Bifurcations and global dynamics of a predator–prey mite model of Leslie type

In this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus-type and cusp-type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle-node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.