= 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 $\partial_t\mathbf u=\mathbf f(\mathbf u)+\operatorname{diag}(D_u,D_v)\nabla^2\mathbf u$. At a uniform steady state, write its reaction Jacobian as $J=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$. Without <diffusion> stability requires $a+d<0$ and $\Delta=ad-bc>0$. A Fourier perturbation evolves under $J-q^2\operatorname{diag}(D_u,D_v)$, whose <determinant> is
$$
\Delta(q)=\Delta-(D_va+D_ud)q^2+D_uD_vq^4.
$$
Its <trace> becomes more negative with increasing $q$. A stationary diffusion-driven instability therefore requires
$$
\boxed{D_va+D_ud>0,\qquad (D_va+D_ud)^2>4D_uD_v\Delta,}
$$
with 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.
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