Bifurcations in a Predator–Prey Model of Leslie-Type with Simplified Holling Type IV Functional Response

In this paper, we complete the remaining investigation of local bifurcations in a predator–prey model of Leslie-type with simplified Holling type IV functional response. The system has at most three equilibria, and local bifurcations were completely investigated in the cases of one and three equilibria, but in the case of two equilibria the previous study was only on a fixed parameter. We extend the study in the case of two equilibria for all parameters, and find that the system exhibits Hopf bifurcations of codimensions 1 and 2, and Bogdanov–Takens bifurcations of codimensions 2 and 3. Previous results and our research show that the codimension of local bifurcations is at most 3, and both focus type and cusp type Bogdanov–Takens bifurcations of codimension 3 can occur.