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.

If a one-dimensional reaction-diffusion model has a strict unstable squared-wavenumber band $(q_{-},q_{+})$, its nonzero Neumann modes on $[0,L]$ have $q_n=(n\pi /L{)}^2$. Increasing $L$ from zero first reaches the band with $n=1$ at the displayed threshold. Equality is neutral; the first mode is strictly unstable for $\pi /\sqrt{q_{+}}<L<\pi /\sqrt{q_{-}}$. Thus positive growth has an infimum in domain length rather than an attained smallest length. The zero mode must be checked separately through reaction stability.

For a reaction-stable two-species equilibrium with both Jacobian matrix diagonal entries negative and positive diagonal diffusivities, each spatial mode has more negative trace after diffusion is added. Its determinant is the positive reaction determinant plus positive coefficients of $k^2$ and $k^4$. Hence every mode stays stable. Unequal diffusion alone is insufficient for a Turing instability; the reaction feedback must also permit a destabilizing weighted diagonal sum.

Suppose a two-species reaction-diffusion instability has an unstable squared-wave-number band $(q_{-},q_{+})$. On a rectangle with Neumann boundary conditions, $q=(n_x\pi /L_x{)}^2+(n_y\pi /L_y{)}^2$, with both indices nonnegative. The displayed thin-width condition excludes every $n_y\ge 1$ mode but can leave $n_y=0$ modes in the band for sufficiently large $L_x$. Omitting zero indices would incorrectly remove the longitudinal pattern. Linear onset selects allowed directions; it does not alone establish the final nonlinear pattern.

For $u_t=D{u}_{xx}-u(u-r)(u-1)-\rho (v-u)$ and $0=v_{xx}-v+u$, a Fourier mode perturbation about zero has $v=u/(1+k^2)$ and growth rate $\sigma (k)=-r-D{k}^2+\rho k^2/(1+k^2)$. If $\rho >D$, its maximum is $-r+(\sqrt{\rho }-\sqrt{D}{)}^2$ at $k^2=\sqrt{\rho /D}-1$; if $\rho \le D$, its maximum is $-r$ at zero. Replacing $(u,v,r)$ by $(1-u,1-v,1-r)$ gives the corresponding criterion at the other homogeneous equilibrium.

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.