Mathematical analysis of tri-trophic food webs with carrying capacity and Holling-type predation using fractal-fractional Caputo derivatives

Amjad E. Hamza; Arshad Ali; Khaled Aldwoah; Hicham Saber; Ria Egami; Amel Touati; Amal F. Alharbi; Amjad E. Hamza; Arshad Ali; Khaled Aldwoah; Hicham Saber; Ria Egami; Amel Touati; Amal F. Alharbi

doi:10.3934/math.2025589

AIMS Mathematics

2025, Volume 10, Issue 6: 13130-13150. doi: 10.3934/math.2025589

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Research article

Mathematical analysis of tri-trophic food webs with carrying capacity and Holling-type predation using fractal-fractional Caputo derivatives

1.
Department of Mathematics, College of Science, University of Hail, Hail 55473, Saudi Arabia
2.
Department of Mathematics, University of Malakand, Lower Dir, Chakdara 18000, Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia
4.
Department of Mathematics, College of Science and Humanity, Prince Sattam bin Abdulaziz University, Sulail, Al-Kharj 11942, Saudi Arabia
5.
Mathematics Department, College of Science, Northern Border University, Arar 91431, Saudi Arabia
6.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received: 02 March 2025 Revised: 13 May 2025 Accepted: 23 May 2025 Published: 06 June 2025
MSC : 26A33, 34A08, 34A12

This research investigated a tri-trophic food chain model, incorporating carrying capacity and Holling-type predation, formulated using the fractal-fractional Caputo derivative. The four equilibrium states---trivial, prey-only, prey-predator, and coexistence---presented and their stability was discussed. Existence and uniqueness of solutions were established using Schaefer's and Banach's fixed point theorems. Also, stability requirements in the sense of Hyers-Ulam (H-U) were investigated. Numerical simulations were performed using the extended numerical method of Adams-Bashforth-Moulton (ABM), and comparative results were graphically presented to demonstrate the impact of varying fractal-fractional orders. A sensitivity analysis revealed how perturbations in individual parameters influence the model's outcome. The model accounts for memory and hereditary effects in ecological interactions. The proposed method enhances accuracy, stability, and convergence for long-time simulations compared to classical models.
- tri-trophic food chain models,
- carrying capacity,
- Holling type predation,
- equilibrium points, existence and stability analysis,
- numerical analysis
Citation: Amjad E. Hamza, Arshad Ali, Khaled Aldwoah, Hicham Saber, Ria Egami, Amel Touati, Amal F. Alharbi. Mathematical analysis of tri-trophic food webs with carrying capacity and Holling-type predation using fractal-fractional Caputo derivatives[J]. AIMS Mathematics, 2025, 10(6): 13130-13150. doi: 10.3934/math.2025589

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Abstract

This research investigated a tri-trophic food chain model, incorporating carrying capacity and Holling-type predation, formulated using the fractal-fractional Caputo derivative. The four equilibrium states---trivial, prey-only, prey-predator, and coexistence---presented and their stability was discussed. Existence and uniqueness of solutions were established using Schaefer's and Banach's fixed point theorems. Also, stability requirements in the sense of Hyers-Ulam (H-U) were investigated. Numerical simulations were performed using the extended numerical method of Adams-Bashforth-Moulton (ABM), and comparative results were graphically presented to demonstrate the impact of varying fractal-fractional orders. A sensitivity analysis revealed how perturbations in individual parameters influence the model's outcome. The model accounts for memory and hereditary effects in ecological interactions. The proposed method enhances accuracy, stability, and convergence for long-time simulations compared to classical models.

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