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<p>This research analyzes a predator–prey model with hunting cooperation and an alternative food supply for predators. We demonstrate the presence of a region of invariance first, followed by the boundedness of the system’s trajectories and the permanence of the solutions. Additionally, we show that the equilibrium point (0,&#xa0;0) possesses the property of being a repeller. The requirements for the occurrence of two positive equilibrium points are explicitly given. The first equilibrium point has the characteristic of being a weak focus, repeller, or attractor, and the second equilibrium point is a saddle. In addition, we identify two crucial scenarios: (i) A separatrix curve <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\bar{\Sigma }\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi mathvariant="normal">Σ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation> divides the solutions into qualitatively different sectors, and (ii) a positive saddle point generates a homoclinic curve through the stable to unstable manifolds. These facts highlight how sensitive the system is to beginning conditions, especially in proximity to the separatrix. Furthermore, the dynamics of the model can be strongly affected by Hopf, Bogdanov–Takens, and transcritical bifurcations that the system may experience. Finally, we validate our analytical results using bifurcation diagrams and numerical simulations.</p>

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Bistability and Periodicity in an Amended May–Holling–Tanner Model Involving Generalist Predators and Hunting Cooperation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月1日

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  • Francisco Javier Reyes-Bahamón,
  • Alix Dayana Galindo-Leiva,
  • Julio Cesar Duarte-Vidal,
  • Eduardo González-Olivares

摘要

This research analyzes a predator–prey model with hunting cooperation and an alternative food supply for predators. We demonstrate the presence of a region of invariance first, followed by the boundedness of the system’s trajectories and the permanence of the solutions. Additionally, we show that the equilibrium point (0, 0) possesses the property of being a repeller. The requirements for the occurrence of two positive equilibrium points are explicitly given. The first equilibrium point has the characteristic of being a weak focus, repeller, or attractor, and the second equilibrium point is a saddle. In addition, we identify two crucial scenarios: (i) A separatrix curve \(\bar{\Sigma }\) Σ ¯ divides the solutions into qualitatively different sectors, and (ii) a positive saddle point generates a homoclinic curve through the stable to unstable manifolds. These facts highlight how sensitive the system is to beginning conditions, especially in proximity to the separatrix. Furthermore, the dynamics of the model can be strongly affected by Hopf, Bogdanov–Takens, and transcritical bifurcations that the system may experience. Finally, we validate our analytical results using bifurcation diagrams and numerical simulations.