Generalized Flip Bifurcation, Strong Resonances, Arnold Tongue, and Marotto's Chaos in a Discrete Predator‐Prey Model

Abstract

This paper mainly studied the dynamic properties of a discrete predator‐prey system with a weak Allee effect on the predator. We first provided the topological structure of orbits in the vicinity of each fixed point. Then, through center manifold reduction, polynomial complete discrimination system, and bifurcation theory, we precisely proved the emergence of codimension‐2 bifurcations, including generalized flip bifurcation, strong resonances of 1:2, 1:3, 1:4, along with the Arnold tongue in the weak resonances. Additionally, we also discussed the chaotic behavior in the sense of Marotto of the model. Finally, we performed numerical simulations of the model's dynamics using Maple 2024 and Matlab R2024a to further validate the correctness of the aforementioned theoretical results.