Bifurcation Analysis of a Two-Dimensional Cancer–Immune Model with Antigenicity and Strong Allee Effect

Mathematical modeling of tumor–immune dynamics has become an essential tool for understanding the nonlinear behavior of cancer progression and control. In this talk, we investigate a two-dimensional cancer growth model that has been enhanced by incorporating two biologically relevant mechanisms: tumor antigenicity and a strong Allee effect governing cancer cell survival thresholds. These additions allow for a more realistic representation of tumor establishment and immune system recognition. Our analysis begins with the identification and classification of the model’s equilibrium points across biologically feasible parameter ranges. We employ analytical and numerical techniques to determine stability and construct bifurcation diagrams. Particular attention is given to codimension-1 bifurcations, such as saddle–node and Hopf bifurcations, which signal qualitative transitions in tumor dynamics. We further extend the analysis to codimension-2 bifurcations, specifically Bogdanov–Takens, Bautin, and cusp bifurcations, to delineate regions in parameter space with complex dynamic behavior. We derive explicit conditions on key parameters that delimit dynamic regimes of tumor suppression, coexistence, and growth. Critical threshold curves are obtained that partition the phase space, revealing how the interplay between antigenicity and the Allee threshold influences tumor fate. The bifurcation structure identified in this enhanced cancer model provides a mechanistic framework for interpreting the three phases of cancer immunoediting, elimination, equilibrium, and escape in terms of underlying mathematical thresholds. These insights improve our theoretical understanding of tumor progression and may guide future therapeutic strategies that exploit immune response and tumor viability thresholds.