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.