Bifurcation mechanism, speed feedback controller, and hybrid controller design in a delayed tumor-immune competitive model

In this study, we formulate a delayed tumor-immune competitive model by incorporating dual time delays into the biological interactions, extending prior modeling frameworks. By employing fixed point theory, inequality techniques, and construction of functions, we explore the well-posedness of solutions, including the existence and uniqueness, non-negativity, and boundedness. The new stability and Hopf bifurcation conditions of the proposed model are analytically derived via the Routh–Hurwitz criterion and the bifurcation theory of delayed differential systems. We design two control strategies (namely, a hybrid controller with a state feedback and parameter perturbation with delay and a speed feedback controller) to control the stability domain and bifurcation behavior of the formulated model. These controllers effectively delay or advance bifurcation onset and expand or narrow the system’s stability domain. Numerical simulations validate the analytical findings, illustrating how dual delays influence the stability of equilibrium and bifurcation patterns. The results gained from this article can provide theoretical support for optimizing cancer treatment, emphasizing the critical role of time delays in the tumor immune dynamics.