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Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative

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  • Balcı, Ercan
  • Öztürk, İlhan
  • Kartal, Senol

Abstract

In this paper, tumor-immune system interaction has been considered by two fractional order models. The first and the second model consist of system of fractional order differential equations with Caputo and conformable fractional derivative respectively. First of all, the stability of the equilibrium points of the first model is studied. Then, a discretization process is applied to obtain a discrete version of the second model where conformable fractional derivative is taken into account. In discrete model, we analyze the stability of the equilibrium points and prove the existence of Neimark-Sacker bifurcation depending on the parameter σ. Moreover, the dynamical behaviours of the models are compared with each other and we observe that the discrete version of conformable fractional order model exhibits chaotic behavior. Finally, numerical simulations are also presented to illustrate the analytical results.

Suggested Citation

  • Balcı, Ercan & Öztürk, İlhan & Kartal, Senol, 2019. "Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 43-51.
    • Handle: RePEc:eee:chsofr:v:123:y:2019:i:c:p:43-51
    DOI: 10.1016/j.chaos.2019.03.032
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    References listed on IDEAS

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    1. Chen, Wei-Ching, 2008. "Nonlinear dynamics and chaos in a fractional-order financial system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1305-1314.
    2. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
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    Cited by:

    1. 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).
    2. Liaqat, Muhammad Imran & Khan, Adnan & Akgül, Ali, 2022. "Adaptation on power series method with conformable operator for solving fractional order systems of nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    3. Martynyuk, Anatoliy A. & Stamov, Gani Tr. & Stamova, Ivanka M., 2020. "Fractional-like Hukuhara derivatives in the theory of set-valued differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    4. Ndenda, J.P. & Njagarah, J.B.H. & Shaw, S., 2021. "Role of immunotherapy in tumor-immune interaction: Perspectives from fractional-order modelling and sensitivity analysis," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).

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