Periodic And Asymptotic Properties Of Solutions Of Impulsive And Nonautonomous Mathematical Biological Models

The need for describing more actual natural system impels the evolution of mathematical biological models. In recent years, the researches in mathematical biology which modeled by normal differential equations are mainly concentrated on two branches: 1) continuous biological dynamical systems; 2) impulsive semi-dynamical systems. The discussions of continuous biological dynamical systems were main research direction in the past decades; autonomous systems were modified into nonautonomous since people found the factors which affect the systems can be various with the time or the subrogation of the four seasons. People also find recently that continuous biological dynamical systems can not represent some natural phenomena and control behavior of human accurately; impulsive semi-dynamical systems then turn out to be the hotspot of mathematical biology because the relatively instantaneous behavior mentioned above can be described well in impulsive differential equations. Our three models in this paper which have different applicative background respectively are of these two kinds of differential system.Since periodicity exists in nature and human society generally, it has also existed in these three models because of the effect of periodic environment and manual behavior. This dissertation discussed the given impulsive or nonautonomous mathematical biological models and studied the existence and globally asymptotic behavior of periodic solutions of these models. Moreover, the possible complexity of impulsive differential equations is discussed numerically by using software such as Maple or Matlab. The result of this dissertation can be summarized as following:Chapter two modified a state-dependent impulsive differential equations which has a background of pest control in agriculture by adding term of density dependence to make it more actual, and, also more difficult. The system became into the one which has not explicit solutions by this modification. Then we researched the existence of periodic solution of order one of this modified model by using theories of invariant set of impulsive semi-dynamical system and Brouwer's fixed point. Moreover, the attraction of periodic solution of order one and positive invariant of system has be discussed numerically.Chapter three considered an epidemic dynamics model with nonlinear infection rate, in which birth rate equals to death rate and the infectious individuals will get immunity naturallyafter recovery but will also lose it after a period of time. We also considered the factor of manual vaccination control. This paper analyzed the effect of continuous vaccination control and pulsive vaccination control of this system, and gave the reproductive number of the system with continuous vaccination as well as the system of pulsive vaccination. Three types of pulsive vaccinations are analyzed in this paper including proportional type, constant type and the second type of constant vaccination. The local stability and globally asymptotic behavior of border periodic solution (epidemic-elimination solution) of these three types of vaccination have been researched. On the other hand if the epidemic is turn out to be endemic, we studied numerically the influences of impulsive vaccination on the periodic oscillation of the system which is without impulsion and found phenomenon of chaos in this case.In chapter four, we considered two two-species nonautonomous models with stage structure. The models of population dynamics with stage structure can be applied well in management and usage of renewable resource. Nonautonomous system is more actual because it assumes that all the factories which affect the advance of system can be various with the time or the subrogation of four seasons. The first model considered in this chapter is a two-species nonautonomous competitive model with stage structure, in which one species competes with the mature individuals of anthor species. We studied the boundedness, permanence, existence and globally asymptotic stability of periodic solution of this model. The second model we considered is a two-species nonautonomous predator-prey model with stage structure and delay, in which predator species prey on only immature individuals of prey species. We studied the boundedness, permanence, existence of periodic solution of this model.