| This thesis involves two types of bio-dynamic models:A class of predator-prey model on zooplankton with Holling type and a class of SIR epidemic model. By using the knowledge of the nonlinear analysis and nonlinear partial differential equations, particularly, the theories and methods of the parabolic equation and the corresponding elliptic equation, the coexistence, positivity, boundedness, the bifurcation and stability of solutions of the two models are discussed.By the fixed point index theory and the bifurcation theory, a predator-prey model on zooplankton with homogeneous Dirichlet boundary conditions is studied. Then, a SIR epidemic model with homogeneous Neumann boundary conditions is discussed by using the Hurwitz-Rouche criterion, the method of upper and lower solutions, the comparison principle and the lyapunov function method.The main contents in this thesis are as follows:In chapter1, the background of predator-prey models and epidemic models are introduced. Some research works and results in the related field are also given there.In chapter2, a predator-prey model on zooplankton with homogeneous Dirichlet boundary conditions is investigated in four parts. First, by using the comparison principle, a priori estimate of positive solutions is made. Then, according to the fixed point index theory, the existence of the solutions is proved. What’s more, based on treating b as the bifurcation parameter, the local bifurcation is extended to global bifurcation. Finally, the stability of the solutions is proved.In chapter3, an epidemic SIR model is considered. Because the model is complex, we can just discuss the first two equations. This chapter can be divided into two parts. In the first part, by using the lyapunov function theory, locally and the globally asymptotical stability of the disease-free equilibrium and the endemic equilibrium are discussed; In the second part, the locally asymptotical stability of the endemic equilibrium and the disease-free equilibrium are proved. |
