 Solution

At a positive spatially homogeneous equilibrium,soThe Jacobian matrix of the reaction terms there isThe linear stability analysis of a two-dimensional system requires negative trace and positive determinant. Since $a>0$, the homogeneous equilibrium is stable exactly when

For a spatial Fourier mode proportional to $\cos (kx)$, put $q=k^2$. The linearized matrix isIts trace is still negative when $0<a<1$, whileA Turing instability occurs for some $q>0$ exactly when this quadratic has a negative minimum. This requiresWriting $d=\sqrt{D}$, the second inequality reduces to $d<a(\sqrt{2}-1)$. Hence, for continuously available wave numbers, the condition isOn a bounded spatial domain, one additionally requires an allowed wave number $k$ for which $\det J_{k^2}<0$.