Qualitative Study Of Several Classes Of Delay Differential Equations

Based on the theory of linear functional analysis and the techniques of qualitative and stability analysis of differential equations,this dissertion studies the dynamical behaviors of several classes of delayed differential equations.Our results are the complement or improvement of the previous literature.In Chapter 1,we give the background,previous works and significance of the models described by delayed differential equations.In addition,we outline our motivations and works in this chapter.In Chapter 2,in view of the effect of delay in the transmission of infectious diseases,we establish a class of infected bacteria model described by delayed reaction diffusion equations under Dirichlet boundary condition.By developing the basic theory of partial functional differential equations and applying the method of the upper and lower bounds of the estimated solutions,we give some properties of solutions of the models,such as the existence,uniqueness and convergence.Thus,with the help of our analysis of dynamical behavior for the model,the mechanism of infectious diseases is showed.Moreover,our conclusion will provide a theoretical support for the prevention and control of infectious diseases in practice.In Chapter 3,by introducing time delay influence,we establish a class of delayed reaction-diffusion systems under the Neumann boundary condition.By developing the iterations of interval mapping and analyzing techniques of dynamical system and improving the global attractiveness theory of interval mapping,we obtain the positive invariant set and the attracting domain of the system,and further sufficient conditions under which the equilibrium is locally or globally attractive.Furthermore,we show the transmission mechanism of the “infectious force”of bacteria and infected people in some infectious disease models with different nonlinear “bacteria to the population”,which will be helpful for the prevention and control of infectious diseases.In Chapter 4,in order to explain the propagation mechanism of infectious diseases,we study the bifurcation problem of Neumann boundary value problem with reaction-diffusion system with constant incubation period.By deepening the research methods and techniques for bifurcation problems of functional differential equations without diffusion terms,the existence and no-existence of Hopf bifurcation around the positive equilibrium of the reaction-diffusion systems with Neumann boundary condition and constant incubation period are obtained.In Chapter 5,in the framework of Filippov method,we first introduce the effects of discontinuous functions.Then by constructing a new generalized Lyapunov function,we obtain the existence and global exponential stability of positive periodic solutions for a model of hematopoiesis with time-varying delays and discontinuous harvesting.Moreover,as numerical simulations,an example is presented to demonstrate the effectiveness and feasibility of theoretical results.Finally,we give a summary of the thesis work and prospects to the further research.