Codimension-2 Bifurcations in an Angiogenesis–Tumor–Immune Model Predict Lymphocyte Extinction

In this paper, we extend the bifurcation analysis of the classical DeLisi–Rescigno minimal ordinary differential equation model in cancer dynamics with the immune system by explicitly incorporating capillary vascularization into a spherical tumor. Unlike prior treatments where higher-codimension phenomena remained implicit, we provide an explicit and comprehensive bifurcation diagram that identifies Bautin and nondegenerate Bogdanov–Takens bifurcations. In contrast to the nonvascular model, there is a lymphocyte suppression regime, an immune-deactivation state driven by vascular–immune coupling, and introduces the concept of a Bautin bifurcation at infinity within this oncological context. A description of the general bifurcation analysis is provided, thus supplying a biological interpretation.