Mathematical analysis of a diffusive predator-prey model with herd behavior and prey escaping

¹ Biomathematics Laboratory, University of Djillali Liabes, Sidi-Bel-Abbès, Algeria.
² Faculty of Exact Sciences and Informatics, Mathematic Department, University of Hassiba Benbouali, Chlef, Algeria.
³ Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria.
⁴ Faculty of sciences, Mathematic Department, University of Moulay Tahar, Saida, Algeria.
⁵ Laboratoire de Géométrie, Analyse, Contrôle et Applications, University of Moulay Tahar, Saida, Algeria.

^* Corresponding author: djilali.salih@yahoo.fr

Received: 27 December 2018
Accepted: 2 October 2019

Abstract

In this paper, we consider a new approach of prey escaping from herd in a predator-prey model with the presence of spatial diffusion. First, the sensitivity of the equilibrium state density with respect to the escaping rate has been studied. Then, the analysis of the non diffusive system was investigated where boundedness, local, global stability, Hopf bifurcation are obtained. Besides, for the diffusive system, we proved the occurrence of Hopf bifurcation and the non existence of diffusion driven instability. Furthermore, the direction of Hopf bifurcation has been proved using the normal form on the center manifold. Some numerical simulations have been used to illustrate the obtained results.