= Autocatalytic enzyme clearance with a saddle threshold
{title2=$u'=\eta u-u+uv+(K-\lambda)v,\quad\epsilon v'=u-uv-Kv$}
<Substrate> replication competes with saturating <enzyme> clearance in this fast-slow model, where $K>\lambda>0$ and $\epsilon\ll1$. The physical strip $u\geq0$, $0\leq v\leq1$ is invariant. If $0<\eta<\lambda/K$, the origin is attracting and a positive saddle exists at $u_e=\lambda/\eta-K$, $v_e=1-\eta K/\lambda$. Eliminating the fast complex fraction gives $u'\simeq\eta u(u-u_e)/(u+K)$, separating extinction from runaway. For initially zero complexes, the stable-manifold threshold is $u_*=u_e+\epsilon(1-\eta)v_e+O(\epsilon^2)$, away from singular <parameter> limits. To derive the correction, write the <stable manifold> as $u=u_e+\epsilon\phi(v)$; its invariance equation gives $\phi'=-(1-\eta)$ and $\phi(v_e)=0$. A Dulac factor $1/u$ has strictly negative weighted divergence, excluding a periodic alternative to the two basins. Saturated clearance and autocatalytic growth can therefore create an unstable <concentration> threshold rather than a stable nonzero balance.
Back to article page
