Turing instability

A Turing, or diffusion-driven, instability occurs when a spatially homogeneous equilibrium is stable to homogeneous perturbations but unstable to a nonzero spatial Fourier mode after unequal diffusion is introduced.

Let a stable two-species reaction equilibrium have Jacobianand diffusion matrix $\mathrm{diag}(D_u,D_v)$. A mode with $q=k^2$ has determinantSince its trace decreases with $q$, diffusion-driven instability occurs exactly when

When $D_u=D_v=D$, every spatial Fourier mode replaces the reaction Jacobian $J$ by $J-D{k}^{2}I$. Each eigenvalue is shifted left by $D{k}^2$, so diffusion cannot destabilize a stable homogeneous equilibrium.

For reaction Jacobianthe critical diffusivity ratio isIt approaches one as the stable reaction system approaches marginality.