Complex dynamics of a predator-prey model with constant-yield prey harvesting and Allee effect in predator
Jianhang Xie, Changrong Zhu
TL;DR
The paper analyzes a Holling type I Leslie-Gower predator-prey model with predator Allee effect and constant prey harvesting, deriving the full equilibrium structure and conducting a detailed bifurcation analysis. It identifies boundary and interior equilibria and establishes conditions for their existence and stability, including thresholds $h=\tfrac{1}{4}$ and discriminants $\Delta_1,\Delta_2$. The study demonstrates saddle-node bifurcations at boundary points, Hopf bifurcations at interior equilibria with direction determined by the first Lyapunov coefficient, and a codimension-2 Bogdanov-Takens bifurcation when two parameters vary, revealing rich dynamical behavior and periodic oscillations. These results have ecological implications for harvesting strategies and the management of predator-prey systems with 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.