A predator–prey model incorporating fear effect, prey vigilance costs, and predator hunting cooperation: analysis and simulations

Document Type : Research Article

Author

Ayodeji Quadri Akinsanya

Department of Mathematics, University of Tennessee, Knoxville, USA.

10.22067/ijnao.2026.97683.1832

Abstract

In this study, we propose and analyze a predator-prey model that incorporates prey vigilance, fear effects, and predator cooperative hunting to capture key behavioral mechanisms influencing population dynamics. The model explicitly accounts for non-consumptive effects of predation risk, whereby prey fear and vigilance reduce effective prey growth and modify predator-prey interactions even in the absence of direct predation. We first establish the mathematical structure of the model and identify all biologically meaningful equilibria, including the extinction equilibrium, the predator-free equilibrium, and the coexistence equilibrium. Local stability analysis reveals that fear and vigilance significantly alter classical stability thresholds by reducing prey reproductive potential and limiting predator access to prey. Using Lyapunov-based methods, we derive sufficient conditions for the global asymptotic stability of the extinction and predator-free equilibria, demonstrating that strong fear effects combined with prey vigilance or elevated predator mortality can suppress predator persistence regardless of initial population densities. For the coexistence equilibrium, we show that stable coexistence occurs under balanced levels of fear, vigilance, and predator cooperation, while excessive cooperation destabilizes the equilibrium. This destabilization is characterized analytically by a Hopf bifurcation with respect to the predator cooperation parameter, leading to the emergence of sustained population oscillations. Numerical simulations, including phase-plane portraits and time-series analyses, are presented to support the theoretical results and confirm the analytical stability and bifurcation findings.

Keywords

Predator– prey model

Prey vigilance (fear effect)

Predator cooperation

Global stability

Hopf bifurcation

Subjects

• Modeling and Simulation

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