Stability and bifurcation analysis of a Michaelis–Menten-type predator harvesting predator–prey system with strong Allee effect on prey

In this paper, a Michaelis–Menten-type predator harvesting predator–prey system with strong Allee effect on prey is proposed. The boundedness of the system is studied, and the existence of all possible equilibria is investigated. In particular, the system has at most four positive equilibria. The stability analysis of the coexistence positive equilibria is studied with the help of the graphical Jacobian method. Under some conditions, the bifurcations exhibited by the system are researched. With the support of Sotomayor’s theorem, saddle-node bifurcation and transcritical bifurcation are proven. The existence and the direction of the Hopf bifurcation have also been determined. The universal unfolding of the Bogdanov–Takens bifurcation of codimension 2 is calculated near the cusp. Our theoretical analysis is verified by numerical simulations. The results show that the strong Allee effect plays an important role in the stability of the system with respect to the bifurcation phenomenon, compared to the system without the strong Allee effect.