Bifurcation Analysis Of A Predator-prey Model

Through the theoretical analysis of predator-prey model and numerical simulation,we can know that there is a kind of existence mode which is both dependent and restricted between predator and prey populations in nature.The analysis of these models provides reference and guidance for the protection of population,the decision of economic benefit,the development and management of nature,etc.This paper mainly considers a class of predator-prey models with constant harvest rate and fear effect of prey population.The existence and stability of equilibria,Hopf bifurcation and Bogdanov-Takens bifurcation at positive equilibrium points are discussed in detail by using differential equation qualitative and bifurcation theory.The Leslie-Gower predator-prey model and the predator-prey model with Holling class II functional response are studied respectively.For Leslie-Gower predator-prey model,we first discuss the conditions of the existence of boundary equilibrium points and positive equilibrium points of the system,and analyze the local stability of equilibrium points.After that,the direction of the Hopf bifurcation at the positive equilibrium point and the stability of the limit cycle are discussed,and the numerical verification is given.Finally,we get the conditions of Bogdanov-Takens bifurcation near the positive equilibrium point with the corresponding saddle-node bifurcation curves,Hopf bifurcation curves and homoclinic bifurcation curves.For another predator-prey model with Holling II functional response,the existence and stability of the equilibria are discussed,and the conditions of the existence of Hopf bifurcation at the positive equilibrium point and the stability of the nearby limit cycle are analyzed.Finally,the Hopf bifurcation near the positive equilibrium point of the system are simulated by two specific sets of parameter values.