The Stability Of Predator-prey Model With Refuge And Nonlinear Harvest Effect

In this paper,the stability and Turing instability of predator-prey model with shelter and nonlinear harvesting effect are studied.The model consist of three parts.In Section 1,a predator-prey model with uniform spatial distribution with shelter and nonlinear harvesting effect is discussed.Firstly,the existence and local asymptotic stability of positive equilibrium point are analyzed,the global stability of equilibrium points of system is analyzed by Dulac criterion,and then the existence and uniqueness of limit cycle are proved by Pioncare-Bendixson ring theorem.The optimal harvesting strategy for two species is obtained by using the Pontryagin maximum principle.In Section 2,the stability of predator-prey diffusion model with shelter and nonlinear harvesting effect is discussed.The existence and uniform boundedness of the global solution of the model are studied.The local stability of the positive equilibrium point is proved by linearizatio method.The global stability of the positive equilibrium point is proved by constructing the Lyapunov function,and it is proved that if the diffusion coefficient is large enough,the Turing instability will be caused.In Section 3,the Turing instability of predator-prey cross-diffusion model with shelter and nonlinear harvesting effect is discussed.