Coexistence Of Two Classes Of Biological Diffusion Model And Qualitative Analysis

This thesis involves two types of bio-dynamic models:A class of predator-prey model with Leslie item and a class of SIR epidemic model with non-monotonic inci-dence. Using the knowledge of the nonlinear analysis and nonlinear partial differen-tial equations, particularly, the theories and methods of the parabolic equation and the corresponding elliptic equation, we have discussed the coexistence, positivity, boundedness and stability of solutions of the two models.By the bifurcation theory and the energy method, we study a predator-prey model with homogeneous Neumann boundary conditions We discuss a SIR epidemic model with homogeneous Neumann boundary conditions by using the Hurwitz-Rouche criterion, the method of upper and lower solutions and the comparison principleThe main contents in this thesis are as follows:In chapter1, we introduce the background of predator-prey models and epi-demic models. Some research works and results in the related field is also given there.In chapter2, a class of ratio-dependent Holling-Leslie type predator-prey model, with homogeneous Neumann boundary condition, is considered. Let d2be as the bifurcation parameter, we study the bifurcation at positive constant equilibrium by means of the bifurcation theory and the Leray-Schauder degree theory; Then the local bifurcation which can be extended to the global bifurcation is proved and the fact that the global bifurcation joins up with infinity in the case of one-dimension is obtained; Lastly, the condition for nonexistence of positive solutions is given.In chapter3, an epidemic SIR model with nonmonotone incidence rate is con-sidered. Firstly, the positivity, uniform boundedness and global attractivity of the solutions are given. What’s more, by using the Hurwitz-Rouche criterion, we dis-cuss the locally asymptotical stability of the disease-free equilibrium and the en-demic equilibrium. Consequently, by the method of upper and lower solutions and the comparison principle, our results show that:If the constant input rate is big enough, the endemic equilibrium is globally asymptotically stable; If the constant input rate or the contact rate is small enough, the disease-free equilibrium is globally asymptotically stable.