Hopf Bifurcation Analysis Of Two Kinds Of Nonlinear Delay Differential Equation Models

In this paper,we mainly study the stability of two kinds of nonlinear delay differential equations at positive equilibrium,the conditions for the existence of Hopf bifurcations and the direction and properties of Hopf bifurcations.The full text is divided into four parts.The first part introduces the background and significance of this research,the research status at home and abroad and some related theoretical knowledge,including system stability and some related definitions and theorems used in Hopf bifurcation theory.In the second part,we study the dynamics of a class of population models with time-delay in polluted environment.Considering the two delays of population response to poisons and poisons intrusion into human body,an appropriate model for describing the system is proposed.By using the stability theory of functional differential equation and Hopf bifurcation theory,the local stability of positive equilibrium points,the conditions for the occurrence of Hopf bifurcation and thedirection and properties of the bifurcation are analyzed.Finally,the conclusions are verified by numerical simulation.In the third part,we consider a predator-prey system with two delays and four groups.The stability theory of delay differential equation is used to analyze the local stability of positive equilibrium point and the conditions for the existence of Hopf bifurcation.Finally,numerical simulation is carried out.The fourth part is the overall summary of the work of this paper and the outlook for the future.