Dynamical Analysis Of A Diffusive Predator-Prey Model With Crowley-Martin Functional Response

In the natural ecological systems,the relationship between the predator and prey is a basic interaction among species,which is closely related with the functional response of the predator.Different functional responses can induce different dynamical behaviors and spatiotemporal patterns,which can be used to explain the ecological complexity.A diffusive predotor-prey system with predator interference and Neumann boundary condition is considered in this paper.We derive some results on the existence and stability of constant stationary solutions,and the existence and nonexistence of nonconstant stationary solutions.The main contents are follows.We first introduce the existence and stability of constant positive equilibria of the system.Through analyzing the relations between the conversion rate of the predator and the constant positive equilibrium of system,we can obtain the number of constant positive equilibria.Then through analyzing the characteristic equations,we obtain the conditions that the positive equilibrium of the system is locally asymptotically stable.Finally,by constructing the well known Lyapunov functional,we obtain the condtions that the unique positive equilibrium of the system is globally asymptotically stable.Then,we investigate the nonexistence and existence of nonconstant steady states of the system.For the nonexistence,we first give two equivalent systems,and derive the existence of upper bounds and lower bounds through the maximum principle and priori estimates for positive solutions of two equivalent systems.Then through the regulariy theory,we obtain the limit profile of the positive solutions of two equivalent systems.Finally,combined with the implicit function theorem and Poincaré theorem,we give some sufficient conditions for the nonexistence of nonconstant positive steady states of the system.Moreover,for the existence of nonconstant positive equilibria,we also derive a priori upper and lower bounds for positive solutions of system through the regularity theory.Then by using of the Leray-Schauder degree theory,we obtain the existence of nonconstant steady states of the system.It is shown that there exist no nonconstant positive steady states when the effect of the predotor interference is strong or the conversion rate of the predator is large,and nonconstant positive steady states emerge when the diffusion rate of the pretador is large.