Qualitative Analysis Of A Class Of Predator-prey Model With Delay And Nonlocal Competition

From physics,chemistry to biology,reaction-di?usion equation has been widely used to describe various phenomena in modern science.Di?usion seems to be a process of ’trivialization’,in which any initial state,spread over a long period of time,always reaches a state of constant everywhere.Thus,Turing’s work of describing the formation of patterns in nature with reaction-di?usion in 1952 was a pioneering work.Population ecology is a branch of ecology.It is also a branch of ecology in which mathematics has been applied most widely and deeply,developed most systematically and mature.The predator-prey relationship is one of the most important relationships in ecology.More and more researchers wish to explore the relationship between them by establishing ordinary di?erential or partial di?erential models.In recent years,due to the wide application of biological models such as predator-prey model,the research on it has attracted the attention of many researchers.In order to make the model more realistic,we need to consider not only the current situation of the species population in the modeling,but also the ’historical’ state of the species population.For example,pregnancy,digestion and so on all have an impact on the current population growth rate.Therefore,introducing time delay into the model would be more realistic.In addition,based on their previous memories,predators tends to live in areas with high prey density and prey tends to stay away from area with high predator density.Prey’s perception of danger depended not only on the environment,but also on their own memories.Therefore,it is natural to introduce spatial memory into the model.The main work of this paper is as follows:In chapter 1,we introduce the research background,research status,research characteristics of this paper and introduce some definitions in this paper.In chapter 2,we investigate the stability and Hopf bifurcation of a di?usive predator-prey system with herd behaviour.The model is described by introducing both time delay and nonlocal prey intraspecific competition.Compared to the model without time delay,or without nonlocal competition,thanks to the together action of time delay and nonlocal competition,we prove that the first critical value of Hopf bifurcation may be homogenous or non-homogeneous.We also show that a double-Hopf bifurcation occurs at the intersection point of the homogenous and non-homogeneous Hopf bifurcation curves.Furthermore,by the computation of normal forms for the system near equilibria,we investigate the stability and direction of Hopf bifurcation.Numerical simulations also show that the spatially homogeneous and non-homogeneous periodic patters.In chapter 3,we introduce spatial memory into a predator-prey model with herd behavior.Taking the di?usion coe cient memory-based and the average memory period of predators as control parameters,we obtain the stable conditions of the positive equilibrium of system and prove the existence of Hopf bifurcation.In addition,a double Hopf bifurcation occurs at the intersection of the nonhomogeneous Hopf bifurcation curves,and a spatially nonhomogeneous quasi-periodic pattern can be observed near the double Hopf bifurcation point by numerical simulation.In chapter 4,we summarize the research results of chapter 2 and chapter 3.