Qualitative Analysis Of Positive Steady States For Diffusive Predator-Prey Systems

In the past decades, ecological models with spacial structure have been one of the most active fields and received extensive concerns from many mathematicians and biologists. In particular, due to the differences in energy transformation, the long-time behaviors and the existence/nonexistence of nonconstant positive steady states for diffusive predator-prey systems with different functional responses are im-portant themes in studying population dynamical models. This thesis is concerned with the qualitative property for some diffusive predator-prey systems with different functional responses.Firstly, we study two classes of predator-prey models with Robin boundary conditions. One incorporates the modified Leslie-Gower functional response for the predator and the general functional response p(u) for the prey. The other one involves the Beddington-DeAngelis functional response. By applying the theory of fixed point in cones, we give some sufficient and necessary conditions for the existence and nonexistence of coexistence solutions to these systems, which are dependent on some parameters and the principle eigenvalues under Robin boundary conditions. Furthermore, we will discuss the stability of coexistence solutions and the asymptotic behavior for the parabolic systems. In particular, we give an application of the first class of predator-prey system, that is to say, we replace the functional response p(u) with Holling-II type functional response and obtain the corresponding results.Secondly, we study a kind of modified Holling-Tanner type predator-prey sys-tem under homogeneous Neumann boundary conditions and discuss the global at-tractivity and persistence property of the parabolic system. By using the linearized method and Lyapunov function method, we investigate the locally and globally asymptotic stability of the positive constant steady state when the parameters sat-isfy suitable conditions. After this, we apply the maximal principle and Harnack inequality to give a priori upper and lower bounds estimate for the positive solu-tions to elliptic system. Then, we establish the nonexistence and existence of the nonconstant positive steady states by energy method and Leray-Schauder degree theory, respectively, in the case of large diffusion. Furthermore, we study a class of three-species predator-prey system with diffusion, in which the two predators con-sume the common prey. The locally and globally asymptotic stability of the unique positive constant steady state is discussed. Through giving a priori estimate on the positive solution to the corresponding elliptic problem, the nonexistence of noncon- stant positive steady state is established by applying the energy method when the diffusion coefficient d1 is sufficiently large.Finally, in view of the important role of impulsive differential equations in mod-eling some ecological systems, we discuss a class of ratio-dependent Holling-III type predator-prey system with diffusion and impulses. Some sufficient conditions for the existence of positively invariant set, ultimate boundedness of solutions, persis-tence property and the extinction of predator population are given. Furthermore, we generalize these results to multi-species predator-prey systems.