= Solution
For spatially homogeneous <perturbations> write $q=1+u$ and $c=\eta+v$. The linearized reaction equations are
$$
\begin{pmatrix}\dot u\\\dot v\end{pmatrix}
=J\begin{pmatrix}u\\v\end{pmatrix},\qquad
\boxed{J=\begin{pmatrix}-1&0\\\kappa\eta&-\kappa\end{pmatrix}.}
$$
This <triangular matrix> has <eigenvalues> $-1$ and $-\kappa$, both negative. Hence the <homogeneous steady state> is linearly <asymptotically stable> against homogeneous <perturbations>. Even when $\kappa=1$, where the two <eigenvalues> coincide and a <polynomial> factor may occur in the <matrix exponential>, all such <perturbations> still decay.
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