Past exam of the soft matter and biological physics course of the University of Cambridge / 2015 / Paper 1 / 2 / c / Solution

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.