= Solution
After an <initial layer> of duration $O(\epsilon)$ in slow time, $v\simeq u/(u+K)$ while $u$ changes only slightly. The reduced <substrate> equation is
$$
\boxed{u'\simeq\eta u-\frac{\lambda u}{u+K}
=\frac{\eta u(u-u_e)}{u+K}.}
$$
Below $u_e$ it decreases to zero; above $u_e$ it increases without bound. The exact threshold for initial $v=0$ is the intersection of the <saddle equilibrium>'s <stable manifold> with that axis. For small $\epsilon$ it is
$$
\boxed{u_*\simeq u_e=\frac{k_2e_0}{k_0s_0}-\frac{k_{-1}+k_2}{k_1s_0}.}
$$
In dimensional initial-<substrate> units the leading threshold is $s_0u_*\simeq k_2e_0/k_0-(k_{-1}+k_2)/k_1$. Comparing the actual initial value $u(0)=1$ to this threshold determines its fate; the reference scale can be held fixed when varying the initial <concentration>.
The initial sequestration produces an $O(\epsilon)$ correction if desired. Write the <stable manifold> as $u=u_e+\epsilon\phi(v)+O(\epsilon^2)$. The invariance equation at leading order gives $\phi'(v)=F(u_e,v)/G(u_e,v)=-(1-\eta)$ and $\phi(v_e)=0$. Hence
$$
\boxed{u_*=u_e+\epsilon(1-\eta)v_e+O(\epsilon^2),}
$$
for fixed <parameters> away from degenerate limits. The positive correction reflects the <substrate> consumed in building complexes from the initially <enzyme>-free-complex state.
The physical strip $u\geq0$, $0\leq v\leq1$ is invariant, and the off-diagonal <derivatives> are nonnegative there, so ordered initial <substrate> values remain ordered. There are only the attracting origin and the <saddle equilibrium>. With <Dulac function> $1/u$, the divergence of the weighted vector field is $-(K-\lambda)v/u^2-(1+K/u)/\epsilon<0$, excluding <periodic orbits> in its interior. The <saddle equilibrium> <separatrix> therefore gives the unique threshold on the initial axis: below it solutions tend to extinction, at it they tend to the <saddle equilibrium>, and above it they escape to infinity. Growth is at most exponential, so this is $u\to\infty$ as time tends to infinity, not a <finite-time blowup>. This is <autocatalytic enzyme clearance with a saddle threshold>.
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