= Chemotactic pattern-instability criterion
{title2=$\alpha\chi>(\sqrt{D_a}+\sqrt{\beta D_n})^2$}
For $n_t=D_nn_{xx}-\chi(na_x)_x+n(1-n)$, $a_t=D_aa_{xx}+\alpha n-\beta a$, the equilibrium $(1,\alpha/\beta)$ is stable without transport. A spatial mode of wavenumber $k$ has negative trace and <determinant> $D_nD_ak^4+(D_a+\beta D_n-\alpha\chi)k^2+\beta$. 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.
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