Stability and Bifurcation Analysis of a Predator–Prey Model with Allee Effect and Predator Harvesting

In this paper, we investigate a predator–prey model with Holling-II functional response, Allee effect and constant-yield predator harvesting, by comparing the differences between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the harvesting rate of predators. For system without harvesting, the Allee effect leads to population extinction. The system has at most one positive equilibrium and has a supercritical Hopf bifurcation which depends on the natural mortality rate of predators. Besides, by using normal form theory, we show that the system with [Formula: see text] reveals rich dynamic properties, including saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, where numerical simulations are presented to demonstrate the Bogdanov–Takens bifurcation of codimension 2 with a limit cycle and a homoclinic cycle. The system can generate up to two positive equilibria with the changes of [Formula: see text], which indicates that appropriate predator harvesting can assist in regulating the ecosystem. We then give the optimal harvesting strategy by using Pontryagin’s maximum principle. Finally, numerical simulations are performed to validate the functions of Allee effect and harvesting. Theoretical studies and numerical simulations demonstrate that the Allee effect can lead to species extinction and highlight the role of appropriate harvesting in controlling the stability of the system.