First domain length for an unstable Neumann band ($L_c=\pi /\sqrt{q_{+}}$)

If a one-dimensional reaction-diffusion model has a strict unstable squared-wavenumber band $(q_{-},q_{+})$, its nonzero Neumann modes on $[0,L]$ have $q_n=(n\pi /L{)}^2$. Increasing $L$ from zero first reaches the band with $n=1$ at the displayed threshold. Equality is neutral; the first mode is strictly unstable for $\pi /\sqrt{q_{+}}<L<\pi /\sqrt{q_{-}}$. Thus positive growth has an infimum in domain length rather than an attained smallest length. The zero mode must be checked separately through reaction stability.