Impacts of cooperative hunting and Allee effect on the dynamics of a three-species system with sexually reproducing top predators.

This study investigates the complex dynamics of a tri-trophic predator-prey system incorporating the Allee effect in prey, cooperative hunting by mesopredators, and sexual reproduction in the apex predator. Using bifurcation analysis and numerical simulations, we examine how these factors-individually or combinedly-shape ecosystem stability, both in the presence and absence of the top predator. In the absence of the apex predator, species survival depends critically on prey's productivity and mesopredator cooperation strength: lower values lead to extinction, while moderate prey growth supports survival, induce bistability, or stable coexistence. Higher growth rates induce Hopf bifurcations, sustained oscillations, and homoclinic dynamics, whereas excessively high productivity unexpectedly drives the system back to extinction. Prey herd behavior acts as a stabilizing force, counterbalancing cooperative predation. With the inclusion of the top predator, the system becomes highly sensitive to parameter variations. Increasing mesopredator growth drives transitions through extinction, coexistence, bistability, periodic and chaotic oscillations, and eventual collapse via period-doubling cascades. Cooperative hunting intensity induces multiple Hopf bifurcations, functioning as either stabilizing or destabilizing factor depending on its magnitude. Similarly, variations in apex predator productivity cause additional routes to chaos through saddle-node and period-doubling bifurcations. Consequently, small changes in biological traits can trigger major ecological shifts, ranging from stable coexistence to chaos or sudden extinction. Environmental stochasticity further alters system fragility. Without the apex predator, low noise levels maintain coexistence, but increasing variability can eliminate both species or selectively drive predator toward extinction. When the apex predator is present, the system becomes extremely fragile, with even weak noise capable of pushing populations beyond critical thresholds, often resulting in extinction.