Bifurcations of higher codimension in a Leslie-Gower predator-prey model with Holling II functional response and weak Allee effect

A Leslie-type predator-prey system with Holling II functional response and weak Allee effect in prey is analyzed deeply in this paper. Through rigorous analysis, the system can undergo a series of bifurcations such as cusp type nilpotent bifurcation of codimension 4 and a degenerate Hopf bifurcation of codimension up to 3 as the parameters vary. Compared with the system without Allee effect, it can be concluded that weak Allee effect can induce more abundant dynamics and bifurcations, in particular, the increase in the number of equilibria and the appearance of multiple limit cycles. Moreover, when the intensity of predation is too high, the prey affected by the weak Allee effect will also become extinct, and eventually lead to the collapse of the system. Finally, we present some numerical simulations by MATCONT to illustrate the existence of bifurcations and some phase portraits of the system.