Bifurcation Analysis of Tumor-Immune Dynamics and the Role of Strong and Weak Allee Effects in Cancer Immunoediting Process

We introduce a mathematical model to study tumor–immune interactions and analyze the effects of strong and weak Allee dynamics to capture the cooperative behavior inherent in tumor growth. In this work, we observe that the presence of Allee dynamics delays tumor proliferation at low cell densities, significantly influencing the critical thresholds that regulate tumor cell cooperation. We examine key dynamical behaviors through bifurcation analysis, including saddle-node, Hopf, Bautin, and Bogdanov–Takens (BT) bifurcations, using detailed analytical expressions and numerical continuation methods. Our model demonstrates that a strong Allee effect raises the critical threshold for tumor cell cooperation, leading to greater tumor control, as a higher antigenicity value is required to maintain equilibrium thresholds. In contrast, a weak Allee effect has a less pronounced impact on the model compared to a strong Allee effect. We also explore the case where the weak Allee effect corresponds to the parameter representing the half-saturation cancer clearance level, revealing a key dynamic behavior of the system under this condition. In this case, we prove the existence of BT bifurcation of co-dimension-three-type cusp. These findings could provide valuable insights into the transitions between elimination, equilibrium, and escape phases in the cancer immunoediting process. Moreover, our results could serve as a valuable tool for precisely calibrating these effects in the development of more effective therapeutic strategies.