The Study For The Mathematical Ecology Models With Diffusion And Cross-diffusion

This paper deals with such a class of predator-prey models that the predator, in addition to the prey considered, has other natural sources of food. we investigate these predator-prey models with homogeneous Neumann boundary conditions, or with mixed boundary conditions. For the predator-prey models with homogeneous Neumann boundary conditions, since diffusions don't create that the various species co-exist in non-constant time-independent positive solutions, we must introduce cross-diffusion effects. And the cross-diffusion means that, in a certain kind of prey-predator relationships, a great number of prey species form a huge group to protect themselves from the attack of predator, so that the predator moves away from a large group of preys. It is a general phenomenon in the nature. In this paper, by making use of the Degree theory, we prove the existence and non-existence of the positive steady-state solutions to these models on a certain of conditions. Furthermore, by making use of the bifurcation theory and the stability theory, we also investigate local or global bifurcation of the positive steady-state solutions, local stability and the asymptotic behavior of the positive solutions to some predator-prey models. This paper consists of the following eight chapters.In Chapter 1, we introduce the background of mathematical ecology models, results and advance in studying. In Chapter 2, we introduce some preliminaries. We will use these knowledge to study the existence and non-existence of the positive steady-state solutions of the predator-prey models in the paper.In Chapter 3, we investigate a predator-prey model with diffusion and with mixed boundary conditions, where let cu/(r+u~2) be the conversion rate of the prey captured to the predator, and cu/(r+u~2) exhibits that the conversion rate cu/(r+u~2) is approximately equal to cu/γ when u is small enough, but the conversion rate is inhibited when u is large enough. Inthis model, the prey is with homogeneous Neumann boundary condition and the predator is with homogeneous Robin boundary condition. We firstly obtain the following results: if 6 > ^2^1, then a > /xiC^Mi is the sufficient and necessary condition for the existence of positive steady-state solution of the predator-prey model. And we also investigate the existence and nonexistence the positive steady-state solutions for b < Aiefe. Next, we also prove that the local stability and uniqueness of the positive steady-state solution. Last, we discuss the limit of positive steady-state solutions and the asymptotic behavior of the positive solutions to the predator-prey model when one of diffusion coefficients approaches to oo.In Chapter 4, we continue to investigate the predator-prey model in Chapter 3, but the boundary conditions of the prey and the predator are exchanged. By a similar argument as in Chapter 3, we obtain that a > mb + diAi is the sufficient and necessary condition for the existence of positive steady-state solutions of the predator-prey model.In Chapter 5, we investigate a Lotka-Volterra predator-prey model with cross-diffusion. First, suppose that a satisfies that mxb < a < l^ll^n2 when 0 < mira2 < 1, or a> m\b when m\m,2 > 1, such that m,\V > u, where (u,v) is the positive constant solution of the model. Let d\ < mi*~u and d4 > mi~_g, then there exists (^1,^2,^3)^4) such that the model has non-constant positive steady-state solutions, which means that the cross-diffusion d\ may be helpful to create more positive steady-state patterns. Whereas, if o doesn't satisfy the condition above, or d\ > miJ~", or d4 < m ~_~, then the model has no non-constant positive steady-state solution bifurcating from the constant positive solution.In Chapter 6, we investigate a predator-prey model with cross-diffusion, where the conversion rate -^ is monotone increasing and bounded. Suppose that rriib < a <+ mim2 - 7) + y/(mib + mim2 - t)2 + 47mi& such that mrf > u, where (u,v) isthe positive constant solution of the model. Let d\ < mi? " and d\ > m ~_g, then, by a similar argument as in Chapter 5, there exists (di,d2,dz,d^) such that the predator-prey model exists non-constant positive steady-state solutions, which implies that the cross-diffusion d4 may be helpful to create more positive steady-state patterns.In Chapter 7, we investigate a ratio-dependent predator-prey model with diffusion. We prove that there exists a positive number ao(b) depending on b such that, if b < mi and ao(b) < a < mi, then there exists {di,d2) such that the predator-prey model exists non-constant positive steady-state solutions. Whereas, for a > mi, the model has no non-constant positive steady-state solution bifurcating from w, where w is the positive constant solution of the model.In Chapter 8, we introduce a cross-diffusion effect into the second equation of the predator-prey model in Chapter 7. Moreover, We prove that there exist positive constants a2(b) and D%. If max{mi~m;},0} < b < 2mx such that mi < a2(b), and mi < a < a2(b), then there exists (rfi, d2, d3) satisfying di < (^ffi -u) x ^ and rf3 > D%, where w = (u, v) is the positive constant solution of the model, such that the model has non-constant positive steady-state solutions. Whereas, if d3 < D°, or a > mx and a > a2(b), or di > (?￡+-& — u) x -7, then the model has no non-constant positive steady-state solution bifurcating from w.