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July  2009, 12(1): 151-168. doi: 10.3934/dcdsb.2009.12.151

Bifurcation analysis in models of tumor and immune system interactions

Dan Liu 1, , Shigui Ruan 2, and  Deming Zhu 1,

1. 

Department of Mathematics, East China Normal University, Shanghai 200062, China
2. 

Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124

Received  July 2008 Revised  September 2008 Published  May 2009

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.
Keywords: oscillation, lymphocyte., vascularization, tumor, Bogdanov-Takens bifurcation, saddle-node.
Mathematics Subject Classification: Primary: 34C23, 34C60; Secondary: 37G1.
Citation: Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151
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