The Stability Of Positive Equilibrium And Analysis Of Hopf Bifurcation For Several Classic Of Delay Differential Systems

There exist delays in the various activities of life, such as the processes of gene transcription, translation and protein formation of gene regulation system. Some research shows that the delay will increase the stability of systems of gene regulation and make it appear new features.In this paper, the Hopf bifurcation problems of three types of the gene regulatory networks model with time delay are investigated. By choosing the delay as a bifurcation parameter, it is shown that Hopf bifurcation occurs as the delay passes through a sequence of critical values. Finally, using of standard theory to simplify the system, and using of Hopf bifurcation theory gives direction and stability of bifurcation formula. This article consists four sections.Chapter one is an introduction, Introduced topic background and meaning, Time delays differential calculus square the distance bifurcation and application of research present condition, textual and main contents and this text research problem demand of some prepare knowledge.In chapter two, it constructed a two-delay model of gene expression model. And then by regarding two time delays as bifurcation parameter, Firstly, analysis and determination of the stability of positive equilibrium is discussed, Later the use of normal form and center of the Hopf bifurcation theorem popular direction and stability of bifurcation formula.In the chapter three, the Hopf bifurcation of three genes expression model with three time delays have been investigated. By regarding three time delays as bifurcation parameter and analyzing the characteristic equation of the linear system of the original system at the positive equilibrium, local stability of positive equilibrium and existence of Hopf bifurcation are constructed. Finally, theory has been popular with central direction and bifurcation branch expression.In the chapter four, it constructed a class of gene interaction models with a single time delay. Taking time delayτas the bifurcation parameter, the stability and existence of Hopf bifurcation are obtained. Firstly, we analyze the stability of positive equilibrium. By Center Manifold Theorem and Standard Methods, Hopf bifurcation are derived, which determines the direction and stability of bifurcating periodic solutions formula.