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Dispersion relation after fast-inhibitor elimination ( $σ (k)$ )

For activator diffusion

D

, local linear decay

r

, and coupling

ρ (u - v)

, eliminating the fast inhibitor gives

σ (k) = - D k^{2} - r + ρ \frac{k ^{2}}{1 + k ^{2}} .

Its pattern-forming threshold is

ρ_{c} = (r + D)^{2}, k_{c} = (r / D)^{1/4} .

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