A Study Of Two Types Of Predator-prey Models

In the development of biological mathematics,the reaction-diffusion equation has an extremely important position and is applied in many fields.Many scholars use it to establish many mathematical models to solve some problems in real life.Two types of reaction-diffusion equations are mainly studied in this paper.The first chapter introduces the main content of this paper by introducing the development process of the reaction-diffusion equation.A kind of predator-prey model that uses Monod-Haldane type functional response function to defend predators and reduces the prey growth rate in the form of fear under homogeneous Neumann boundary conditions is mainly studied in chapter 2.Firstly,we get a prior estimate of positive solutions,and analyze the local stability of the positive constant solution.Secondly,we provide the conditions for the nonexistence and existence of non-constant positive solutions of the system.Then,on the basis of the bifurcation theory we have learned,the local bifurcation and the global bifurcation of the system are given,and the stability of the local bifurcation solutions is analyzed.Finally,numerical simulations are performed by Matlab.The other predator-prey model with Michaelis-Menten type prey harvesting term under Dirichlet boundary conditions is mainly discussed in chapter 3.Firstly,employing the maximum principle,we get a priori estimate of the positive solutions.Secondly,by means of the degree theory we study the necessary and sufficient condition for the existence of positive solutions of equilibrium equations.Then,based on the bifurcation theory,the local bifurcation of the model is discussed by treating d as a parameter.Finally,we use Matlab to verify theoretical results.