Asymptotic Behaviors On Population And Epidemic Systems With Impulsive Effects

The theories and methods of impulsive differential equations have got great development and become a whole system in recent thirty years. It is widely applied in various domains such as population dynamics, epidemiology, medicine dynamics, bio-cybernetics, biostatistics, genetics and chemical response. Our main purposes in the paper are to investigate the dynamic properties of predator-prey, Chemostat and epidemic systems with impulsive or periodic varying effects. In these models, by using the theories and methods of impulsive, discrete and continuous dynamical systems, arithmetic operators, optimization and numerical simulation, we study local and global bifurcations and asymptotical stability of their positive equilibria, existence and stability of periodic solutions, permanence, attractor, chaotic phenomenon and optimal control. The whole thesis is divided into five chapters.In chapter 1, we introduce concisely the present development of relevant subjects about population and epidemic dynamics as well as the main work done in this thesis. Moreover, we give some definitions and fundamental theories of differential equations and impulsive differential equations.In chapter 2, based on the background of pest control in agriculture, we put forward and investigate two three-dimension predator-prey systems. In section 2.1, we construct a two-prey one-predator system with Ivlev's functional response and impulsive perturbation on predator at fixed moments. Conditions for the extinction and permanence of the system are established via the comparison theorem. Numerical simulations are carried out to explain the conclusions we have obtained. Furthermore, the resulting bifurcation diagrams clear exhibit that the impulsive system takes on many forms of complexities including period-doubling bifurcation, period-halving bifurcation and chaos. In section 2.2, since a new species is provided with quite abundant food once it invades a new environment, we need describe its dynamic behavior with a ratio-dependent predator-prey system. In order to control the increase of the new species, we introduce its natural enemy into the new environment,thereby obtain a hybrid ratio-dependent three species food chain model, which is constituted by a hybrid type subsystem of prey and middle-predator and a middle-top predators' subsystem with Holling II functional response. For the model, we obtain condition for top-predator invading, the permanence of system, the stability of positive equilibrium and Hopf bifurcation occurring as the positive equilibrium loses its stability. Numerical simulations show that the continuing system exists the chaotic attractor.In chapter 3, we introduce and study three food chain Chemostat models, which are Monod-Haldene type with periodic varying input, Lotka-Volterra type with periodic varying dilution rate and Monod type with k times pulsed varying input. We investigate the behavior of the substrate bacterium subsystem and obtain the critical value condition for the prey and predator cultured successfully. Using the standard techniques of bifurcation theory, we prove the system exists a stable positive periodic solution as the bifurcation parameter value is more than the critical value but not very large. Furthermore, choosing the different bifurcation parameters, we numerically analyze the bifurcation and the complexity of the periodically variable and impulsive food chain Chemostat systems.In chapter 4, we discuss two SIRS epidemic models with saturated contact rate, respectively. In section 4.1, we study the SIRS epidemic model with pulse vaccination. Sufficient conditions for global asymptotically stability of the infection-free periodic solution and uniform persistence of this model are obtained. Using bifurcation theory the bifurcation parameter for existence of the positive periodic solution is given. In section 4.2, considering that people need go through a certain period before they get well if they fall sick, we add a delay item in the SIRS epidemic model with pulse vaccination and obtain a delayed SIRS epidemic model with pulse vaccination and saturated contact rate. By using the comparison theorem, we investigate the global attractivity of the infection-free periodic solution and the permanence of the system, and obtain the condition for the disease eradicated and becoming endemic.In chapter 5, we introduce a viral disease pest-control model with impulsive effect, which is based on constant impulsive releasing the infected to cause a viral disease in the susceptible pest. By using Floquet theorem and small amplitude perturbation method, we obtain the critical value for the stability of the suspected pest eradication periodic solution. Choosing the pest birth rate as bifurcation parameter and using the standard techniques of bifurcation theory, we prove that above this threshold but not very large, there exists a stable positive periodic solution. Furthermore, we numerically analyze the bifurcation and the complexity of the impulsive system. Numerical results show that the impulsive effect brings the system many complex phenomena, which are quasi-period oscillations, period-doubling and chaos.