Fold-Flip and Bogdanov–Takens Bifurcations of a Discrete Predator–Prey System with Cooperative Hunting

This paper proves the occurrence of the codimension-2 bifurcations — the fold-flip and Bogdanov–Takens in the discrete predator–prey model with predator cooperative hunting, which is not obtained by discretizing any continuous predator–prey model. The normal form is computed for each bifurcation to check nondegeneracy conditions and forecast possible dynamical behaviors near the bifurcation points. The theoretical results are validated by plotting the continuation curve and comparative phase portraits of approximating vector fields, maps, and normal forms using the packages Matlab-Matcontm, and Mathematica. Our results indicate that a high level of cooperative hunting and minor changes in other system parameters can lead to long-term destabilization and produce highly complex behaviors in the system.