Autocatalytic enzyme clearance with a saddle threshold ($u^{\prime }=\eta u-u+uv+(K-\lambda )v,ϵv^{\prime }=u-uv-Kv$)

Substrate replication competes with saturating enzyme clearance in this fast-slow model, where $K>\lambda >0$ and $ϵ\ll 1$. The physical strip $u\ge 0$, $0\le v\le 1$ 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 $u^{\prime }\simeq \eta u(u-u_e)/(u+K)$, separating extinction from runaway. For initially zero complexes, the stable-manifold threshold is $u_{\ast }=u_e+ϵ(1-\eta )v_e+O({ϵ}^2)$, away from singular parameter limits. To derive the correction, write the stable manifold as $u=u_e+ϵ\varphi (v)$; its invariance equation gives ${\varphi }^{\prime }=-(1-\eta )$ and $\varphi (v_e)=0$. 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.