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A chaos study of fractal–fractional predator–prey model of mathematical ecology

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  • Kumar, Ajay
  • Kumar, Sunil
  • Momani, Shaher
  • Hadid, Samir

Abstract

This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on fractal–fractional Atangana–Baleanu (AB) and Caputo operators, we present a newly developed system of differential equations for the predator–prey system. Our study utilized the fixed point postulate to investigate the uniqueness and existence of solutions. Additionally, Ulam’s type of stability of the proposed model is established with the help of nonlinear functional analysis. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyze its behavior. The generalized non-linear system with fractal–fractional Atangana–Baleanu (AB) and Caputo non-integer operators have been solved numerically via the Toufik–Atangana (TA) scheme respectively. We have demonstrated the applicability and effectiveness of these methods by analyzing numerical simulations for the fractal–fractional predator–prey ecological model and the numerical simulation has been calculated by MATLAB programming.

Suggested Citation

  • Kumar, Ajay & Kumar, Sunil & Momani, Shaher & Hadid, Samir, 2024. "A chaos study of fractal–fractional predator–prey model of mathematical ecology," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 857-888.
    • Handle: RePEc:eee:matcom:v:225:y:2024:i:c:p:857-888
    DOI: 10.1016/j.matcom.2023.09.010
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    References listed on IDEAS

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    1. Atangana, Abdon & Qureshi, Sania, 2019. "Modeling attractors of chaotic dynamical systems with fractal–fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 320-337.
    2. Agrawal, Khushbu & Kumar, Ranbir & Kumar, Sunil & Hadid, Samir & Momani, Shaher, 2022. "Bernoulli wavelet method for non-linear fractional Glucose–Insulin regulatory dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Ali, Zeeshan & Rabiei, Faranak & Hosseini, Kamyar, 2023. "A fractal–fractional-order modified Predator–Prey mathematical model with immigrations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 466-481.
    4. Xu, Changjin & Liu, Zixin & Pang, Yicheng & Akgül, Ali & Baleanu, Dumitru, 2022. "Dynamics of HIV-TB coinfection model using classical and Caputo piecewise operator: A dynamic approach with real data from South-East Asia, European and American regions," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    5. Kumar, Ajay & Kumar, Sunil, 2022. "A study on eco-epidemiological model with fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    6. Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    7. Kumar, Sunil & Kumar, Ajay & Samet, Bessem & Gómez-Aguilar, J.F. & Osman, M.S., 2020. "A chaos study of tumor and effector cells in fractional tumor-immune model for cancer treatment," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
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