Solution
= Solution
Since $f(0)=0$ and $f'(0)=-r$, the <linearization> is
$$
f(u)=-ru+O(u^2).
$$
For a spatial <Fourier mode> of the perturbation, part a gives
$$
\frac{\partial\widehat u}{\partial t}
=\left[
-Dk^2-r+\rho\left(1-\frac1{1+k^2}\right)
\right]\widehat u.
$$
Thus $\widehat u(k,t)\propto e^{\sigma(k)t}$ with the <dispersion relation after fast-inhibitor elimination>
$$
\boxed{\sigma(k)=-Dk^2-r+\rho\frac{k^2}{1+k^2}.}
$$