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Bifurcation analysis in models of tumor and immune system interactions. (English) Zbl 1179.34047
Summary: The purpose of this paper is to present qualitative and bifurcation analysis near the degenerate equilibrium in models of interactions between lymphocyte cells and solid tumor and to understand the development of tumor growth. Theoretical analysis shows that these cancer models can exhibit Bogdanov-Takens bifurcation under sufficiently small perturbation of the system parameters whether it is vascularized or not. Periodic oscillation behavior and coexistence of the immune system and the tumor in the host are found to be influenced significantly by the choice of bifurcation parameters. It is also confirmed that bifurcations of codimension higher than 2 cannot occur at this equilibrium in both cases. The analytic bifurcation diagrams and numerical simulations are given. Some anomalous properties are discovered from comparing the vascularized case with the avascular case.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92C50 Medical applications (general)
34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37G10 Bifurcations of singular points in dynamical systems
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