Bifurcation Structures and Persistence Conditions in a Nonlinear Predator–Prey Model with Allee and Switching Effects

Resumen

This paper studies the dynamics of a nonlinear ecological model with one predator and two prey species governed by several biologically relevant factors such as prey-switching, refuges, and Allee effects. The first prey species receives a regular refuge, while the second prey species is affected by a strong Allee effect on its population growth at low density. Furthermore, the predator has switching dynamics of prey availability, which improves the survival strategy as well. The equilibrium points of the model are investigated to study the existence and biological relevance of these points. The local dynamics of the interior and boundary equilibria are examined through a combination of analytical methods and bifurcation theory. Using Sotomayor's theorem, we give necessary and sufficient conditions for transcritical and Hopf bifurcations. The analysis is based on the Dulac–Bendixson criterion and Lyapunov functions to ensure that all species have uniform persistence. Finally, persistence is discussed as a global property of the switching functional response to a single predator's consumption of two prey. This has happened with the presence of the refuge effect on the first prey and the Allee effect on the second prey only, regardless of mortality and saturation rates concerning the predator. Numerical results for a set of default parameters confirm our analytical results, visualizing the occurrence of local bifurcation and the persistence of this model. The results show that the system's stability is affected by the prey switching and refuge provision, depending on the strength of the Allee effect and harvesting pressure.