Stability And Hopf Bifurcation For Two Classes Of Delay Differential Equation Models

In this paper,two classes of delay differential equation models are considered.By using the stability theory of delay differential equation and numerical simulation,we study the dynamics behavior for the two classes of delay differential equation systems.That include the stability of thesystems,the existence of the Hopf bifurcation and the bifurcation properties.This article is divided into four chapters.In chapter one,the background and significance,our main works,and some basic knowledge of delay differential equation are introduced.Inchaptertwo,the existing model is corrected by considering the infection delay.Then,by the stability theory of delay differential equation,the stability and Hopf bifurcation for the positive equilibrium are studied.Finally,the numerical simulations are carried out to explain our theorems.In chapter three,we propose the Logistic model with multi-delay by considering the environmental factors and explain the realistic signification.Then,by using the stability theory of delay differential equation and the zero point theorem of continuous functions we study the characteristic equation of the linear system.Furthermore,by using the center manifold theorem and normal form theory,the existence of the Hopf bifurcation and the bifurcation properties are also obtained.Finally,our theorems are verified by some groups of numerical simulations.In chapter four,we do a summary description for our research.