Eur. Phys. J. Plus (2022) 137: 704
https://doi.org/10.1140/epjp/s13360-022-02926-x

Regular Article

Prey group defense to predator aggregated induced fear

¹ Department of Mathematics, DePauw University, 46135, Greencastle, IN, USA
² Department of Mathematics and Computer Science, Samford University, 35229, Birmingham, AL, USA

^b kantwifo@samford.edu

Received: 20 April 2022
Accepted: 5 June 2022
Published online: 17 June 2022

Abstract

In this article, we investigate the influence of fear in a predator interference predator-prey model, where the interactions between the species follow a non-monotonic response function. Unlike previous studies, we consider that fear of predator is triggered by the aggregation of the predators which in turn induces some anti-predator response in the prey. The conditions under which all biologically meaningful equilibrium points exist are discussed in detail. The global dynamics of the model is determined when there exists a unique positive interior equilibrium point. By changing the strength of fear of predator and predator aggregate sensitivity, the model admits several interesting codimension one bifurcation results such as the existence of multiple supercritical Hopf and saddle-node bifurcations around an interior equilibrium point. To determine the direction and stability of the limit cycles emitted by the Hopf bifurcation, the first Lyapunov coefficient is computed in detail. It is conjectured that the model exhibits a stable limit cycle, and bistability via supercritical Hopf bifurcation between two interior equilibrium points as the strength of fear increases beyond the formation of a homoclinic loop. Our numerical simulations present some rich codimension two bifurcations including Bogdanov–Takens and cusp bifurcations. The theoretical findings and conjectures obtained in this work are substantially corroborated with numerical simulations.