The Qualitative Analysis Of The Biology System With Time Delay

In this paper the population dynamics model and eco-epidemic model have been discussed, based on the traditional differential model, we adding in time delay, and thus we propose the differential equations with delay. This paper arranged as the following chapters:In the first chapter, the background and development of population dynamics system are given, and the major work of this paper is introduced.In the second chapter, the Hopf bifurcation of a predator-prey system has been studied which changes the roles of predator and prey when the prey becomes adult and with delay effect, we factor in age structure as the delay to establish the corresponding delay differential equations. We have discussed the system boundedness and persistence. By applying the theorem of Hopf bifurcation, Hopf bifurcation occurs as the delay crosses the critical value. Numerical simulation supported the academic results.In the third chapter, the persistence of an eco-epidemic model with distributed delay is investigated. Time delay plays an important role in eco-epidemic model; it may be affect the extinction and permanence. At last, sufficient condition of the permanence of the model is obtained.In chapter four, an eco-epidemic model with infinite delay and integrated pest management is proposed, the sufficient conditions for the stability of pest eradication periodic solution and permanence of the system are obtained by using Floquet's theory and then rich dynamic phenomena such as bifurcation and chaos and so on was showed by numerical simulation.