Gamma-Distributed Memory and Hopf Stabilization in a Tumor–Macrophage Interaction Model

Maturation and signaling in tumor-associated macrophages are distributed over time rather than concentrated at a single instant. We replace the discrete maturation lag in a tumor–macrophage model with weak and strong gamma memory kernels. Because the kernels are normalized, the equilibria of the original model are preserved, whereas the linear chain trick yields distinct polynomial characteristic equations for the two memory shapes. The analysis identifies kernel-dependent local stability thresholds for the coexistence equilibrium and shows that the same memory scale can correspond to different local regimes under weak and strong memory. We establish simple imaginary-axis crossings and verify transversality, so the results identify local Hopf thresholds. Sensitivity and two-parameter stability diagrams further describe how these thresholds depend on M2 growth, phenotype conversion and M2 loss.