The Influence of Generalist Predator and Michaelis–Menten Harvesting in a Holling–Tanner Model

In this paper, a Holling–Tanner predator–prey model with generalist predators and Michaelis–Menten-type prey harvesting is investigated. We analyze the existence and stability of equilibria and find the system has at most three positive equilibria. The double positive equilibrium belongs to the cusp type, with its codimension being at least 5. We then prove that the triple positive equilibrium is either a nilpotent focus (or elliptic point) of codimension 3, or a nilpotent elliptic equilibrium with codimension no less than 4. Additionally, the system undergoes two types of bifurcations: a cusp-type degenerate Bogdanov–Takens bifurcation (codimension 3) and a Hopf bifurcation. Using numerical simulations, the system has two limit cycles, which indicates that Michaelis–Menten-type prey harvesting makes the system’s dynamics more complex.