Permanence, Stability, Extinction and Chaos in a Nonautonomous Predator–Prey Model with Additive Allee Effect

In this work, we investigate the qualitative dynamics of a nonautonomous predator–prey model incorporating a saturated Holling-type functional response and an additive Allee effect in the prey population. Such a framework is biologically relevant since ecological parameters often fluctuate in time due to seasonal or environmental variations, while Allee mechanisms may strongly influence prey growth at low population densities.

First, by applying the comparison principle and constructing suitable differential inequalities, we establish sufficient conditions ensuring the positivity and permanence of the system. In particular, we prove that both prey and predator populations remain uniformly bounded
away from extinction whenever appropriate persistence thresholds are satisfied. Next, a Lyapunov-type function based on logarithmic distance is introduced in order to derive explicit criteria for the global stability of positive solutions. This analysis guarantees that all trajectories starting from positive initial values asymptotically approach each other, showing that the system exhibits robust long-term behavior under the permanence regime.

Furthermore, we obtain threshold conditions under which the predator population becomes extinct asymptotically, revealing how the combined effects of mortality, predation saturation, and the Allee mechanism can lead to predator disappearance even when prey survive.

In addition to the theoretical results, numerical simulations are performed to illustrate the validity of the analytical findings and to explore the richness of the model’s dynamics. In particular, for certain parameter regimes, especially in the region of weak Allee effect, the system exhibits chaotic dynamics characterized by irregular oscillations, sensitive dependence on initial conditions, and complex attractors.