yields on the microorganism and the bacteria. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear stochastic eco-epidemic model is presented to study the dynamics of harvested prey–predator model with group defence. One of the key roles of forming a group is to defend themselves from the attack of the predators. Since prey population spreads infection through direct contact; we consider two groups of prey population. Using some certain assumptions, a stochastic model is formulated by considering noisy intrinsic growth rate of susceptible pest population, death rate of infected pest and predator populations, respectively. Next, the existence and uniqueness of positive solution are performed using comparison theorem of stochastic differential equations (SDEs). The boundedness and uniform continuity of the system are analyzed. Then, the bionomic equilibrium and optimal harvesting policy are obtained by using stochastic maximum principle. We verify the conditions of extinction and persistence in mean with the numerical results. To verify our consideration with the reality, the effects of harvesting effort and catchability coefficient are observed. In addition, infected and uninfected regions are reported to specify the infection free system. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of solution to obstacle problem for quasilinear stochastic partial differential equations with Neumann boundary condition. Our method is based on the analytical techniques coming from parabolic potential theory. The solution is expressed as a pair (u , ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random regular measure satisfying minimal Skohorod condition. [ABSTRACT FROM AUTHOR]

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function. [ABSTRACT FROM AUTHOR]

By means of Itô's formula and a comparison theorem for integral equations, we study the blow up in finite time of semilinear stochastic differential equations of the form Here, is non-negative and non-decreasing by components, is a predictable and continuous process, is an -Brownian motion and is an -measurable random variable. The results of this paper can be seen as extensions of the Feller and Osgood criteria. [ABSTRACT FROM AUTHOR]

The dynamical properties of a stochastic non-autonomous ratio-dependent predator-prey system are studied by applying the theory of stochastic differential equations, Itô's formula and the method of Lyapunov functions. First, the existence, the uniqueness and the positivity of the solution are discussed. Second the boundedness of the moments and the upper bounds for growth rates of prey and predator are studied. Moreover, the global attractivity of the system under some a weaker sufficient conditions are investigated. Finally, the theoretical results are confirmed by the special examples and the numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic model for algal bloom with nutrient recycling is investigated. The model incorporates a white noise term in the growth equation of algae population to describe the effects of random fluctuations in the environment, and a nutrient recycling term in the nutrient equation to account for the conversion of detritus into nutrient. The existence and uniqueness of the global positive solution of the model is first proved. Then we study the long-time behavior of the model. Conditions for the extinction and persistence in the mean of the algae population are established. By using the theory of integral Markov semigroups, we show that the model has an invariant and asymptotically stable density. Numerical simulations illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic single-species system with Lévy jumps in a polluted environment. Some sufficient conditions on extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean, stability in the mean and stochastic permanence are obtained. The threshold between extinction and weak persistence in the mean is established. At the same time, under a simple condition, it is proved that this threshold also is the threshold between extinction and stability in the mean. The results reveal that Lévy jumps have significant effects to the persistence and extinction results. [ABSTRACT FROM AUTHOR]

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier. [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, 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]

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to establish a new LaSalle-type theorem for stochastic differential delay equations of neutral type. From the theorem follow some useful criteria on the orbital asymptotic stability. Several examples are also given for illustration. This shows clearly the power of our new theorem. [ABSTRACT FROM AUTHOR]

In this paper we review our recent research on stochastic delay population dynamics. The main aim is to reveal that a different type of environmental noise has a different effect on delay population systems. [ABSTRACT FROM AUTHOR]