= Two-species diffusion-driven instability criterion
= Two-species Turing criterion
{synonym}
Let a stable two-species reaction equilibrium have Jacobian
$$
J=\begin{pmatrix}a&b\\c&d\end{pmatrix},
\qquad \operatorname{tr}J<0,
\qquad \det J>0,
$$
and diffusion matrix $\operatorname{diag}(D_u,D_v)$. A mode with $q=k^2$ has determinant
$$
\det(J-qD)=\det J-(D_va+D_ud)q+D_uD_vq^2.
$$
Since its trace decreases with $q$, diffusion-driven instability occurs exactly when
$$
D_va+D_ud>0,
\qquad
(D_va+D_ud)^2>4D_uD_v\det J.
$$
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