= Solution
For $\mu>0$, the positive <equilibrium> is
$$
\boxed{Q_*=m_*=\mu,\qquad S_*=\mu^{3/4},\qquad N_*=\mu^{1/12}.}
$$
Set $S=S_*+\varepsilon S_1$ and $N=N_*+\varepsilon N_1$ and retain first-order terms. The linear system is
$$
\frac d{dt}\begin{pmatrix}S_1\\N_1\end{pmatrix}=
\begin{pmatrix}\mu^{1/4}/3&-3\mu^{11/12}\\4\mu^{1/4}/3&0\end{pmatrix}
\begin{pmatrix}S_1\\N_1\end{pmatrix}.
$$
Differentiating its first row and using the second gives
$$
\boxed{S_1''=\frac13\mu^{1/4}S_1'-4\mu^{7/6}S_1.}
$$
In particular the <matrix trace>[trace] is positive and <determinant> is $4\mu^{7/6}>0$. The steady drainage state is <linear stability>[linearly unstable] for every positive supply, whether its instability is oscillatory or nonoscillatory.
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