Eur. Phys. J. Plus (2021) 136: 940
https://doi.org/10.1140/epjp/s13360-021-01940-9

Regular Article

Delayed predator–prey model with prey social behavior

¹ Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria
² Department of Mathematics, Faculty of Exact Sciences and Computer Science, Hassiba Benbouali University, Chlef, Algeria
³ Engineering School, DEIM, Tuscia University, 01100, Viterbo, Italy
⁴ Department of Mathematics, Visva-Bharati, 731235, Santiniketan, West Bengal, India

^a djilali.salih@yahoo.fr, s.djilali@univ-chlef.dz

Received: 12 June 2021
Accepted: 2 September 2021
Published online: 14 September 2021

Abstract

In this paper, we propose a delayed predator–prey model in the presence of prey herd behavior. The main goal of this research is to provide the bifurcation analysis of the suggested model and highlight the importance of the prey herd behavior and its influence on the reproduction and the evolution of predator densities. The prey in herd can cooperate to defend the group, where this defense mechanism can lead to a deadly injures for predator. It has been shown that the investigated system can undergo Hopf bifurcation in both the absence and presence of time lags, and forward bifurcation in the absence of time delay. For determining the direction of the Hopf bifurcation in the presence of time lags, the normal form on the center of manifold is applied. The obtained theoretical results are checked using numerical simulations with a brief discussion on the biological relevance. Further, the effect of some crucial parameters on the temporal behavior of solution is discussed.