The hunting cooperation of a predator under two prey's competition and fear-effect in the prey-predator fractional-order model

Ali Yousef; Ashraf Adnan Thirthar; Abdesslem Larmani Alaoui; Prabir Panja; Thabet Abdeljawad; Ali Yousef; Ashraf Adnan Thirthar; Abdesslem Larmani Alaoui; Prabir Panja; Thabet Abdeljawad

doi:10.3934/math.2022303

AIMS Mathematics

2022, Volume 7, Issue 4: 5463-5479. doi: 10.3934/math.2022303

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Research article

The hunting cooperation of a predator under two prey's competition and fear-effect in the prey-predator fractional-order model

1.
Department of mathematics, Kuwait College of Science and Technology, 2723 Kuwait City, Kuwait
2.
Department of Studies and Planning, University of Fallujah, Anbar, Iraq
3.
Moulay Ismail university, FST Errachidia, MAIS Laboratory, MAMCS Group, Morocco
4.
Department of Applied Science, Haldia Institute of Technology, Purba Midnapore-721657, West Bengal, India
5.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 30 October 2021 Revised: 17 December 2021 Accepted: 26 December 2021 Published: 07 January 2022
MSC : 37N25, 39A30, 39A60, 92D25, 92D40

This paper investigates a fractional-order mathematical model of predator-prey interaction in the ecology considering the fear of the prey, which is generated in addition by competition of two prey species, to the predator that is in cooperation with its species to hunt the preys. At first, we show that the system has non-negative solutions. The existence and uniqueness of the established fractional-order differential equation system were proven using the Lipschitz Criteria. In applying the theory of Routh-Hurwitz Criteria, we determine the stability of the equilibria based on specific conditions. The discretization of the fractional-order system provides us information to show that the system undergoes Neimark-Sacker Bifurcation. In the end, a series of numerical simulations are conducted to verify the theoretical part of the study and authenticate the effect of fear and fractional order on our model's behavior.
- fear effect,
- Caputo fractional order,
- predator-prey model,
- stability,
- Neimark-Sacker bifurcation
Citation: Ali Yousef, Ashraf Adnan Thirthar, Abdesslem Larmani Alaoui, Prabir Panja, Thabet Abdeljawad. The hunting cooperation of a predator under two prey's competition and fear-effect in the prey-predator fractional-order model[J]. AIMS Mathematics, 2022, 7(4): 5463-5479. doi: 10.3934/math.2022303

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Abstract

This paper investigates a fractional-order mathematical model of predator-prey interaction in the ecology considering the fear of the prey, which is generated in addition by competition of two prey species, to the predator that is in cooperation with its species to hunt the preys. At first, we show that the system has non-negative solutions. The existence and uniqueness of the established fractional-order differential equation system were proven using the Lipschitz Criteria. In applying the theory of Routh-Hurwitz Criteria, we determine the stability of the equilibria based on specific conditions. The discretization of the fractional-order system provides us information to show that the system undergoes Neimark-Sacker Bifurcation. In the end, a series of numerical simulations are conducted to verify the theoretical part of the study and authenticate the effect of fear and fractional order on our model's behavior.

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