Hopf Bifurcation And Periodic Solutions For Several Biological Models

Recently,dynamical systems are applied extensively in many fields such as Mechanics,Physics,Chemistry,Biology,Ecology,Control,Numerical calculations, Engineering technology,Economics and social sciences etc,and the study of stability and bifurcation has been one of eternal subjects in differential equation.In this thesis,we firstly study stability and Hopf bifurcation of two models with stagestructure and Holling typeⅡfunctional response.Finally,we consider existence of multiple periodic solutions of a class of impulsive neutral delay differential equation.In Chapter 1,we mainly introduce the background and development situation of the subject,and give some theoretical tools and preliminaries serving the discussion in the paper.In Chapter 2,a predator-prey system with stage-structure for predator is presented and studied,whose coefficients are delay-dependent.The stability and Hopf bifurcation of positive equilibrium are studied by using Hopf bifurcation theorem and Geometric switch criterion,which shows that the positive equilibrium is from stable to unstable to stable as time delay increases and Hopf bifurcation occurs. Further,an explicit formula for determining the stability and the direction of periodic solutions bifurcating from positive equilibrium is derived by the normal form theory and center manifold argument.Finally,some numerical simulations are also given to illustrate our analytic results.In Chapter 3,we study a predator-prey system with stage-structure for prey, whose coefficients are also delay-dependent.The local and global stability of boundary equilibria are studied,respectively.More importantly,the stability and Hopf bifurcation of positive equilibrium are studied by using Hopf bifurcation theorem, which shows that the positive equilibrium is always stable if f(0)＞0 and Hopf bifurcation occurs if f(0)＜0,the positive equilibrium is from unstable to stable as time delay increases.Finally,some numerical simulations are also given to illustrate our analytic results.Impulsive delay differential equation may express several real-world simulation processes which are subject to short time disturbances.In recent thirty years, the theory of impulsive delay differential equation has been developed.Hence,in Chapter 4,a class of impulsive neutral delay differential equation is studied,which includes many nonlinear biology models.By using fixed point theorem in cone,some sufficient conditions of the existence for multiple periodic solutions are obtained.