= Solution
If $D\geq1/2$, both nonconstant coefficients of $\Delta(q)$ are nonnegative, so it is positive for $q\geq0$. If $D_c<D<1/2$, its <discriminant> is negative, so it is again strictly positive. Therefore $\boxed{D>D_c\implies\text{no diffusion-driven instability}}$.
At $D=D_c$ the double root can only give a neutral zero <eigenvalue> if it coincides with an allowed spatial mode; it still gives no positive growth rate. This is the finite-<wavenumber> onset of a <Turing instability>, not an oscillatory crossing since the mode <trace> stays negative.
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