Study On The Biology Dynamic System With Time Delay And Impulsive Effect

Population dynamics is a very important branch of biomathematics. In this paper, the population dynamics models with impulsive and time delay on the base of traditional differential model have been considered. This paper arranged as follows:In the first chapter, the development of population dynamics system are given, and the major work of this paper is introduced.In the second chapter, a delayed stage-structured predator-prey model with the immature prey are taken care of by their parents and biological control strategies (that is releasing natural enemies at different fixed time) is considered. By using comparison theorem, the existence of the globally asymptotically stable prey-eradication periodic solution when the number of releasing natural enemies more than some critical value is proved. The conditions for the permanence of the system are given. Numerical simulation proved the academic results.In the third chapter, the stability of a delayed stage-structured predator-prey model is considered. Delay is existed popularly on the action of populations, and it operated important role with populations, it may be affect the stability of positive equilibrium. By using Roth-Hurwitz criterion discussing the existence and stability of positive equilibrium.In the last chapter, a modified Holling-Tanner predator-prey model is studied. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs whenτcrosses some critical value. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and center manifold theorem. From calculate, the characters of Hopf bifurcation on the critical valueτj are obtained. By using Numerical simulation explained the stability of a positive equilibrium, and the rich dynamic behaviors of Hopf bifurcation.