Analysis of food chain mathematical model under fractal fractional Caputo derivative

Adnan Sami; Amir Ali; Ramsha Shafqat; Nuttapol Pakkaranang; Mati ur Rahmamn; Adnan Sami; Amir Ali; Ramsha Shafqat; Nuttapol Pakkaranang; Mati ur Rahmamn

doi:10.3934/mbe.2023097

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 2094-2109. doi: 10.3934/mbe.2023097

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Analysis of food chain mathematical model under fractal fractional Caputo derivative

1.
Department of Mathematics, University of Malakand Chakdara, Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics and Statistics, The University of Lahore, Sargodha 40100, Pakistan
3.
Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand
4.
School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200030, China
Academic editor: Mehmet Yavuz

Received: 07 September 2022 Revised: 02 November 2022 Accepted: 06 November 2022 Published: 14 November 2022

In this article, the dynamical behavior of a complex food chain model under a fractal fractional Caputo (FFC) derivative is investigated. The dynamical population of the proposed model is categorized as prey populations, intermediate predators, and top predators. The top predators are subdivided into mature predators and immature predators. Using fixed point theory, we calculate the existence, uniqueness, and stability of the solution. We examined the possibility of obtaining new dynamical results with fractal-fractional derivatives in the Caputo sense and present the results for several non-integer orders. The fractional Adams-Bashforth iterative technique is used for an approximate solution of the proposed model. It is observed that the effects of the applied scheme are more valuable and can be implemented to study the dynamical behavior of many nonlinear mathematical models with a variety of fractional orders and fractal dimensions.
- food web model,
- fractal-fractional Caputo,
- existence uniqueness and stability,
- fractional Adams Bashforth method
Citation: Adnan Sami, Amir Ali, Ramsha Shafqat, Nuttapol Pakkaranang, Mati ur Rahmamn. Analysis of food chain mathematical model under fractal fractional Caputo derivative[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2094-2109. doi: 10.3934/mbe.2023097

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Abstract

In this article, the dynamical behavior of a complex food chain model under a fractal fractional Caputo (FFC) derivative is investigated. The dynamical population of the proposed model is categorized as prey populations, intermediate predators, and top predators. The top predators are subdivided into mature predators and immature predators. Using fixed point theory, we calculate the existence, uniqueness, and stability of the solution. We examined the possibility of obtaining new dynamical results with fractal-fractional derivatives in the Caputo sense and present the results for several non-integer orders. The fractional Adams-Bashforth iterative technique is used for an approximate solution of the proposed model. It is observed that the effects of the applied scheme are more valuable and can be implemented to study the dynamical behavior of many nonlinear mathematical models with a variety of fractional orders and fractal dimensions.

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