= Solution
A generic chemical <reaction–diffusion system> couples local kinetics to spatial transport:
$$
\partial_tu=f(u,v)+D_u\nabla^2u,\qquad\partial_tv=g(u,v)+D_v\nabla^2v,\qquad D_u,D_v>0.
$$
A homogeneous stationary state satisfies $f=g=0$. Let its reaction Jacobian be $J=\begin{pmatrix}a&b\\c&d\end{pmatrix}$. Local well-mixed stability requires $a+d<0$ and $ad-bc>0$. A Fourier perturbation of wavevector magnitude $q$ evolves under $J-q^2\operatorname{diag}(D_u,D_v)$, whose determinant is
$$
D_uD_vq^4-(D_va+D_ud)q^2+\det J.
$$
Since its trace remains negative, a <diffusion-driven stationary instability> appears if the determinant becomes negative for some allowed wavevector. In a spatially large domain the conditions are
$$
\boxed{D_va+D_ud>0,\qquad(D_va+D_ud)^2>4D_uD_v\det J.}
$$
Together with reaction-only stability, these give a finite unstable wavelength range. Equal <diffusion> coefficients cannot generate this instability: they simply shift each reaction eigenvalue by $-Dq^2$. On a finite domain, an admissible spatial eigenmode must actually lie in the unstable band. Nonlinearities then determine saturation and whether a predicted pattern is stable, so a linear unstable band alone does not calculate a final pattern.<Notes on the Turing Instability and Chemical Instabilities by Michael Cross>[Linear stability of two-component chemical reaction-<diffusion> equations]{ref}
Other kinetics support travelling fronts or oscillatory waves rather than a stationary pattern. For instance, the scalar abstract equation $\partial_tu=D\partial_x^2u+ru(1-u/u_s)$ has a pulled-front speed $2\sqrt{Dr}$ for a sufficiently steep invading initial profile. Here it is used as an abstract kinetics-and-<diffusion> example, without a biological interpretation. The ratio of <diffusion> time $L^2/D$ to reaction time characterizes whether spatial mixing is rapid enough to justify a well-mixed model.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2015/paper-1-reaction-diffusion-spectrum.png]
{title=Differential diffusion can destabilize a finite band even when the uniform reaction system is linearly stable}
{source=paper-1-reaction-diffusion-spectrum.py}
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