Bifurcation Analysis For A Predator-prey Model With Holling Type ? Functional Response

In recent years,with the development of biological mathematics,many scholars have focused on the dynamic properties of predator-prey model.Introducing appropriate functional response function,time delay,diffusion and nonlocal competition terms into the predator-prey model can better describe the dynamic behavior of the population.In this paper,we discuss a predator-prey model with Holling type ? functional response,and analyze the dynamic properties of the model without or with nonlocal competition,respectively.On the one hand,the dynamic properties of the predator-prey model without nonlocal competition are analyzed.First,the stability of equilibrium without delay and diffusion is discussed.The existence and properties of Hopf bifurcation at the positive equilibrium are analyzed with the predator's hunting ability as a parameter.Second,the delay and diffusion are introduced into the model,and the influence of parameters on the existence and value of Hopf bifurcation is discussed.Finally,the corresponding theoretical results are illustrated by numerical simulations.On the other hand,we further study the dynamics of the predator-prey model with nonlocal competition.First,by analyzing the distribution of the roots of the characteristic equation at the steady state solution of the model,the stability conditions of the steady state solution of the model and the sufficient conditions for the existence of Hopf bifurcation are given.Second,according to the center manifold theory and the normal form method,the Hopf bifurcation properties at the steady state solution of the normal value are analyzed,including the direction of bifurcation and the stability of the bifurcating periodic solution.Finally,when the Hopf bifurcation of the model is analyzed,it is found that the model have a double Hopf bifurcation point when the time delay and the time required by the predator to deal with the prey change.In the two-parameter plane,the double Hopf bifurcation normal form of the predator-prey model with nonlocal competition is further derived.It is found that there are stable spatially homogeneous periodic solutions and spatially inhomogeneous periodic solutions near the double Hopf bifurcation point.