Chemotactic pattern-instability criterion ($\alpha \chi >(\sqrt{D_a}+\sqrt{\beta D_n}{)}^2$)

For $n_t=D_n{n}_{xx}-\chi (n{a}_x{)}_x+n(1-n)$, $a_t=D_a{a}_{xx}+\alpha n-\beta a$, the equilibrium $(1,\alpha /\beta )$ is stable without transport. A spatial mode of wavenumber $k$ has negative trace and determinant $D_n{D}_a{k}^4+(D_a+\beta D_n-\alpha \chi )k^2+\beta$. A continuous unstable band exists exactly under the displayed inequality. On a finite interval with Neumann boundary conditions, an allowed discrete wavenumber must additionally fall in that band. The high-wavenumber positive term is diffusive smoothing; the negative chemotactic term drives aggregation.