Abstract
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.
Highlights
Cancer still remains one of the most dangerous killers of humankind in the 21th century
The qualitative analysis and some bifurcation results near the degenerate equilibrium have been given for the cancer models (1) and (2) in this paper
By applying the transformation and bifurcation theory in [10] and [29], we have discovered that the degenerate equilibrium is a nondegenerate cusp of codimension two when the parameters take some critical values whether the cancer model suffers the neovascularization or not
Summary
Cancer still remains one of the most dangerous killers of humankind in the 21th century. Millions of people die from this disease every year throughout the world ([9]). Investigation ([19]) showed that about ten percent of patients who have spontaneous immunodeficiency diseases may develop cancer. Clinic and laboratory sources indicate that the immune system plays an important role in controlling and eliminating tumor cells, and decreasing the observed incidence of cancer. This response of immune system to the precancerous and cancerous is the so-called immunosurveillance ([17]). More detailed research about the immune surveillance can be found in [5],[13],[14], and [24]
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