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A physicist whose teaching and research address pattern formation in systems away from equilibrium. His notes on the Turing instability show how a spatially uniform chemical state can resist uniform perturbations yet become unstable when diffusion is included. Linear stability analysis separates the reaction dynamics from the wavelength-dependent effect of diffusion, explaining how a preferred spatial pattern can develop.^[ref]

A generic chemical reaction–diffusion system couples local kinetics to spatial transport:A homogeneous stationary state satisfies $f=g=0$. Let its reaction Jacobian be $J=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Local well-mixed stability requires $a+d<0$ and $ad-bc>0$. A Fourier perturbation of wavevector magnitude $q$ evolves under $J-q^2\mathrm{diag}(D_u,D_v)$, whose determinant isSince 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 areTogether 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 $-D{q}^2$. 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.^[ref]

Other kinetics support travelling fronts or oscillatory waves rather than a stationary pattern. For instance, the scalar abstract equation ${\partial }_{t}u=D{\partial }_x^{2}u+ru(1-u/u_s)$ 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.

A reaction-diffusion system couples local changes in several fields with their spatial spreading. For concentrations $u=(u_1,\dots ,u_m)$, a common model isThe diffusion coefficients $D_i$ describe transport, while the reaction rates $f_i$ describe local conversion, production or loss. Writing $D=\mathrm{diag}(D_1,\dots ,D_m)$ gives the vector equation $u_t=D\Delta u+f(u)$. Chemical concentrations and rates must be modeled consistently; a reaction term can exchange material with an external supply.

A spatially homogeneous equilibrium $u_{\ast }$ satisfies $f(u_{\ast })=0$. Linearizing a spatial mode with $-\Delta \varphi =q\varphi$ givesThus reaction stability tests $J$, while spatial stability tests all boundary-compatible values of $q$. Unequal diffusion can create a Turing instability even when homogeneous perturbations decay.^[ref] The Brusselator and quadratic activator-inhibitor model give explicit two-species examples. Initial and boundary conditions are part of the system; the kinetic equations alone do not select its spatial modes.

Let a homogeneous two-species reaction equilibrium have Jacobian matrixand let $D=\mathrm{diag}(D_u,D_v)$ with $D_u,D_v>0$. A spatial mode with Laplacian eigenvalue $-q$, $q\ge 0$, evolves by $J-qD$. Its trace remains negative, while its determinant isFor continuous wavenumbers, a Turing instability exists exactly whenThese conditions put the minimum of $P$ at positive $q$ and below zero. The unstable band isOn a finite domain, an actual Laplacian mode satisfying the boundary conditions must lie strictly in this band. For example, Neumann boundary conditions on $[0,L]$ allow $q_n=(n\pi /L{)}^2$, rather than every real $q$. The inequalities alone may therefore predict an instability band containing no available mode. At the band endpoints the mode is neutral. This separates the continuous-wavenumber kinetic criterion from finite-domain mode selection.^[ref]