Isola and mushroom dynamics of limit cycles and bifurcations in a predator-prey system with additive Allee effect

Abstract

This paper investigates a predator-prey system with an additive Allee effect and a generalized Holling IV functional response using a dynamical system approach. By means of a mixture of analytical and numerical procedures, we find the existence of codimension two and three Bogdanov-Takens bifurcation, codimension-three generalized Hopf bifurcation, and codimension-two cusp of limit cycles. We also found mushroom and isola bifurcations of limit cycles as the first examples of such phenomena in a predatory interaction. The model predicts that extinction of both populations may only occur if the Allee effect is strong. However, long term coexistence is possible in both weak and strong Allee regimes indicating that predation has a balancing role in the interaction dynamics. Nonetheless, a weak Allee effect can result in complex dynamics as well, including the presence of isolas, mushrooms and cusps of limit cycles.