= Solution
When $D_u=D_v=D$, the linearized operator is
$$
\partial_t
\binom{\widetilde u}{\widetilde v}
=
\left(M+D\nabla^2I\right)
\binom{\widetilde u}{\widetilde v}.
$$
For a Fourier mode of wavenumber $k$, its two growth rates are
$$
\lambda_j(k)=\lambda_j(0)-Dk^2.
$$
Both homogeneous growth rates have negative real part, and subtracting the nonnegative scalar $Dk^2$ moves them farther into the left half-plane. Thus no spatial mode can become unstable. This is the <impossibility of a two-species Turing instability at equal diffusivities>.
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