Stability and bifurcation analysis of fractional-order tumor-macrophages interaction model with multi-delays

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1$$ {M}_1 $$-type macrophages with pro-inflammatory properties), and pro-tumor cells ( M2$$ {M}_2 $$-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters tau 1$$ {\tau}_1 $$ and tau 2$$ {\tau}_2 $$. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters.