Modeling Tumor-Immune Interactions with Hyper-Allee Effects: Formulation, Stability, and Bifurcation Analysis.

Cancer progression emerges from a dynamical interplay between neoplastic growth and immune surveillance, yet most mathematical models reduce this interaction to logistic kinetics or single-threshold Allee effects, overlooking the layered barrier structure that governs tumor establishment. We introduce a three-dimensional immuno-oncology model coupling immune effector cells, cancer cells, and immunotherapy, which we reduce via singular perturbation and scale to a dimensionless planar system. The key novelty lies in endowing both compartments with generalized Allee functions: a single threshold for immune recruitment and a dual hyper-Allee law for tumor growth, reflecting cooperative, autocrine, and threshold-dependent processes at low cell densities. Through bifurcation analysis and numerical continuation in the clinically motivated parameter plane of immune loss versus cytotoxic efficacy, we uncover a global atlas of qualitatively distinct regimes. The organizing center is a cusp-type degenerate Bogdanov-Takens bifurcation of codimension three, from which saddle-node, Hopf, homoclinic, and limit-point-of-cycle curves emanate. Our results provide a quantitative scaffold for the three Es of cancer immunoediting: elimination, equilibrium, and escape, and suggest that patient stratification and therapeutic timing should be viewed as navigation problems in a high-dimensional threshold landscape.