Hopf Bifurcation Analysis Of The Reaction-Diffusion Models With Time Delay

Reaction-diffusion equations are used to describe the practical problems in ecology,chemistry,physics and many other fields frequently,such as the evolution of biological populations,the progress of chemical reactions,etc.These problems are related to both time and space.In addition,the change of states in these problems is sometimes not only related to the state of the current moment,but also depends on the state of the past moment,which needs to be characterized by the reaction-diffusion models with time delay.Studying the reaction-diffusion models with time delay from the perspective of bifurcation can not only help people understand the real process described more clearly,but also provide new methods and new contents for the theoretical improvement of the functional differential equations.Therefore,it is of great practical significance to study the Hopf bifurcation of the reaction-diffusion models with time delay.In this paper,an activation-inhibition model(Lengyel-Epstein system)and a predator-prey model(nutrient-microbial system)are considered.The dynamic behavior of two systems is investigated by using Hopf bifurcation theory,normal form theory,center manifold theorem,steady-state bifurcation theory and some relevant theories of the functional differential equations.Specifically,under the homogeneous Neumann boundary condition,we mainly discuss the influence of the time delay on the stability of the positive equilibrium and Hopf bifurcation of the system,and the Hopf bifurcation and steady-state bifurcation of the diffusion system without time delay are also considered.Some numerical simulations are carried out by MATLAB to verify the theoretical results in the end.The main contents of this paper are as follows:In chapter 1,we introduce the background and research status about the reaction-diffusion models with time delay,and then many research achievements and progress of Lengyel-Epstein system and nutrient-microorganism system by domestic and foreign scholars are also introduced in detail.In chapter 2,the Lengyel-Epstein reaction-diffusion model with time delay is studied.Firstly,taking the time delay τ as the bifurcation parameter,the stability of the positive equilibrium and the existence of Hopf bifurcation of the ODE and PDE systems are discussed by analyzing the characteristic equation.The results show that when the time delay τ passes through the critical valueτ=τ00,the system will change from stable state to unstable state and produce Hopf bifurcation.Secondly,the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the center manifold theorem and the normal form theory.Finally,some numerical simulations are carried out by MATLAB.The results show that the ODE system will produce a stable limit cycle near the positive equilibrium,and the PDE system will show periodic oscillations which can cause spatially nonhomogeneous/homogeneous bifurcating periodic solutions.In addition,the direction of Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable.In chapter 3,the nutrient-microorganism reaction-diffusion model with time delay is considered.Firstly,by using the time delay τ as the bifurcation parameter,the characteristic equation is analyzed by linearization method,and the properties of the time delay τ are explored.The effect of time delay on the stability of the positive equilibrium for ODE and PDE systems is discussed,and the existence of Hopf bifurcation is studied.Secondly,the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are deduced.Finally,some numerical simulations are given by MATLAB to show the dynamic behaviors of system.In chapter 4,the diffusive nutrient-microorganism model without time delay is investigated.Firstly,for the reaction-diffusion system,choosing the parameter b as the bifurcation parameter,the existence of Hopf bifurcation is studied by using Hopf bifurcation theory,and the phase diagram is made to describe the properties of bifurcation parameter.Then,the stability and bifurcation direction of the spatially homogeneous bifurcating periodic solutions at b=b0H are judged by using the normal form theory and the center manifold theorem.Secondly,for the elliptic system,the existence of steady-state bifurcation at the positive equilibrium is studied by using the steady-state bifurcation theory,and some properties of the bifurcation parameter are discussed.In addition,for the reaction-diffusion system,the stability and bifurcation direction of the spatial non-homogeneous periodic solutions when a=ajH(j≥1)are studied by taking the parameter a as the bifurcation parameter.By using MATLAB,the numerical calculations show that the Hopf bifurcation direction of the system is supercritical and the spatial non-homogeneous bifurcation periodic solutions is stable for a=ajH(1≤j≤9).Finally,some numerical simulations are carried out,which shows that the system will produce limit cycles at b=b0H and spatial nonhomogeneous periodic solutions at b<b0H.In chapter 5,we give an induction of the paper.