Stability And Bifurcation Analysis Of Two Kinds Of Dynamical Models With Time Delay

Delay differential equations are widely used in building bio-models with the development of biomathematics.On one hand,researchers should use the mathematical model of ordinary differential equations to study the current state of the dynamic system,on the other hand,it is necessary for people to pay special attention to the influence of time delay on past states.Considering the time delay play a pivotal role in dynamic systems,introducing the delay differential equation is vitally importance.To date,the study of stability and branching problem of time-delay dynamic systems plays a significant role in related practical applications.It's not only involves classical dynamic system theory,algebra theory,functional method and other related knowledge,but also need researchers have a deeper understanding on biology,epidemiology,oncology,and neural networksWe use some important theories and methods including Lyapunov stability theorem,LaSalle invariant set principle,Hopf bifurcation theorem together with numerical simulations method to study the dynamic properties of two kinds of dynamic models with time delay.The main contents are summarized as followsIn Chapter 2,we study a class of SIR epidemic model with logistic growth and obtain the threshold value R0,which determines the extinction and outcome of the disease.When Ro<1,we prove that the disease-free equilibrium is global asymptotically stable for any delay ?,it is means that the disease is died out in the real life.When R0>1,there will be there cases:when ?<?0,endemic equilibrium is unstable;when ?>?0,endemic equilibrium is local asymptotically stable under;When ?=?0,Hopf bifurcation occurs.The results of numerical simulations are performed to verify the conclusions.Also,we make some sensitivity analysis of parameters.In Chapter 3,we consider a dynamic model of oncolytic therapies with time delay.Based on LaSalle invariance principle,we obtain that the boundary equilibrium is global stability when b<1+?(K+?)/?K for any time delay ??0.At the same time,the sufficient condition for producing Hopf bifurcation at the positive equilibrium point has been given,which indicates that the system has periodic solution,the number of oncolytic viruses and tumor cells change periodically.Finally,the numerical simulations show that our mathematical findings are accurate.