Properties Analysis Of Stochastic Predator-prey System With Crowley-Martin Functional Response

Because the interactions between predator and prey are universal and important,the mathematical models established for the dynamic relationship between predator and prey have become one of the important topics in mathematical ecology.In recent years,many scholars have devoted themselves to this topic and got good results.Because the biological populations in nature are inevitably affected by the external environmental factors,in this paper we consider the influence of random factors,add the environmental noise to the Leslie-Gower predator-prey model with Crowley-Martin functional response,and analyze the influence of linear disturbance and nonlinear disturbance on the dynamic properties of the predator-prey model.On the one hand,we perturb the linear term in the system and obtain the existence and uniqueness of the solution and the persistence,extinction,stationary distribution and ergodicity of the system.Firstly,the existence and uniqueness of the global positive solution are proved by constructing an appropriate Lyapunov function and applying It(?)'s formula.Secondly,the sufficient conditions for the permanence and extinction in the sense of time average are obtained by using the comparison theorem of stochastic differential equations and other methods.Finally,the stationary distribution and ergodicity of the system are discussed,and the conclusions are verified by numerical simulations.On the other hand,considering the influence of time on the model parameters and introducing nonlinear term disturbance,the existence and uniqueness of the global positive solution are also obtained.The sufficient conditions for the extinction and weak persistence of the system are obtained by using appropriate Lyapunov functions,quadratic variation,exponential martingale inequality and other methods.Finally,the corresponding numerical simulations are given to verify the correctness of the conclusion by using Milstein method.