Dynamical effects of herd behavior and constant harvesting in a Leslie-type predator-prey model: A bifurcation analysis

In this paper, we investigate a Leslie-type predator-prey model with prey group defense and constant predator harvesting. First, we establish conditions for the existence and stability of all feasible equilibria of the model. We then derive a critical harvesting threshold for the existence of interior coexistence equilibria. A detailed stability analysis of the positive equilibrium shows that the two species can coexist persistently for appropriate parameter values and initial conditions. Furthermore, we systematically analyze the saddle-node and Hopf bifurcations exhibited by the model. The direction and stability of the Hopf bifurcation are determined by computing the first Lyapunov coefficient. By constructing a universal unfolding near the cusp point, we further prove that the model undergoes a codimension-two Bogdanov-Takens bifurcation. Finally, numerical simulations are performed to illustrate our theoretical results.