Asymptotic Behavior of a Stochastic SIRS Model with Non-linear Incidence and Levy Jumps

A stochastic susceptible-infective-recovered-susceptible( SIRS) model with non-linear incidence and Levy jumps was considered. Under certain conditions, the SIRS had a global positive solution. The stochastically ultimate boundedness of the solution of the model was obtained by using the method of Lyapunov function and the generalized Ito's formula. At last,asymptotic behaviors of the solution were discussed according to the value of R0. If R0< 1,the solution of the model oscillates around a steady state, which is the diseases free equilibrium of the corresponding deterministic model,and if R0> 1,it fluctuates around the endemic equilibrium of the deterministic model.

Extinction and Stationary Distribution of a Stochastic SIR Epidemic Model with Jumps

A stochastic susceptible-infective-recovered(SIR)epidemic model with jumps was considered.The contributions of this paper are as follows.(1) The stochastic differential equation(SDE)associated with the model has a unique global positive solution;(2) the results reveal that the solution of this epidemic model will be stochastically ultimately bounded,and the non-linear SDE admits a unique stationary distribution under certain parametric conditions;(3) the coefficients play an important role in the extinction of the diseases.

Dynamical analysis for the sustained harvesting of microorganisms using flocculants in a random environment

The use of flocculants to collect/extract microorganisms is of great practical significance for the development of the application of microorganisms. In this paper, a high-dimensional nonlinear stochastic differential equation model is constructed to describe the continuous culture of microorganisms with multiple nutrients and the flocculation process of microorganisms. The study of the dynamics of this model can provide feasible control strategies for the collection/extraction of microorganisms. The main theoretical results are sufficient conditions for the permanence and extinction of the stochastic differential equation model, which are also extensions of some results in the existing literatures. In addition, through numerical simulations, we vividly demonstrate the statistical characteristics of the stochastic differential equation model.

Dynamics of a stochastic Holling II predator-prey model with Levy jumps and habitat complexity

This paper investigates a stochastic Holling II predator-prey model with Levy jumps and habit complexity.It is first proved that the established model admits a unique global positive solution by employing the Lyapunov technique,and the stochastic ultimate boundedness of this positive solution is also obtained.Sufficient conditions are established for the extinction and persistence of this solution.Moreover,some numerical simulations are carried out to support the obtained results.

Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

In this paper,stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied,and we use distribution delay to simulate the delay in nutrient conversion.By the linear chain technique,we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations.First,we state that this model has a unique global positive solution for any initial value,which is helpful to explore its stochastic properties.Furthermore,we prove the stochastic ultimate boundness of the solution of system.Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established,which means the microorganism will be extinct.Moreover,we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions.Finally,we provide some numerical examples to illustrate theoretical results,and some conclusions and analysis are given.

DYNAMICAL PROPERTIES OF A STOCHASTIC TWO-SPECIES SCHOENER＇S COMPETITIVE MODEL

A stochastic two-species Schoener＇s competitive model is proposed and investigated. Sufficient conditions for the existence of global positive solutions, boundedness~ uniform continuity, global attractivity stochastic permanence and extinction are obtained. More- over, the upper-growth rate and the average in time of the sample paths of solutions are also estimated. Finally, some figures are introduced to illustrate the main results.