Modelling And Research Of Nonlinear Dynamical Systems With Time Delay

Recently, stage-structured population dynamics models have attracted much attention in mathematical biology fields. This is not only because they are simpler than the models governed by partial differential equations, but also they can exhibit phenomena similar to those of partial differential models, and many important physiological parameters can be incorporated. Acquired immunodeficiency syndrom (AIDS), which is caused by the human immunodeficiency virus (HIV), is a kind of immunodeficiency desease. Many infected individuals usually go through longer incubation period (5-12 years) before they develop to AIDS case. HIV can cause infected individuals progressing to weakness of immune function and progressing to AIDS at last. Infected individuals will die of opportunity infection. From the initial simple models of May and Anderson, modelling and analysis on HIV/AIDS transmission have caused many scholars interested. The development of such models is aimed at both better understanding of the observed epidemiological patterns and providing a theoretical basis for prediction and control of HIV/ AIDS incidence.In the light of the recent work in these two kind of biological models, the dissertation provides a systematic study on the dynamical behavior of the nonlinear dynamical systems (the stage-structured predator prey systems and HIV/AIDS transmission systems) built by delay differential equations.The main contents and results in this dissertation contain two parts: Part one (contains Chapter 2-Chapter 3) establishes systematically the predator prey model with stage structure and mutual interference and the stage-structured predator prey model with time delay for the first time. By constructing Lyapunov functions, using LaSalle invari-ance principle, uniform persistence of infinite dimensional systems, characteristic equation theory and bifurcation theory, etc., we discuss the local or global asymptotic behavior of the equilibria, permanence of the system and bifurcation phenomena. The sufficient conditions under which the equilibria are locally or globally asymptotically stable, and the populations are permanent or extinct, Hopf bifurcations occur, etc., are obtained.Part two (contains Chapter 4-Chapter 6) establishes systematically the HIV/AIDS transmission model with treatment, HIV/AIDS transmission model with delay and nonlinear incidence and HIV/AIDS infection of CD4+ T-Cells model with delay for the first time. By using the matrix theory, the generalized Bendixson-Dulac theorem, the properties of measurable function in functional analysis, fluctuation lemma, the qualitative and stability theory, characteristic equation theory, persistence theory and bifurcation theory, etc., we systematically discuss the stability of the equilibria, the persistence of the disease, the delay effect on stability of the system and bifurcation phenomena, etc. The threshold (the basic reproductive number) of the models, the conditions of the disease persistence or extinction, stability of equilibria and bifurcations occurrence of the systems are obtained.The main contents and innovations in this dissertation can be summarized as the following:In chapter 1 (introduction), the background, significance of the theory and practices, research ongoings at Home and Abroad, main contents and methods in this dissertation are briefly introduced.In chapter 2, based on the effect of intraspecific interference among predators and stage structure on the predator prey systems, the predator prey model with Beddington-DeAngelis functional response is established for the first time. By using the uniform persistence theory for infinite dimensional systems and comparison arguments, we discuss the uniform persistence and global asymptotic stability of the boundary equilibrium, and obtain the conditions of permanence and extinction for the populations. By constructing Lyapunov functions and using LaSalle's invariance principle, we analyze the stability of equilibria, and obtain the sufficient conditions under which the equilibria are locally asymptotically stable. By means of the iterative methods, we obtain the sufficient conditions under which the positive equilibrium are globally attractive.In Chapter 3, based on the effect of the stage structure and delay of the population gestation on the classic Lotka-Volterra predator prey, the predator prey model with time delay is established. By qualitative and stability analysis, constructing Lyapunov functions and using LaSalle's invariance principle, the sufficient conditions of permanence and global asymptotic stability of the equilibria are obtained. By characteristic equation theory and the normal form theory and center manifold argument, the distributions of the roots of the state dependent characteristic equation are analyzed, and the conditions of the stability switches and of Hopf bifurcations occurrence in the system are obtained.In Chapter 4, by using some methods in matrix theory and bifurcation theory in delay differential equations, we discuss the dynamical behavior of the HIV/AIDS transmission model with treatment for the first time and obtain the basic reproductive number of the model and the sufficient conditions that determine the endemic equilibrium is locally asymptotically stable and Hopf bifurcation occur in the system. By using the generalized Bendixson-Dulac theorem, the global asymptotic stability of the endemic equilibrium is proved. The obtained results in theory via computer simulation are examined.In Chapter 5, by using the properties of measurable function in functional analysis and fluctuation lemma, the theory of delay characteristic equations, the qualitative theory in delay differential equations, etc., the dynamical behavior of the HIV/AIDS transmission model with nonlinear incidence and time delay due to incubation period are discussed. the basic reproductive number of the model is obtained. The global asymptotic stability of the infection-free equilibrium is proved. The local asymptotic stability of the endemic equilibrium is analyzed by means of the theory of delay characteristic equations. By qualitative analysis, the sufficient conditions for the uniform persistence of the disease are obtained. The effect of the delay on disease transmission is also discussed. The obtained results in theory via computer simulation are examined.In Chapter 6, by using the qualitative and stability theory, bifurcation theory in delay differential equations, etc., the dynamical behavior of the HIV/AIDS infection of CD4+ T-cell model with the proliferation process of infectious cells and delay due to emission new viral particles is analyzed systematically. The threshold that viral particles are released by each lysing cell is obtained. The sufficient conditions that can determine the system is persistence, and the infection equilibrium preserves its stability, and Hopf bifurcations of the system occur, etc., are given. The explicit expression of delay length estimation is estimated by Nyquist criteria. The obtained results in theory via computer simulation are examined.