The Nature Of The Two Types Of Reaction-diffusion Model Analysis Solution

In this paper, we mainly study qualitative properties of solutions for two types of predator-prey model subject to homogeneous Neumann boundary condition:a class of Michaelis-Menten type predator-prey model with harvesting rate for preys and a class of ratio-dependent generalized Holling-Tanner system. By means of the theories and methods of the parabolic equation and the corresponding elliptic equation, we shall discuss the dissipation, stability, existence of solutions of the two models.A Michaelis-Menten type predator-prey model with harvesting rate for preysA ratio-dependent generalized Holling-Tanner systemThe main contents in this paper are as follows:In chapter1, we study qualitative properties of solutions for a Michaelis-Menten predator-prey model with harvesting rate for preys subject to homogeneous Neu-mann boundary condition. Firstly, the dissipation of solution is proved by means of the parabolic equation’s comparison principle. Secondly, the stability of positive constant steady-state solution is discussed by eigenvalue theory is discussed. Third-ly, the local bifurcation at positive constant steady-state solution for N-dimensional case is obtained by local bifurcation theory and the structure of solutions near bifur-cation point is given. Lastly, the local branch can be extended to the global branch through global bifurcation theory. In chapter2, the stability and existence of positive steady-state solution for a ratio-dependent generalized Holling-Tanner system, with homogeneous Neumann boundary condition, are researched. In section one, the global asymptotic stability of positive constant steady-state solution (u*, v*) is obtained by means of specturm theory. In section two, a priori estimates of positive solution are given by the maximum principle and the Harnack inequality. In section three, the non-existence of the non-constant positive solution is proved through the intergral property, ε-Young inequality and Poincare inequality. In section four, the existence of non-constant positive solution is investigated with the help of the priori estimates and Leray-Schauder degree theory. Moreover, the sufficient condition for the existence of positive solution is gained.