 Finite-domain selection of a reaction-diffusion instability

In a bounded domain, boundary conditions select a discrete set of spatial eigenvalues. If $-\Delta {\varphi }_j=q_j^2{\varphi }_j$, the linearized two-species reaction-diffusion equation evolves this mode with matrix $J-q_j^{2}D$, where $J$ is the reaction Jacobian matrix and $D$ the diagonal diffusion matrix. An unstable band of continuous wavenumbers produces an actual growing perturbation only if it contains an allowed $q_j$.

For a one-dimensional interval of length $L$ with Neumann boundary conditions, $q_j=j\pi /L$. An instability band $(q_{-},q_{+})$ is therefore active precisely when some nonnegative integer $j$ has $q_{-}<j\pi /L<q_{+}$. A sufficiently small domain can exclude the entire band even though the corresponding infinite-domain calculation predicts instability.