Fractional dynamics and computational analysis of food chain model with disease in intermediate predator

Jagdev Singh; Behzad Ghanbari; Ved Prakash Dubey; Devendra Kumar; Kottakkaran Sooppy Nisar; Jagdev Singh; Behzad Ghanbari; Ved Prakash Dubey; Devendra Kumar; Kottakkaran Sooppy Nisar

doi:10.3934/math.2024830

AIMS Mathematics

2024, Volume 9, Issue 7: 17089-17121. doi: 10.3934/math.2024830

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Fractional dynamics and computational analysis of food chain model with disease in intermediate predator

1.
Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India
2.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Korea
3.
Department of Mathematics, Kermanshah University of Technology, Kermanshah, Iran
4.
Department of BCA, Laxmi Narain Dubey College, Babasaheb Bhimrao Ambedkar Bihar University, Muzaffarpur, Motihari 845401, Bihar, India
5.
Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India
6.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia

Received: 15 February 2024 Revised: 09 April 2024 Accepted: 19 April 2024 Published: 16 May 2024
MSC : 26A27, 26A30, 26A33, 28A80, 35R11

In this paper, a fractional food chain system consisting of a Holling type Ⅱ functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type Ⅱ functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations.
- predator-prey model,
- fractional food chain model,
- fractional derivative,
- Ulam-Hyers stability analysis,
- Banach space
Citation: Jagdev Singh, Behzad Ghanbari, Ved Prakash Dubey, Devendra Kumar, Kottakkaran Sooppy Nisar. Fractional dynamics and computational analysis of food chain model with disease in intermediate predator[J]. AIMS Mathematics, 2024, 9(7): 17089-17121. doi: 10.3934/math.2024830

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Abstract

In this paper, a fractional food chain system consisting of a Holling type Ⅱ functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type Ⅱ functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations.

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