Bifurcation analysis in models of tumor and immune system interactions

Dan Liu, Shigui Ruan, Deming Zhu

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)151-168
Number of pages18
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number1
StatePublished - Jul 1 2009


  • Bogdanov-Takens bifurcation
  • Lymphocyte
  • Oscillation
  • Saddle-node
  • Tumor
  • Vascularization

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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