 Thin-domain Turing mode selection

Suppose a two-species reaction-diffusion instability has an unstable squared-wave-number band $(q_{-},q_{+})$. On a rectangle with Neumann boundary conditions, $q=(n_x\pi /L_x{)}^2+(n_y\pi /L_y{)}^2$, with both indices nonnegative. The displayed thin-width condition excludes every $n_y\ge 1$ mode but can leave $n_y=0$ modes in the band for sufficiently large $L_x$. Omitting zero indices would incorrectly remove the longitudinal pattern. Linear onset selects allowed directions; it does not alone establish the final nonlinear pattern.