= Solution
At every <equilibrium> the store balance gives $f=0$, and adding $\gamma$ <times> that balance to the cytoplasmic one gives $u=\mu$. The unique-root result therefore gives a single positive <equilibrium> $(\mu,V(\mu))$. Its <Jacobian matrix>[Jacobian] is
$$
A=\begin{pmatrix}-1-\gamma f_u/\epsilon&-\gamma f_v/\epsilon\\f_u/\epsilon&f_v/\epsilon\end{pmatrix},\quad
\boxed{\det A=-f_v/\epsilon>0,\qquad\operatorname{tr}A=-1+(f_v-\gamma f_u)/\epsilon,}
$$
with <derivatives> evaluated at that <equilibrium>. At small $\mu$, $f_u>0$ and the <matrix trace>[trace] is negative; sufficiently large $\mu$ also has $f_u>0$, since $J'$ is of order $u^{-3}$ and $K'$ of order $u^{-5}$. Thus \b[both extreme regimes have a <linearly stable> unique <equilibrium>].
The fast variable is not $v$ alone. Using <total-calcium fast fibres>, introduce total scaled <calcium> $w=u+\gamma v$. Then $\dot w=\mu-u$, while on fast <time> $\sigma=\tau/\epsilon$ the leading <equations> give $du/d\sigma=-\gamma f$, $dv/d\sigma=f$ and $w$ constant. <total-calcium fast fibres>[Fast paths] are therefore straight lines of slope $-1/\gamma$ in the $(u,v)$ plane. Attracting portions of the <critical manifold> satisfy $f_v-\gamma f_u<0$, rather than merely $f_v<0$. Along $v=V(u)$, slow motion obeys
$$
\boxed{\frac{du}{d\tau}=\frac{\mu-u}{1+\gamma V'(u)},\qquad V'=-f_u/f_v.}
$$
Starting with large $u$ and $v=0$, $f>0$ drives a rapid move upward and left, loading the store until the trajectory approaches the right attracting branch. For large $\mu$ in the right stable regime, slow motion on that branch carries it toward $u=\mu$; the direction reverses appropriately if the starting value lies below that <equilibrium>. For sufficiently small $\mu$, slow motion travels left along the right branch until its fold, then a fast approximately constant-$w$ jump carries the trajectory to the left attracting branch. Subsequent slow motion reaches the small-$u$ <equilibrium>. At a fold the slow approximation is singular, so the full <equations> resolve the transition. At very small $u$ an additional loss of time-scale separation is possible because the rates themselves vanish; the fast/slow drawing describes the separated portions, not a uniform estimate arbitrarily near the origin.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2009-calcium-phase-plane.png]
{title=Calcium phase plane: leading fast jumps and slow relaxation for small and large input}
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