Research On The Dynamics Of Predator And Prey System With Cooperative Hunting

Cooperative behavior within the populations is a widespread and important phenomenon in the ecosystem,which is conducive to the reproduction and survival of the populations,and can promote the sustainable survival of natural species.Many interesting biological phenomena in nature can establish appropriate population dynamics models and use mathematical theory to explain.For the cooperative behavior in the population,most of the theories focus on how the cooperative behavior of the prey population affects the dynamic behavior of the population,and little attention is paid to the predator population.Therefore,this paper mainly considers the predator-prey system for cooperative hunting.Firstly,the predator-prey system proposed by Alves M.T.et al.in J.Theor.Biol.in 2017 is analyzed theoretically and completely.It is found that the predator cooperative hunting intensity will affect the survival of the population.And by changing the initial value of the predator density to strengthen or weaken the intensity of its cooperative hunting,and then adjusting the coexistence state of the predator and the predator,so that the system tends to be stable.In this paper,the existence conditions of one or more coexistence equilibrium points of the system are given by changing the threshold of predator survival,Also we analyze the character of the equilibria.It is also proved that the system has Hopf bifurcation and the stability conditions of Hopf bifurcation are given.Secondly,predator-prey model with Smith growth and hunting cooperation is established In this paper.It is found that the hunting cooperation intensity and the net growth rate of the prey can affect the coexistence state of the population.And the condition that one or more coexisting equilibria exist in the system is given,then the system will present a bistable state,that is,the population either keeps stable coexistence or the predators are extinct,and the preys reach saturation;And Hopf bifurcation occurs at some equilibria,the population loses steady state,resulting in continuous predator-prey oscillation.When the two coexisting equilibria coincide,the system presents the BT bifurcation,showing a single steady state,and the predator extinction equilibrium will become the only stable state.At the same time,corresponding numerical simulations and biological interpretation are carried out.