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# Deterministic and Stochastic Fractional-Order Hastings-Powell Food Chain Model

Moustafa El-Shahed1,*, Asmaa M. Al-Dububan2

1 Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Unaizah, 51911, Saudi Arabia
2 Department of Mathematics, Buraydah College of Sciences and Arts, Qassim University, Buraydah, 51431, Saudi Arabia

* Corresponding Author: Moustafa El-Shahed. Email:

Computers, Materials & Continua 2022, 70(2), 2277-2296. https://doi.org/10.32604/cmc.2022.019314

## Abstract

In this paper, a deterministic and stochastic fractional-order model of the tri-trophic food chain model incorporating harvesting is proposed and analysed. The interaction between prey, middle predator and top predator population is investigated. In order to clarify the characteristics of the proposed model, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions of the proposed model are examined. Some sufficient conditions that ensure the local and global stability of equilibrium points are obtained. By using stability analysis of the fractional-order system, it is proved that if the basic reproduction number , the predator free equilibrium point is globally asymptotically stable. The occurrence of local bifurcation near the equilibrium points is investigated with the help of Sotomayor’s theorem. Some numerical examples are given to illustrate the theoretical findings. The impact of harvesting on prey and the middle predator is studied. We conclude that harvesting parameters can control the dynamics of the middle predator. A numerical approximation method is developed for the proposed stochastic fractional-order model.

## Keywords

M. El-Shahed and A. M. Al-Dububan, "Deterministic and stochastic fractional-order hastings-powell food chain model," Computers, Materials & Continua, vol. 70, no.2, pp. 2277–2296, 2022.

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