AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026330207
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RESEARCH ARTICLE

A dynamic study of cooperative hunting and a protected zone prey–predator model

Mustafa M. Subhi1 ,  Raid K. Naji1*
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1 Department of Mathematics, College of Science, University of Baghdad, Baghdad , Iraq
IJOCTA, 026330207 https://doi.org/10.36922/IJOCTA026330207
Received: 11 August 2026 | Revised: 25 September 2026 | Accepted: 28 September 2026 | Published online: 3 November 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Predator–prey interactions are influenced by various ecological factors, including cooperative hunting, prey refuges, and harvesting, which can shape population dynamics, ecosystem stability, and species survival. This study proposes and studies a prey–predator system that interacts in two zones: the protected and the unprotected. It is assumed that the predator has the capability of cooperative hunting behavior. The objective of this study is to understand the role of a predator’s capabilities in the dynamical behavior of the system. All the solution properties are studied. The stability analyses of the equilibria are investigated. The impact of perturbing the parameters is assessed using local bifurcation analysis. Persistence conditions of the system are established. Numerical simulations using the fourth-order Runge-Kutta method in MATLAB confirm all analytical results. It was shown that the system has at most two boundary equilibrium points, and the positive equilibrium point, if it exists, is multiple. In addition, a numerical study shows that the system with only point attractors exhibits a wide range of bistability behaviors.

Graphical abstract
Keywords
Protected and unprotected zones
Cooperative hunting
Stability
Bifurcation
Persistence
Funding
None.
Conflict of interest
The authors declare they have no competing interests.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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