Dynamical Analysis Of Two Classes Of Predator-prey Models With Time Delay

In this paper,we mainly study the dynamics of two classes of predator-prey model with time delay.The article includes three chapters.The preface is in chapter 1,we introduce the research background of this article,the main task and some important preliminaries.In Chapter 2,This section is concerned with zooplankton-phytoplankton-toxic systems with the Holling-II functional response.we induced a gestation delay to the Holling-II functional re-sponse function to describe the delay in the conversion of nutrient consumed to species.Firstly,the existence and asymptotically stability of the equilibrium were researched;Next we discussed the occurrence of the Hopf bifurcation by choosing the time delay ? as a bifurcation parame-ter and obtained the properties of Hopf bifurcation such as the direction and the stability by using the Poincare normal form theory and center manifold theorem;Finally,some numerical simulation are presented to justify the theoretical results.In Chapter 3,a predator-prey-mutualist system with digestion delay is studied.We prove that the positive equilibrium is locally asymptotically stable when 0 ??<?0 and the system undergoes a Hopf bifurcation at the positive equilibrium when ? = ?0.By applying the Poincare normal form theory and center manifold theorem,we investigate the properties of Hopf bifurca-tion,such as the direction and stability.Some interesting numerical simulations are carried out to verify the main theoretical conclusions in the last.