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Stability and bifurcation analysis of a mathematical model for tumor–immune interaction with piecewise constant arguments of delay

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  • Gurcan, Fuat
  • Kartal, Senol
  • Ozturk, Ilhan
  • Bozkurt, Fatma

Abstract

In this paper, we propose and analyze a Lotka–Volterra competition like model which consists of system of differential equations with piecewise constant arguments of delay to study of interaction between tumor cells and Cytotoxic T lymphocytes (CTLs). In order to get local and global behaviors of the system, we use Schur–Cohn criterion and constructed a Lyapunov function. Some algebraic conditions which satisfy local and global stability of the system are obtained. In addition, we investigate the possible bifurcation types for the system and observe that the system may undergo Neimark–Sacker bifurcation. Moreover, it is predicted a threshold value above which there is uncontrollable tumor growth, and below periodic solutions that leading to tumor dormant state occur.

Suggested Citation

  • Gurcan, Fuat & Kartal, Senol & Ozturk, Ilhan & Bozkurt, Fatma, 2014. "Stability and bifurcation analysis of a mathematical model for tumor–immune interaction with piecewise constant arguments of delay," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 169-179.
    • Handle: RePEc:eee:chsofr:v:68:y:2014:i:c:p:169-179
    DOI: 10.1016/j.chaos.2014.08.001
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    References listed on IDEAS

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    1. Radouane Yafia, 2006. "Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response," Discrete Dynamics in Nature and Society, Hindawi, vol. 2006, pages 1-13, July.
    2. F. Bozkurt, 2013. "Modeling a Tumor Growth with Piecewise Constant Arguments," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-8, May.
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    Cited by:

    1. Liu, Xiangdong & Li, Qingze & Pan, Jianxin, 2018. "A deterministic and stochastic model for the system dynamics of tumor–immune responses to chemotherapy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 162-176.
    2. S. Kartal & M. Kar & N. Kartal & F. Gurcan, 2016. "Modelling and analysis of a phytoplankton–zooplankton system with continuous and discrete time," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 22(6), pages 539-554, November.
    3. Duan, Wei-Long & Lin, Ling, 2021. "Noise and delay enhanced stability in tumor-immune responses to chemotherapy system," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    4. Kaya, Guven & Kartal, Senol & Gurcan, Fuat, 2020. "Dynamical analysis of a discrete conformable fractional order bacteria population model in a microcosm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).

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