Coexistence Analysis Of Two Types Of Reaction Diffusion Equations

In natural science in practical problem solving, many of them can be studied through the establishment of the mathematics model. The scholars found that many of the models can be attributed to the reaction-diffusion equation after a large number of research. Through the research of the reaction-diffusion equation, people can explain and predict some ecological problems and natural phenomena more scientifically, and provide a reasonable way for the problem solving. At the same time in a more in-depth research, scholars have proposed many challenging problems, is attracting a lot of lovers to explore.In this paper, the dynamical properties of the two kinds of reaction-diffusion equation are studied. One is the coexistence state problem of a modified Holling-? type predator-prey model with Michaelis-Menten type harvesting: The other is a HBV infection model with CTL immune response under homogeneous Neumann boundary conditions: w(x,0)=w0(x)?0,x??.The main contents are organized as follows: In chapter 1, we introduce the background and research results of predator-prey models and HBV infection model with CTL immune response, and give the main results of this thesis.In chapter 2, a modified Holling-IV type predator-prey model with Michaelis-Menten type harvesting under homogeneous Neumann boundary conditions is stud-ied. Firstly, by the maximum principle and Harnack inequality, a priori estimate of non-negative solutions is given; the local asymptotic stability, uniform asymptotic stability and global asymptotic stability of the constant solution are proved. Sec-ondly, using Poincare inequality and other knowledge, study the positive equilibrium solution does not exist, using the fixed point index theory and homotopy invariance, given the positive balance of existence for the solution of sufficient conditions. Third-ly, With the help of bifurcation theory and L-S degree theory, the local bifurcation of steady-state system at the positive constant solution U1 is obtained by treating d,2 as bifurcation parameter; it is shown that under certain conditions, the local bifurcation generated from (d23, U1) can be extended to global bifurcation. Finally, regarding h as a Hopf bifurcation parameter, the existence of periodic solutions near positive constant equilibrium (u1,vi) is presented.In chapter 3, the stability of a HBV infection model with CTL immune response under the homogeneous Neumann boundary conditions is analyzed. It contains three parts:in part one, a priori estimates of positive solutions are obtained by using maximum principle; in part two, the local asymptotic stability of two constant equilibrium solutions is discussed by using Hurwitz theorem; and in part three, the global asymptotic stability of the disease-free equilibrium point Uo is proved by constructing the upper and lower solutions and corresponding monotone iterative sequences.