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Mathematical Biosciences and Engineering

2012, Volume 9, Issue 2: 347-368. doi: 10.3934/mbe.2012.9.347
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Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions

  • 1.
    Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071
  • 2.
    Department of Mathematics, University of Miami, Coral Gables, FL 33124
  • 3.
    Department of Mathematics, East China Normal University, Shanghai 200062
  • Received: 01 January 2011 Accepted: 29 June 2018 Published: 01 March 2012

  • MSC : Primary: 34C23, 34C60; Secondary: 37G10.

  • This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions between lymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions. We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerate Hopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated from the interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained for the two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parameters and demonstrate the phenomenon of long-term tumor relapse in the model.

    Citation: Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions[J]. Mathematical Biosciences and Engineering, 2012, 9(2): 347-368. doi: 10.3934/mbe.2012.9.347

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  • This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions between lymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions. We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerate Hopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated from the interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained for the two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parameters and demonstrate the phenomenon of long-term tumor relapse in the model.


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