Influence of fear and Allee effects on stability and species survival in a tri-trophic food chain

• Published in: Communications in nonlinear science & numerical simulation Vol. 152; p. 109115
• Main Authors: Kumar, Vinay, Bairwa, Suneel Kumar, Tanvi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2026
• Subjects: Allee effect; Bifurcation; Chaos control; Coexistence; Fear effect; Bifurcation; Coexistence; Fear effect; Chaos control; Allee effect
• ISSN: 1007-5704
• DOI: 10.1016/j.cnsns.2025.109115

This study investigates the impact of fear and Allee effects on the qualitative behavior of a tri-trophic food chain model. We formulate a mathematical model where the fear effect influences the bottom prey population while the top predator is subject to the Allee effect. We establish conditions for the boundedness and persistence of the model. The existence and stability of equilibrium points are analyzed. Comprehensive numerical simulations are performed to verify the analytical findings. Our study reveals that fear and Allee effects significantly impact stability and chaos within the system. In the absence of the Allee effect, the system exhibits more complex dynamics and a broader stability range concerning the strength of the fear effect. Conversely, the presence of the Allee effect, coupled with increased fear strength, leads to a steep decline in the top predator population. The interplay between fear and Allee effects promotes the coexistence and long-term survival of the three species within the system. •Dynamics of a tri-trophic food chain model with fear and Allee effects are analyzed.•Boundedness, positivity, and local stability conditions are established.•Numerical simulations reveal diverse dynamical behavior of the system.•Fear and Allee effects, individually or in combination, help in controlling chaos.•Without Allee effect, fear induces more complex and broader stable dynamics.