Dynamic Analysis Of Stochastic Predator-prey System With Holling ? Functional Response

In the ecosystem,there is ubiquitous stochastic noise,which may lead to changes in the interaction between populations and between populations and environment.Recently,the population system with all kinds of stochastic noises has become an important research topic in biological mathematics.On basis of stochastic disturbance(white noise and colored Levy noise)and Holling ? nolinear functional response,this paper mainly study the dynamic behaviors such as well-posedness of the solution,stochastic stability,boundedness,extinction and persistence for two types of stochastic predator-prey system.The results show that stochastic noises have a significant effect on dynamic behaviors of the system and can change its dynamic properties.In the first part of this paper,we investigate a class of stochastic chemostat model with Michaelis-Menten food chain in which the dilution rate is disturbed by white noise.Firstly,the global existence and uniqueness of positive solution is proved by comparison principle.Then,by constructing Lyapunov function and using Ito formula,sufficient condition for stochastic global asymptotic stability of washout equilibrium is obtained.Under different conditions for white noise,we reveal that oscillatory behaviors of the solution which around predator-free equilibrium and positive equilibrium for deterministic model.Finally,some examples explain that asymptotic properties of the model can be changed by white noise.In the second part of this paper,we investigate a class of three-species prey-predator system with Holling ? functional response and stochastic perturbations involving white noise and Levy noise.Firstly,the global existence and uniqueness and stochastic ultimate boundedness of positive solution are proved by comparison principle and Lyapunov function theory.Then,by constructing Lyapunov function and using Ito formula,Jansen inequality,chebyshev inequality and stochastic persistence theory,sufficient conditions ensuring stability,extinction,strongly persistence in the mean and stochastic permanence in the sense of probability are established for the system.The results show that both white noise and Levy noise may change asymptotic properties of the system.Finally,it is verified by examples that stochastic noises can suppress chaotic dynamics.