Type Ii Holling Functional Response Of Predator - Prey System

Population ecology is an important branch of ecology science. Since the complexity of ecological relations, mathematical methods and results have been used in and have emerged from ecology. Now population ecology has become the branch that mathematics is most deeply applied in and which is the most systematic one. Early population studies concentrated on local population dynamics. However, it is not enough that populations of organisms are considered only in time. Many ecological processes that are distributed over some spaces should be considered in space. Therefore, ecological models of PDE have attracted considerable attention in recent years.In this paper, we discuss a predate-prey model with Holling type II functional response.First, the weakly coupled reaction-diffusion system describing two interacting species with homogeneous Neumann boundary conditions is studied. We derive the prior estimates for the solutions of parabolic system, and the global existence and uniqueness results of solution are given by upper and lower solutions. A sufficient condition for the global asymptotical stability is given by Lyapunov function and the local asymptotical stability. It is revealed that if the intrinsic growth rate of prey is slow, or if the capturing rate of predator is slow, or if the intra-specific competition of predator is strong enough, positive constant solution is globally asymptotically stable. Also, we show that the steady state has no non-constant positive solution if one of the diffusion rates is large enough.Second, we consider the strongly coupled elliptic system with homogeneous Dirichlet boundary conditions. The prior estimates for thesolution of nonlinear elliptic system are derived. It is shown that there is no coexistence state if diffusion rates are strong, or if the intrinsic growth rates are slow. Making use of the Schauder fixed point theory, we derive some sufficient conditions to have a coexistence state for the strongly coupled elliptic problem. Moreover, our results reveal that this problem possesses at least one coexistence state if the intra-specific competition of predator is strong and cross-diffusions are relatively weak, or if the capturing rate is slow and cross-diffusions are relatively weak.