Hopf Bifurcation And Turing Bifurcation Of Diffusion Predator-prey System With Time Delay

There are many relationships among various species in ecology,in which the predatorprey relationship is particularly important.Because this relationship promotes the flow of energy and biomass from low trophic level to high trophic level,and plays a role in regulating population size.The influence of the predator on the prey may be direct,indirect,or both.Direct influence means that the predator directly preys on the prey,and indirect influence refers to the fear of the prey to the predator during the predation process.At the same time,in order to better describe the relationship of predator and prey,time delay and diffusion mechanism should be considered in the modeling process.Thus,two kinds of multi-factor diffusion predator-prey systems are established in this paper,and the dynamic behavior of the two systems are studied.In Chapter 2,a toxic phytoplankton-plankton system with time delay and diffusion is established.Firstly,when delay and diffusion are not considered into system,the existence of all positive equilibria and its local stability are obtained.Secondly,when only time delay is considered,regarding time delay as the bifurcation parameter,the existence of Hopf bifurcation is obtained.And the direction of Hopf bifurcation and the stability of the periodic solutions are studied by using the center manifold theorem and normal form theory.Then,in order to study the original system,the existence of the Hopf bifurcation is gained.By using the center manifold theorem and normal form theory in partial differential equations,we study the properties of Hopf bifurcation.Finally,the correctness of the above theoretical results are verified by numerical simulations.In Chapter 3,a diffusive predator-prey system with time delay and hunting cooperation is established.First,when system does not include time delay and diffusion,the existence of all positive equilibria and their local stability are obtained.Secondly,the existence of Hopf bifurcation at the positive equilibrium is studied by taking the time delay as the bifurcation parameter,and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the center manifold theorem and the normal form theory in partial differential equations.Then,taking the intraspecific competition rate of the prey population as the bifurcation parameter,the conditions for the occurrence of Turing bifurcation is obtained by using the theory of Turing bifurcation.Furthermore,the corresponding amplitude equation is obtained by using the standard multi-scale analysis method.Finally,numerical simulations check the theoretical results.