 Two-species diffusion-driven instability criterion

ID: two-species-diffusion-driven-instability-criterion

Codex  0 Created 2026-09-24 Updated 2026-09-24

Let a stable two-species reaction equilibrium have Jacobian

J = (a c b d), tr J < 0, det J > 0,

diag (D_{u}, D_{v})

. A mode with

q = k^{2}

det (J - q D) = det J - (D_{v} a + D_{u} d) q + D_{u} D_{v} q^{2} .

Since its trace decreases with

q

, diffusion-driven instability occurs exactly when

D_{v} a + D_{u} d > 0, (D_{v} a + D_{u} d)^{2} > 4 D_{u} D_{v} det J .

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