Bifurcation dynamics in a Leslie-type model: interaction between generalist predator and weak Allee effect

This study investigates the bifurcation dynamics driven by the interaction between a generalist predator and a weak Allee effect in a Leslie-type model through a rigorous theoretical analysis. We reveal that the interplay between the generalist predator (parameter e) and the weak Allee effect (parameter β) leads to a rich bifurcation structure. Specifically: (1) when β = e, the model exhibits a nilpotent cusp of codimension 2, accompanied by degenerate Bogdanov-Takens and Hopf bifurcations of codimension 2; (2) when β ≠ e, the nilpotent cusp can reach codimension 3, and the model undergoes degenerate Bogdanov-Takens and Hopf bifurcations of codimension up to 3. Nearly all conditions guaranteeing these dynamics are derived explicitly and precisely. Our results demonstrate that the generalist predator and weak Allee effect not only enrich the system’s dynamical behaviors and bifurcation patterns but may also lead to prey extinction for certain positive initial densities. Furthermore, smaller values of e favor the coexistence of predator and prey, while larger values of e are detrimental to the prey population. Numerical simulations, including prey extinction and the coexistence of multiple steady states (e.g., homoclinic orbits, limit cycles, and double limit cycles), provide intuitive illustrations of the theoretical findings. This work advances the understanding of higher-codimension bifurcation dynamics in ecological systems. continue to represent an active area of research focus due to their ability to capture fundamental ecological processes while remaining mathematically tractable [1, and h represents the quality of the prey as food for the predator. The study of predator-prey dynamics provides crucial insights into the conditions governing species coexistence and extinction, offering valuable guidance for ecosystem management and conservation policymaking. These models demonstrate that system outcomes are highly sensitive to various biological factors, particularly the functional response characterizing predator feeding behavior. The different functional response types (Holling type I, II, III, IV, and so on) are known to generate distinct dynamical regimes, ranging from stable equilibria to complex oscillatory patterns, each with important ecological implications [15, 16, 17, 18, 19, 20, 21]. Note that in the model (1), the predator is called a specialist predator, since it only relies on a single prey species to survive and will die out in absence of this prey. While in some situations, especially when the prey is severely scarce, the predator can turn to other alternative prey species for food and can persist by switching to other food sources, 2020 MSC: 34C07; 34C25; 34D15; 37G15