Dynamics and bifurcation of a hybrid competition model with applications to adaptive cancer therapy

Adaptive therapy is a novel cancer treatment strategy that proposes to tackle cancer drug resistance by leveraging resource competition between drug-sensitive and resistant cells. Because the underlying mathematical mechanisms of adaptive therapy remain unclear, determining the most effective rates of intervention is a significant challenge. In this paper, we propose a competition model incorporating fixed-time periodic tumor measurements with impulsive interventions performed if the number of tumor cells exceeds a threshold value. For the proposed model, we find a novel type of periodic solutions. Specifically, we demonstrate the existence of various (ℓ, m)T boundary periodic solutions and rigorously analyze their stability. Using bifurcation theory, we further prove the existence and stability of positive periodic solutions. Further, we perform numerical simulations to study these bifurcations with respect to key parameters such as the threshold value (TV) and the monitoring period (T). Numerical studies find that time to treatment failure exhibits a nonlinear dependence on the killing rate of drug-sensitive cells, i.e., it initially increases, reaches a plateau, and subsequently declines as the killing rate increases, revealing that maximizing the killing rate does not yield optimal therapeutic outcomes. The finding indicates that incorporating a threshold can extend patient's survival time, implying the therapeutic benefit of threshold-based adaptive therapy for tumor control.