| By means of the theory of nonlinear partial differential equations and nonlinear analysis, we studied the existence and stability of coexistence solutions of two kinds of predator-prey models.At first, by applying the bifurcation theory and the fixed point index theory in cone, the existence of coexistence solutions of the predator-prey model with Holling type Ⅲ functional response is discussed Secondly, we study the existence and stability of coexistence solutions of the predator-prey model on plankton with Holling type II functional response by means of the Routh-Hurwitz theorem and the bifurcation theoryThe main contents are organized as follows:In chapter1, we introduce the background and research results of predator-prey models, and give the main results of this thesis.In chapter2, a class of diffusive predator-prey model with Holling type III func-tional response is discussed. Firstly, by the maximum principle, a priori estimate of non-negative solutions is given. Secondly, by means of the bifurcation theory, the positive steady-state solutions bifurcating from the semi-trivial solution branch {(α,θα,0):α> λ1} are given by treating a as the bifurcation parameter. Fur-thermore, the local bifurcation branch can be extended to the global one. Finally, the sufficient conditions for the existence of the positive steady-state solutions are obtained by using the fixed point index theory in cone.In chapter3, a predator-prey model on plankton in a marine ecosystem with Holling type II functional response is studied. Firstly, we study the existence of the positive constant equilibrium. Secondly, by the Routh-Hurwitz theorem, the stability of positive constant steady-state solution is discussed. Finally, by means of the bifurcation theory, non-constant steady-state solutions bifurcating from the positive constant steady-state solution are given by treating the diffusion coefficient D as the bifurcation parameter, which implies the existence of non-constant positive steady-state solutions. |
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