Qualitative Properties Of Some Predator-prey Models With Prey Refuge

Predator-prey models can be used as the description of the interaction between prey and predator in biological population dynamics.Prey refuge is established in the evolutionary process of prey to avoid predation,which has practical significance for maintaining ecological balance and protecting endangered populations.Qualitative properties of some predator-prey models with prey refuge are studied in this paper.In Chapter 1,the background and history about the related work are introduced.In Chapter 2,we discuss a diffusive predator-prey model under the homogeneous Neumann boundary condition with Holling-Ⅱ functional response incorporating prey refuge which is proportion to encounters between the prey and predator.First,we study the qualitative properties of solutions to this reaction-diffusion system,which including dissipation and persistence,and the stability of non-negative constant solutions.Then,the nonexistence of nonconstant positive steady-states is obtained by using energy analysis.Finally,the existence of the nonconstant positive steady-states is proved by topological degree theory.In Chapter 3,cross-diffusion is introduced into the model of Chapter 2,the existence of nonconstant positive solutions for strongly coupled systems is proved by topological degree theory.It shows that cross-diffusion may cause Turing Pattern in a certain range of parameters.In Chapter 4,we consider a diffusive predator-prey model under homogeneous Dirichlet boundary contion with modified Leslie-Gower and Beddington-De Angelis functional response incorporating prey refuge which is proportion to the prey density.The existence of positive solutions is derived from the topological degree theory on cones,and exact results on regions in parameter space which have a unique positive solution are also described.