Bifurcations of codimension 4 in a Leslie-type predator-prey model with Allee effects

In this paper, we explore a Leslie-type predator-prey model with simplified Holling IV functional response and Allee effects in prey. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and saddle-node types nilpotent bifurcations of codimension four and a degenerate Hopf bifurcation of codimension up to four as the parameters vary. Our results indicate that Allee effects can induce richer dynamics and bifurcations, especially high sensitivities on both parameters and initial densities for coexistence and oscillations of both populations. Moreover, strong Allee effects ($M>0$) (or ‘transitional Allee effects’ ($M=0$) with large predation rates) can cause the coextinction of both populations with some positive initial densities, while weak Allee effects ($M<0$) (or transitional Allee effects with small predation rates) make both populations with positive initial densities persist. Finally, numerical simulations present some illustrations scarce in two-population models, such as the coexistence of three limit cycles and three positive equilibria.