Asymptotic Properties Of Two Stochastic Predator-prey Models With Functional Response

The coexistence and mutual restriction among populations are the key issues in controlling biological invasion and maintaining ecological balance,while the functional response is a mathematical representation in describing such relationships.Therefore,various functional responses are put forward,such as the Holling type,the Beddington type and so on.In this paper,the Crowley-Martin and the Hassel-Varley type functional responses which depend on predator density in different ways are used.In addition,as the stochastic environmental noise has an unavoidable impact on populations,research on the corresponding stochastic systems plays a vital role in either theoretical analysis or practical applications.Firstly,the existence of stationary distribution,recurrence,the strongly persistence and extinction,the stochastic Hopf bifurcation are investigated for a stochastic predator-prey model with the Crowley-Martin type functional response.First of all,it shows that there exists a unique stationary distribution and the stochastic system is recurrent by using the relevant stochastic differential equation theories and constructing the appropriate Lyapunov function.Furthermore,sufficient conditions for the existence of stationary distribution are obtained and it's interesting to see that the they are not strong.Additionally,numerical simulations are carried out to support the results.Then complete threshold analysis of coexistence and extinction is given,which can be provided as guidelines to practical production.Moreover,the results point out that the stochastic predator-prey model undergoes a stochastic Hopf bifurcation by the point of phenomenological bifurcation.Secondly,the stochastic persistence and the existence of stationary distribution of a stochastic predator-prey model with the Hassel-Varley functional response are discussed.The relevant conclusions are obtained by choosing the appropriate Lyapunov function,using the It? formula and the Chebyshev inequality etc.The validity of these conclusions are verified by numerical simulations in the end.