Qualitative Analysis Of A Predator-Prey Model With Holling ? Functional Response

Population dynamics model is a dynamic model describing the interaction between population and population and between population and environment.Its display mor-phology is mainly manifested in the change of the number of the population with time,and it can accurately describe biological phenomena and predict its development rules.The prey-predator model with functional response function is more in line with the actual background of population.The introduction of time delay will lead to the loss of stability and the generation of bifurcation of the system,and the emergence of bifurcation will complicate the law of natural development.If time delay is neglected,the results will be seriously affected.This thesis is arranged as follows:The first chapter briefly describes the research background,current situation and progress of the predator-prey model with functional response function,the main contents are pointed out and some definitions and theorems used in this paper are introduced.Chapters 2 and 3 consider respectively a predator-prey system with Holling-V func-tional response and no density restriction for predator and the Holling ? predator-prey system with double density Restrict in_+~2.The existence of positive equilibrium by using the equal dip line properties,and the global asymptotic stability of positive equilibrium by the Lyapunov function method are analyzed.Furthermore,we obtain the conditions of existence and uniqueness of limit cycles through Bendixson theorem and Zhang Zhifen uniqueness theorem.Finally,numerical simulations show the validity of our results.In Chapter 4,a class of time-delay Holling ? predator-prey system with stage struc-ture is discussed.Firstly,the local asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation are studied by qualitative analysis.Then,the Hopf b-ifurcation direction and the stability of the bifurcation periodic solution are studied by using the normal form method and the center manifold technique for delay ordinary dif-ferential equation.Finally,numerical simulations are also provided in order to check the obtained theoretical conclusions.