 Diffusion-driven stationary instability

For reaction Jacobian $J=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with $\mathrm{tr}J<0$ and $\det J>0$, a Fourier mode evolves under $J-q^2\mathrm{diag}(D_u,D_v)$. Its determinant is $D_u{D}_v{q}^4-(D_{v}a+D_{u}d)q^2+\det J$. A finite unstable band requires $D_{v}a+D_{u}d>0$ and $(D_{v}a+D_{u}d{)}^2>4D_u{D}_v\det J$, with an allowed domain mode in that band.