Complex Dynamics Analysis of a Tumor–Immune Interaction Model Including Chemotherapy and the Allee Effect

In this study, we investigate the threshold-dependent dynamics of brain metastasis under chemotherapy using a nonlinear mathematical model that incorporates proliferating tumor cells, treatment-damaged cells, nonviable/necrotic cells, an associated immune response, and time-dependent treatment activity. The main objective is to determine how the Allee effect and treatment intensity jointly influence tumor persistence and elimination. The model is analyzed through equilibrium and local stability analyses, center manifold theory, and sensitivity and bifurcation analyses, complemented by numerical simulations under different treatment and initial tumor-burden scenarios. The theoretical analysis reveals that the Allee threshold plays a central role in determining the long-term tumor dynamics. In particular, increasing treatment intensity raises the smaller positive equilibrium, interpreted as a treatment-dependent effective Allee threshold, while decreasing the larger stable positive equilibrium, thereby narrowing the interval between the two positive equilibria. The two positive equilibria coalesce through a saddle-node bifurcation at a critical treatment level; for the baseline parameter set, this occurs at vc≈11.3349, beyond which no positive equilibrium exists. Sensitivity analysis identifies the parameters that most strongly influence the unstable and stable positive equilibria. Numerical simulations further reveal bistable behavior and demonstrate that, for the same treatment level, different initial tumor burdens can lead to either the tumor-free or a positive equilibrium. The numerically determined outcome boundary and phase-plane analysis further characterize this dependence on treatment intensity and initial tumor burden. Overall, the proposed framework provides a threshold-based characterization of the interaction between the Allee effect and time-dependent chemotherapy and demonstrates how treatment can influence both the equilibrium structure and the basin-dependent long-term dynamics of the system.