Stability analysis and Hopf bifurcation in a delayed nonlinear tumor-macrophage model

Macrophages have a dual effect on tumor cells. This paper investigates a delayed nonlinear model describing the interactions between tumor cells (T), classically activated macrophages (M1), and alternatively activated macrophages (M2). The existence and non-negativity of the solutions are established, and the stability of equilibrium points is analyzed. It is found that as the time delay increases, the stability of the coexisting equilibrium point may switch through a Hopf bifurcation. Then, the multiple time scales method (MTS) is employed to derive the normal form of the Hopf bifurcation. Numerical simulations validate our theoretical analysis. Additionally, controlling the parameter (the ratio of activation rates by tumor cells M2/M1 macrophages) can also trigger a Hopf bifurcation. The emergence of stable limit cycles implies the long-term coexistence of tumor cells and macrophages, as well as the long term recurrence of tumors. Sensitivity analysis further confirmed the importance of macrophages in tumor treatment. These results may provide clues for combating cancer.