Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 13C / b / Solution

{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
$$
\rho_c=(\sqrt r+\sqrt D)^2,
\qquad
k_c=(r/D)^{1/4}.
$$


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 Incoming links: Dispersion relation after fast-inhibitor elimination