Chemotactic instability with a wavenumber cutoff ($B>{\inf }_{q>q_0}(Dq+A\gamma /q)$)

For a homogeneous state with linear decay rate $A>0$, chemical decay $\gamma >0$, chemical diffusivity $D>0$ and chemotactic coupling $\beta {\chi }_0{n}_0$, a spatial mode $q=k^2$ has determinant $Q(q)=D{q}^2-Bq+A\gamma$, where $B=\beta {\chi }_0{n}_0-AD-\gamma$. 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.