Global dynamics of a predator-prey model with a Smith growth function and the additive predation in prey

Abstract

We propose and study a predator-prey model with a Smith growth function and the addition predation term described by a Holling Type Ⅱ functional response in prey. This additive predation term can lead to Allee effects in prey population dynamics that can generate complicated dynamics in the corresponding predator-prey model. We provide a through analysis of the global dynamics of the proposed model, including the equilibrium stability, Hopf bifurcation and its directions, existence of a heteroclinic orbit loop and limit cycles. We show that when the predator-prey model exhibits Allee effects, Hopf bifurcation is either backward and supercritical or forward and subcritical. In the strong Allee effect case, the model has a heteroclinic orbit loop connecting two boundary saddle points. Our results show that the coexistence can be achieved by controlling the attack rate of other potential predators so that the model exhibits weak Allee effects or no Allee effect. Both the small additional predation rate and the large replacement rate of mass can improve the coexistence probability of two species. The main difference of dynamics between the model exhibiting weak Allee effect and no Allee effect lies in the pattern of coexistence: If no Allee effect, the coexistence can be a steady state while in the weak Allee case, the coexistence may be periodic.