Dynamics of a Harvested Leslie-Gower Predator-Prey Model with Simplified Holling Type IV Functional Response

This paper investigates a Leslie-Gower predator-prey model with simplified Holling type IV functional response and constant-yield prey harvesting. We analyze conditions for the existence of positive equilibria and prove that the system has at most four positive equilibria. The results show that the double positive equilibrium is a cusp of codimension at most 4, the triple positive equilibrium is a degenerate saddle or nilpotent focus of codimension-3, and the quadruple positive equilibrium is a nilpotent cusp of codimension-5. In addition, as the parameters vary, the system can undergo a cusp-type (or focus-type) degenerate Bogdanov-Takens bifurcation of codimension-4 (or codimension-3). Furthermore, the positive equilibrium is a weak focus of order at most 4, and the model can undergo a degenerate Hopf bifurcation of codimension-4. Finally, our main results are verified by some numerical simulations, which also reveal that there exist three limit cycles containing one positive equilibrium, or one (or two) limit cycles containing three positive equilibria, or a limit cycle as well as a homoclinic loop.