 Solution

TODO: user-deferred biological kinetic mechanism and its biochemical pattern-forming applications. The remainder treats generic two-field reaction-diffusion physics only.

Let $\mathbf{u}=(u,v)$ obey ${\partial }_t\mathbf{u}=\mathbf{f}(\mathbf{u})+\mathrm{diag}(D_u,D_v){\nabla }^2\mathbf{u}$. At a uniform steady state, write its reaction Jacobian as $J=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Without diffusion stability requires $a+d<0$ and $\Delta =ad-bc>0$. A Fourier perturbation evolves under $J-q^2\mathrm{diag}(D_u,D_v)$, whose determinant isIts trace becomes more negative with increasing $q$. A stationary diffusion-driven instability therefore requireswith an allowed domain mode in the resulting negative-determinant band. Equal diffusivities cannot satisfy this while $a+d<0$. The physical mechanism is that transport can destabilize coupled fields even when the local kinetics relaxes a spatially uniform perturbation. Faster diffusion of an inhibitory response is one possible realization.

The selected wave numbers depend on growth rates and boundaries; nonlinear saturation determines the final stripes, spots or other states. Linear instability alone neither supplies a biochemical mechanism nor uniquely predicts a final finite-amplitude pattern.