The Bifurcation Of Predator-prey Model With Cross-diffusion

As an important method in mathematical application areas,the reaction diffusion model combines mathematical theory with practical problems and its research findings in describing dynamical properties of population makes great contribution.Especially,predator-prey model with cross-diffusion plays an irreplaceable role in the study of population relationship.By using the theories and methods of the reaction diffusion equations,three types of predator-prey models with cross-diffusion are basically analyzed and some properties of solution can be obtained in this paper.They are predatorprey system with cross-diffusion and Michaelis-Menton type prey harvesting,Variable-Territory predator-prey model with Holling ? functional response and predator-prey system with cross-diffusion and self-diffusion.There are five chapters in total in this paper.The biological background and research achievements are raised in the first chapter for studying three predator-prey models above.Afterwards,we also provide some basic knowledge.In the second chapter,we study a predator-prey model with Michaelis-Menton type prey harvesting under homogeneous Dirichlet boundary conditions.Under the influence of cross-diffusion,we obtain a priori estimate and sufficient conditions of positive solution by using the upper and lower solution method and Crandall-Rabinowitz bifurcation theory.Thus,we can extend the local bifurcation solution to the global one and make the continuum go to infinity.In chapter three,a Variable-Territory-type predator-prey model with cross-diffusion is discussed and all the researches are established under homogeneous Dirichlet boundary conditions.A priori estimate is obtained by using the upper and lower solution method at first.Then,by applying the spectrum analysis method and bifurcation theory,we can prove the existence of local bifurcation of semi-trivial solution.Finally,extend ]ocal bifurcation to the global bifurcation and prove that the continuum shall go to infinity.A predator-prey model with cross-diffusion and self-diffusion is discussed in the fourth chapter.The model is established under Bazykin type functional response and homogeneous Dirichlet boundary conditions.We can obtain a priori estimate by using the Maximum principle and prove the existence of local bifurcation solutions by applying the bifurcation theory when we treat the growth rate of prey as bifurcation parameter.In chapter five,some results of this paper are summarized and problems to be discussed and solved further more are presented.