Bifurcation Analysis On Several Predator-prey Models

Predation and prey,competition and mutual benefit among species are the basic relationships of common species in nature.Studying the dynamic behavior of predator and prey systems will help us to understand and grasp these relationships.The Rosenzweig-Macarthur(RM)model describes this ecological process well.Taking more external factors into Rosenzweig-Macarthur model can make the model more realistic.Through the research and analysis of predator-prey model,it can play a better role in reminding and guiding people to protect and transform the environment,reveal the changing law of biological population,and predict,manage and regulate the ecosystem.It is of great significance to realize rational utilization of resources,promote ecological balance and maintain the sustainable and stable development of ecological environment.This paper mainly studies the local dynamics and global dynamics of RosenzweigMacarthur model and host-generalist parasitoid model under different conditions,the full thesis is divided into five chapters:The first chapter is the introduction,which explains the research background and current situation at home and abroad,then briefly introduces the research contents of this paper,and lists the basic definitions and theorems involved in this thesis.In the second chapter,we studies the dynamics of Rosenzweig-MacArthur predator and prey models with strong Allee effects and hyperbolic tangent forms(called trigonometric)as a function of predator response,including predator and prey.Firstly,the nullcline of the model,and the stability of the equilibrium point of the model are analyzed.The Hopf bifurcation is judged according to the analysis of nullcline,the existence of Bogdanov-Takens bifurcation and heteroclinic orbit,as well as the coexistence of limit cycle and heteroclinic cycle are proved.Finally,numerical simulation is used to verify the theoretical results.In the third chapter,we study the dynamic behavior of Rosenzweig-MacArthur models with predator intraspecific competition and Holling type II functional response functions with prey refuge.We study the number of positive equilibria,the local and global dynamics including Hopf bifurcation,saddle-node bifurcation,Bautin bifurcation and Bogdanov-Takens bifurcation.The stability and the direction of Hopf bifurcation are proved.The normal form of Bogdanov-Takens bifurcation at the positive equilibrium is derived.In particular,the coexistence of stable limit cycles and unstable limit cycles are also given.Interestingly,the famous Hydra effect appears in this system,and the hydra effect is found which describes the positive effect of the death rate of predators having on the population abundance.Particularly,the positive effects of prey refuge,intraspecific competition among predators are also found,which are similar to the hydra effect caused by the death rate of predators.Multiple hydra effects could occur without the bistability or Allee effect.Further,numerical simulations are used to demonstrate the theoretical results including the existence of the hydra effect region.Finally,some conclusions and discussions are given.In the fourth chapter,we study the dynamic behavior of a host-generalist parasitoid model with Allee effect.The stability of the boundary equilibrium,the existence condition and quantity of the positive equilibrium is obtained,and by using the Lyapunov function to prove the global stability of the positive equilibrium.At the same time,the existence conditions of Hopf bifurcation,Bogdanov-Takens bifurcation and Bautin bifurcation are obtained.Finally,numerical simulation is used to prove the previously discovered results and draw some relevant conclusions.The fifth chapter is a comprehensive summary of the research results,and the future research direction is also prospected.