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Chaos in a Tumor Growth Model with Delayed Responses of the Immune System

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  • M. Saleem
  • Tanuja Agrawal

Abstract

A simple prey‐predator‐type model for the growth of tumor with discrete time delay in the immune system is considered. It is assumed that the resting and hunting cells make the immune system. The present model modifies the model of El‐Gohary (2008) in that it allows delay effects in the growth process of the hunting cells. Qualitative and numerical analyses for the stability of equilibriums of the model are presented. Length of the time delay that preserves stability is given. It is found that small delays guarantee stability at the equilibrium level (stable focus) but the delays greater than a critical value may produce periodic solutions through Hopf bifurcation and larger delays may even lead to chaotic attractors. Implications of these results are discussed.

Suggested Citation

  • M. Saleem & Tanuja Agrawal, 2012. "Chaos in a Tumor Growth Model with Delayed Responses of the Immune System," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:891095
    DOI: 10.1155/2012/891095
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    References listed on IDEAS

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    1. Menchón, S.A. & Ramos, R.A. & Condat, C.A., 2007. "Modeling subspecies and the tumor-immune system interaction: Steps toward understanding therapy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(2), pages 713-719.
    2. El-Gohary, Awad, 2008. "Chaos and optimal control of cancer self-remission and tumor system steady states," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1305-1316.
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    Cited by:

    1. Zhen Wang & Huitao Zhao & Xiangyu Kong, 2013. "Delayed Feedback Control and Bifurcation Analysis of an Autonomy System," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).

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