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

For homogeneous fields the gradient terms vanish, leavingThusThe homogeneous Jacobian is triangular,with strictly negative eigenvalues $-\delta$ and $-\beta$. The fixed point is therefore linearly stable to homogeneous perturbations.

To first order about the fixed point,For a mode proportional to $e^{\sigma t+ikx}$, the linear matrix isIts trace is always negative, so instability occurs exactly when its determinant is negative. Writing $q=k^2$,At onset this quadratic touches zero, requiringThe repeated positive root isand therefore

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.