Global Dynamics Of A Predator-prey System With Cooperative Hunting

Each organism occupies a certain position in the food chain,and each organism depends on and restricts each other.The predator-prey relationship is one of the most basic relationships among populations.Social interaction between individuals within a population is an composition part of the life characteristics of many species,and cooperative hunting within a population in particular is a widespread and important phenomenon in ecosystems.More and more research results show that cooperative hunting plays an important role in the evolution of populations and the stability of ecosystems.In this paper,we study the dynamical behavior of a predator-prey system with cooperative hunting.We divide the parameter space of the system into a number of di?erent regions.In each parameter region,the global dynamic of the system is studied,including the existence of equilibrium points,stability,Hopf bifurcation and its direction,and the existence of limit cycles,etc.Numerical simulations are used to confirm our theoretical results.Finally,by comparing with the system without cooperative hunting,it is found that cooperative hunting beneficials the coexistence of predators and preys.Our results also show that when predator mortality is small,cooperative hunting does not a?ect population coexistence,but does a?ect the coexistence form,which exhibits either a steady-state pattern(the inner equilibrium point is globally asymptotically stable)or an oscillatory pattern(the system has a stable limit cycle).The main contents of this paper are as follows:The first chapter is the introduction,which introduces the research background,significance and development status at home and abroad of the predatorprey system of cooperative hunting,as well as the system studied in this paper and its main research contents.In chapter 2 analyzes the positive invariance,boundedness and the exact number of equilibrium points in different regions of the system.In chapter 3,the stability of equilibrium and bifurcation are studied.We select the death rate of predation population as the bifurcation parameter,discuss the stability of equilibrium point,give the explicit bifurcation value of Hopf bifurcation,and prove the existence of a supercritical and backward Hopf bifurcation.In addition,the saddle-node bifurcation and Bogdanov-Takens bifurcation of the system are proved by using the central manifold theorem and blow-up technique.In chapter 4,by theoretical analysis and numerical simulation,the global stability and the existence of limit cycles are considered,and the global dynamic of the system in each parameter region is given.In chapter 5,the main contents of this paper are summarized and compared with the famous Lotka-Volterra model(i.e.predators have not cooperate hunting).