 Solution

If $D=1$, both species have the same diffusion coefficient. For a perturbation proportional to $e^{\sigma t+ikx}$, diffusion replaces $J$ by $J-k^{2}I$, shifting both reaction eigenvalues to the left by $k^2$. Since the homogeneous equilibrium is already stable, every spatial mode remains stable.

For general $D$, put $q=k^2$. The mode matrix isIts trace iswhile its determinant isWith negative trace, instability occurs exactly when $\Delta (q)<0$ for some $q>0$. The two-species diffusion-driven instability criterion says that this upward-opening quadratic has a negative minimum precisely whenEquivalently, the condition isThis is the Turing instability region.