Stability And Bifurcation Analysis In A Type Of Predator-prey Model

With the continuous exploitation of social resources,the ecological balance of nature is destroyed,and natural disasters also occur.In the process of exploring the population dynamics characteristics of natural species and the interaction between related populations,people pay more and more attention to the importance of sustainable development,which not only meets human needs but also maintains the ecological balance.In population dynamics,there are many kinds of relationships between two populations,but the most popular one is predator-prey relationship.In this paper,a predator-prey model with simplified Holling type IV functional response and the nonlinear harvesting term is studied,and the dynamic properties and numerical simulation results of predator-prey system under artificial hunting are discussed,so as to better explore the population structure and development trend.The structure of this paper is as follows:The first chapter is the introduction,which mainly introduces the research background,source,research status at home and abroad,and the main conclusions of this paper are given.The second chapter is the preparatory knowledge,which mainly introduces the stability judgment methods of the equilibrium and bifurcation methods used in this paper.The third chapter constructs a new predator-prey model by introducing the harvesting term,and studies the positiveness and boundedness of the solutions and the existence and stability of the equilibrium points of the system.The fourth chapter is to explore the bifurcation phenomena that occur at the nonhyperbolic equilibrium point and the degenerate equilibrium point,including transcritical bifurcation,saddle-node bifurcation,Hopf bifurcation and Bogdanov-Takens bifurcation.First,the Sotomayor's theorem is used to prove the existence of the transcritical bifurcation and saddle node bifurcation.Second,the Hopf bifurcation theory is used to obtain the existence conditions of the Hopf bifurcation,and then the parameter domains of the existence of subcritical Hopf bifurcation and supercritical Hopf bifurcation and the stability of the limit cycle are obtained by calculating Lyapunov constant.Third,by calculating the universal unfolding near the cusp of codimension 2,the existence of repelling and attracting Bogdanov-Takens bifurcations of codimension 2 and homoclinic bifurcation are proved.Finally,the existence of heteroclinic bifurcation is analyzed.Numerical simulations further verify the correctness of the results.