Global Analysis In A Specialist (Or Generalist) Predator-prey System With Piecewise-smooth Holling-? Functional Response

In this paper,we study the global dynamics of a specialist(or generalist)predator-prey model with a piecewise-smooth Holling-I functional response function(p(X)=min(?X,M)).The full text is divided into two parts.In the first part?we study a specialist predator-prey model with piecewise-smooth Holling-I functional response function.First,we study the existence of boundary-equilibria and positive equilibria,and then discuss the type and stability of equilibria.When the equilibrium is located at the non-smooth boundary,we study its complex state by the method of "blow up".And then we construct the Dulac function to give the sufficient condition of the nonexistence of the closed orbits and the necessary condition for the existence of the closed orbits of system.Finally,we prove that there has saddle node bifurcation and transcritical bifurcation in the system.Our study shows that the prey will not extinct for all positive initial values,but for some positive initial values?specific predators tend to extinct.In the second part,we study the generalist predator-prey model with piecewise-smooth Holling-I functional response function.At this time,the predator logistic growth without prey.We find that the system always has three boundary equilibria and has at most three positive equilibria.Similarly,we study the type and stability of equilibria,the global stability of boundary equilibria or positive equilibria,the sufficient conditions for the nonexistonce of closed orbits of system,saddle node bi-furcation and transcritical bifurcation.Our study shows that the generalist predators will not extinct for all positive initial values,but for some positive initial values,the prey tend to extinct.