Bifurcations of a Leslie-type model with herd behavior and predator harvesting

It was demonstrated that the herd behavior in prey, described by a square root functional response, induces a Hopf bifurcation in the Leslie-type predator–prey model. This paper is devoted to discussing bifurcations of the same model, but with constant-rate predator harvesting. We show that the system exhibits complicated bifurcations driven by the predator harvesting. More concretely, we first prove that the system has at most two equilibria, which consist of a saddle and either a focus or a node, respectively. Then we verify that the system undergoes the attracting and repelling Bogdanov–Takens bifurcations of codimension two. Additionally, we show that a degenerate Hopf bifurcation of codimension three can also occur at a weak focus. The theoretical findings, which are further illustrated by numerical simulations, reveal that the two populations can coexist periodically under constant-rate predator harvesting.