Eur. Phys. J. Plus (2021) 136: 905
https://doi.org/10.1140/epjp/s13360-021-01900-3

Regular Article

A “Double” fear effect in a tri-trophic food chain model

¹ Department of Mathematics and Astronomy, University of Lucknow, 226007, Lucknow, India
² Department of Mathematics and Computer Science, Samford University, 35229, Birmingham, AL, USA
³ Department of Mathematics, Visva-Bharati, 731235, Santiniketan, West Bengal, India
⁴ Department of Mathematics, Iowa State University, 50011, Ames, IA, USA

^b kantwifo@samford.edu

Received: 20 May 2021
Accepted: 25 August 2021
Published online: 3 September 2021

Abstract

Fear of predators as an indirect effect of predation is a well-observed phenomenon in natural predator–prey communities. After being popularized by Wang et al. in 2016 (J. Math. Biol., 73:1179–1204), the indirect effect of fear has been studied through its incorporation into several mathematical models in different ecological settings. The impact of fear effect in a three-species food chain model, where the top predator is a generalist predator, is not yet investigated. In the present work, a three-species food chain model incorporating the fear of predators is investigated, where the fear of generalist predator impedes the growth rate of the specialist predator, and the fear of specialist predator reduces the birth rate of the prey. We derive criteria for the existence of all possible non-negative equilibrium points and provide some remarks on their local stability. Additionally, the conditions for the occurrence of various local bifurcations have been established. In the absence of fear, we show that the model exhibits chaotic dynamics via periodic-doubling route when the birth rate of the prey is varied. Furthermore, we observe that fear can alter the stability of the model from a chaotic zone to a stable zone via period-halving route. It is also observed that with an appropriate choice of parameter values, fear can act as a damping mechanism, preventing the unwanted blow-up of the generalist predator in finite time. We provide numerical simulations to support our analytical findings.