Multiple parameter bifurcations in a modified Gower-Leslie predator-prey system with addictive Allee effect

In this paper, we explore a modified Leslie-Gower type predator-prey model with Holling I functional response and addictive Allee effect in prey. It is shown that the highest codimension of a nilpotent cusp 4, and the model can undergo degenerate Bogdanov-Takens bifurcation of codimension 4. Besides, when the model has a center-type equilibrium, we show that the model can admit five limit cycles bifurcated from Hopf bifurcation. Our results indicate that addictive Allee effect can induce not only richer dynamics and bifurcations, but also the coextinction of both populations with some positive initial densities. Finally, numerical simulations, including three limit cycles and four limit cycles, are presented to illustrate the theoretical results.