Bifurcation analysis of tumor-immune dynamics under the dual Allee effects

In this work, we investigate the impact of the dual Allee effects on tumor-immune interactions using an ordinary differential equation model. We analyze how the strength of the Allee effect in both effector and cancer cell populations influences the stability of equilibrium points. Our results suggest that moderate positive values of Allee effects can promote rapid population growth and complex population dynamics. In contrast, larger values of the Allee effects reduce the system's dynamical complexity. The model exhibits a rich bifurcation structure, including saddle-node and Hopf bifurcations (co-dimension one) as well as generalized Hopf and Bogdanov-Takens bifurcations (co-dimension two). These findings highlight the importance of identifying critical thresholds in tumor-immune interactions, which could be leveraged for personalized antitumor treatments.