A study on a class of predator-prey models with Allee effect
Jianhang Xie, Changrong Zhu
TL;DR
This work analyzes a Holling Type I Leslie-Gower predator-prey system with predator Allee effect and constant prey harvesting to map how equilibria and their stability evolve across parameter regimes. By nondimensionalizing the model to $\dot{x}=x(1-x)-qxy-h$ and $\dot{y}=s\,y\left(1-\frac{y}{x}\right)(y-m)$, it derives explicit existence criteria for boundary and interior equilibria using discriminants $\Delta_1$ and $\Delta_2$ and classifies their stability via the Jacobian. The study identifies a saddle-node bifurcation at $h=\frac{1}{4}$, a Hopf bifurcation near an interior equilibrium $E_8$ with the direction determined by the first Lyapunov coefficient $\sigma$, and a codimension-2 Bogdanov-Takens bifurcation arising from two-parameter perturbations around cusp dynamics. These results reveal rich dynamical behavior, including possible stable or unstable limit cycles and complex multi-parameter transitions, with implications for managing predator-prey systems exhibiting Allee effects.
Abstract
This paper investigates the dynamical behaviors of a Holling type I Leslie-Gower predator-prey model where the predator exhibits an Allee effect and is subjected to constant harvesting. The model demonstrates three types of equilibrium points under different parameter conditions, which could be either stable or unstable nodes (foci), saddle nodes, weak centers, or cusps. The system exhibits a saddle-node bifurcation near the saddle-node point and a Hopf bifurcation near the weak center. By calculating the first Lyapunov coefficient, the conditions for the occurrence of both supercritical and subcritical Hopf bifurcations are derived. Finally, it is proven that when the predator growth rate and the prey capture coefficient vary within a specific small neighborhood, the system undergoes a codimension-2 Bogdanov-Takens bifurcation near the cusp point.