The Study Of Stability On A Biological Dynamical System With Time Delay

In recent years, with the development of science and technology, mathematics knowledge and thecniques have been introduced to the fields of biological science. Differential equation dynamic system has been widely used in the fields of environmental protection,infectious disease control and prevention,medicine dynamics,population ecology and microorganisms cultivation. Although the models with delays will make the dynamic properties of a system much more complex, it can truly reflect the real natural phenomenon. This paper mainly focuses on some types of the biological system.1,This chapter considers a type of infectious disease model with virus spontaneous mutation and delays. The delay is considered as a parameter to study the stability of the system. we analyze the local stability and the global stability of the disease-free equilibrium in detail through studying the characteristic equations and we at last consider the stability of the three equilibrium points under different conditions.2,In this chapter we descript a type of infection disease model with temporarily immunity and analyze the infection disease model with nonlinear incidence. Then we give the locally asymptotically stable conditions of the disease-free equilibrium and prove that no other equilibriums existing; At suitable conditions the stability of the disease-free equilibrium is not existing, but a local disease equilibrium will occur. Through analyzing the relative characteristic equation, we conclude that the stability switching phenomenon of this local disease equilibrium may took place when the parameter and the delays have been restricted.3,In this chapter we study the stability of a prey system with delays. We judge the local stability of positive equilibrium through considering the root's positive, negative and size of the characteristic equations of the linearing system at equilibrium point; and we get the sufficient condition of the global stability at positive equilibrium by constructing suitable V function.