The Dynamics Of Two Types Of Biological Models

The reaction-diffusion equation reflects the relationship between related vari-ables and time,space,it has many applications in the field of ecology.In order to better understand the natural phenomena in ecology,the reaction-diffusion equa-tion was established to study the change rule between populations in ecology and verified by numerical simulation.In this paper,the kinetic problems of two kinds of biological models under the second boundary conditions are studied by using the theory and method related to parabolic and elliptic equations in reaction-diffusion equations and numerical simulation techniques.This article is divided into three chapters to discuss:In Chapter 1,the background,basis and significance of model are introduced and some common lemmas are given.In Chapter 2,a kind of optimal predator-prey model is studied.Firstly,the existence of a unique positive constant solution for the system is proved by means of the heavy factor.Then,the stability conditions of the positive constant solution of the system are given by using the method of linearization and Lyapunov function.Next,the long time behavior of positive solutions of the system under some condi-tions is discussed.At the same time,the prior estimate,existence and nonexistence of positive solutions for equilibrium systems are investigated by using the compari-son principle,the maximum principle,the degree theory and the energy integration method.Finally,the long time behavior of the solution is verified and explained by numerical simulation.It should be noted that the positive constant solution of the system is used in the numerical simulation,because it involves a univariate quintic equation,the expression of the solution is not easy to find,and the positive solution of the system needs to be found by MATLAB.In Chapter 3,a predator-prey model with shelter is studied.Firstly,the con-ditions for the stability of the positive constant solution of the system are given.Then,take the diffusion rate as bifurcation parameter,the existence condition of bifurcation solution is obtained by using bifurcation theory,and it is shown that the bifurcation curve from the positive constant solution can be extended to infinity in one-dimensional case.Next,conditions of the nonexistence of positive solutions of the system are given.The results show that when the diffusion coefficient is large enough,the system has no positive solution.Finally,the stability and bifurcation conditions of the solution are verified by numerical simulations.