= Chemotactic instability with a wavenumber cutoff
{title2=$B>\inf_{q>q_0}(Dq+A\gamma/q)$}
For a homogeneous state with linear decay rate $A>0$, chemical decay $\gamma>0$, chemical diffusivity $D>0$ and chemotactic coupling $\beta\chi_0n_0$, a spatial mode $q=k^2$ has determinant $Q(q)=Dq^2-Bq+A\gamma$, where $B=\beta\chi_0n_0-AD-\gamma$. The trace is negative, so an instability requires $Q(q)<0$. Without a cutoff, the sharp continuous-wavenumber threshold is $B>2\sqrt{DA\gamma}$. With $q>q_0$, minimize $Dq+A\gamma/q$ over that range; its unconstrained minimizer is $\sqrt{A\gamma/D}$. Discrete geometries additionally require an actual allowed Laplacian eigenvalue in the open unstable band.
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