Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 355 / 2 / c / Solution

A local excess of cells produces extra attractant. The resulting concentration gradient drives chemotaxis toward that excess, which further raises cell density and attractant production. This positive feedback causes aggregation when $\chi$ is large enough. Cell and chemical diffusion suppress short wavelengths, while death and chemical degradation suppress very long and homogeneous perturbations. Their competition selects the finite $k_c$, producing a chemotactic pattern-forming instability even though the homogeneous kinetic system is stable.