Bifurcation Analysis of a Generalist Predator-Prey Model with Holling Type II Harvesting

In this paper, we consider a generalist predator–prey model with nonlinear harvesting, which has at most eight non-negative equilibria. We prove that the double positive equilibrium is a cusp of codimension up to 3; therefore, the system exhibits a cusp-type degenerate Bogdanov–Takens bifurcation of the same codimension. The elementary antisaddle equilibrium can act as a weak focus of the order of no more than two, giving rise to a degenerate Hopf bifurcation of codimension up to two. These high-codimension bifurcations identify organizing centers in parameter space, indicating regions where the ecosystem is highly sensitive and prone to abrupt regime shifts. Our results indicate that the generalist predator can induce a richer bifurcation phenomenon and more complex dynamics and can drive the system to certain desired stable states.