Dynamic Analysis Of Biological Population Model

In biota,there are various complex interactions among species.Among them,the predator-prey relationship between species has been the focus of biologists and mathematicians.In this paper,two kinds of predator-prey models with spatial diffusion term are discussed by using partial differential equation and nonlinear analysis theory etc.under homogeneous Nenmann boundary condition.The main contents of this paper are as follows:In Chapter 1,the related background and research results of models are intro-duced briefly,and we summarize the research content of this paper.In Chapter 2,a new(by way of supply of additional food to the predators)predator-prey biological population model with prey harvesting and ratio-dependent functional response is studied.Firstly,using the comparison principle of partial differential equations and the linearization method,the large-time behavior of non-constant solutions and the stability of constant solutions to the parabolic system are discussed.Secondly,using the maximum principle,Harnack inequality,energy integral method and Leray-Schauder degree theory,we discuss the priori estimation of positive solution of equilibrium state system,as well as the nonexistence and existence condition of nonconstant positive solution.Finally,using the bifurcation theory and degree theory,bifurcation of the equilibrium state system from the posi-tive constant solution is studied.Compared with the existing predator models with prey harvest term,a model for providing additional food to predators is introduced in this paper.The results show that the quality and quantity of additional food have obvious influence on the solutions of parabolic system and equilibrium system.In Chapter 3,a predator-prey population model with Holling ? type response function with constant prey refuge is studied.Firstly,the stability of solution to the system is given by using the linearization method.Secondly,the existence and stability of the Hopf bifurcation periodic solution to the system axe studied by means of space decomposition.Finally,using Matlab tool to carry out numerical simulation,we verify the conclusions on stability and the conditions under which Hopf bifurcation may occur.