Dispersion relation after fast-inhibitor elimination
= Dispersion relation after fast-inhibitor elimination
{title2=$\sigma(k)$}
For activator diffusion $D$, local linear decay $r$, and coupling $\rho(u-v)$, eliminating the fast inhibitor gives
$$
\sigma(k)=-Dk^2-r+\rho\frac{k^2}{1+k^2}.
$$
Its pattern-forming threshold is
$$
\rho_c=(\sqrt r+\sqrt D)^2,
\qquad
k_c=(r/D)^{1/4}.
$$