Discrete-mode Turing threshold for the Brusselator ($b_T={\min }_{k\ne 0\mathrm{allowed}}[1+a^2{D}_u/D_v+D_u{k}^2+a^2/(D_v{k}^2)]$)

On a specified finite domain, only its boundary-compatible wavenumbers can grow. Solve the Brusselator mode determinant for its neutral value of $b$ and minimize over those nonzero modes. Minimizing over all continuous $k$ instead gives the usual threshold and $k_c^2=a/\sqrt{D_u{D}_v}$. The finite-mode minimum must still precede the homogeneous threshold to describe a Turing instability.