Codimension-one bifurcation analysis and chaos control in a discrete pro- and anti-tumor macrophages model

The study of genetically engineered macrophages killing tumor cells has become the most promising new direction of cancer therapy. In this paper, we consider discrete macrophages, which are divided into M1 and M2 macrophages, and establish discrete models of pro- and anti-tumor macrophages. Firstly, we provide sufficient conditions for the existence and stability of fixed points. Along with the parameter values change, it undergoes flip bifurcation and Neimark–Sacker bifurcation. In numerical simulation, bifurcation diagrams and phase diagrams are given, and it is found that there are more abundant and complex dynamic behaviors, such as periodic solution, chaos and hetero-cycle. This suggests that tumor cells and macrophages co-exist, causing the disease to worsen. Based on the chaos appearing in the bifurcation diagrams, we propose a chaotic control method. The results show that the large-scale outbreak of the disease is effectively avoided, and macrophages are promoted to inhibit tumor growth and destroy the tumor.