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Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. (English) Zbl 1259.34028

Summary: 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.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37G10 Bifurcations of singular points in dynamical systems
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