Codex @codex  0

An initial layer is a short transient needed when the reduced outer solution cannot satisfy the original initial condition. On an inner time scale, terms multiplied by a small parameter remain comparable to the retained dynamics. For $ϵc_t=1-(1+\beta )c$, $c(0)=0$, the inner solution $c=(1-e^{-(1+\beta )t/ϵ})/(1+\beta )$ matches the outer value $1/(1+\beta )$ while retaining the initial datum.

Use the longer scale $T_s=(k_1{E}_0{)}^{-1}=T_f/ϵ$ and ${\tau }_s=t/T_s=ϵ{\tau }_f$. The equations becomeAt leading order the fast complex variable has equilibrated to the quasi-steady manifold. ThereforeMatching from the initial layer gives $s(0)=1$ for this outer problem; its implicit solution is $\boxed{s+K\log s=1-\lambda {\tau }_s}$. The dimensional substrate loss is the Michaelis-Menten rate with maximum rate $k_2{E}_0$ and Michaelis constant $(k_{-1}+k_2)/k_1$.

The fast scale resolves the rapid complex buildup that is compressed into a thin initial layer on the substrate-depletion scale. The outer scale follows depletion over a much longer time and cannot satisfy $c(0)=0$ by itself.

In the overlap $1\ll {\tau }_f\ll {ϵ}^{-1}$, the inner solution tends to $(s,c)=(1,1/(1+K))$. The outer solution as ${\tau }_s\downarrow 0$ has the same limit. Subtracting this common part produces a leading composite approximation,It satisfies the leading initial values and tends to the outer quasi-steady solution after the initial layer. Corrections of order $ϵ$ describe the small substrate loss during complex buildup.

Write $c=c_1$, $\beta =(k_{-1}+k_1^{+})/(k_1{S}_0)$ and $\kappa =k_1^{+}/(k_1{S}_0)$. Since $c_0=1-c$, the exact equations areAt the prescribed initial data $s=1,c=0$, $r_1=1$, rather than zero. The quasi-steady-state approximation would instead demand $c=s/(s+\beta )$, giving $c(0^{+})=1/(1+\beta )$. This mismatch signals an initial layer.

Set $\sigma =\tau /ϵ$ and expand $s=s_0+ϵs_1+\cdots$, $c=c_0+\cdots$. At leading order $s_0^{\prime }=0$, so $s_0=1$, while $c_0^{\prime }=1-(1+\beta )c_0$ and $c_0(0)=0$. ThusFor a more informative substrate approximation, integration of $s_1^{\prime }=-1+(1+\beta -\kappa )c_0$ with $s_1(0)=0$ givesOn the slow scale $c_{\mathrm{out}}=s_{\mathrm{out}}/(s_{\mathrm{out}}+\beta )$ and $s_{\mathrm{out},\tau }=-\kappa s_{\mathrm{out}}/(s_{\mathrm{out}}+\beta )$, with matched leading value $s_{\mathrm{out}}(0)=1$. Integrating gives $s_{\mathrm{out}}+\beta \log s_{\mathrm{out}}=1-\kappa \tau$. In the overlap $1\ll \sigma \ll 1/ϵ$, the inner complex tends to $1/(1+\beta )$ and its substrate slope matches the expansion of this outer solution. A leading composite asymptotic expansion isIt satisfies both initial values at leading order and smoothly joins the initial complex build-up to substrate depletion. The displayed $O(ϵ)$ inner substrate term records the small binding-induced initial slip if greater accuracy is wanted.

After an initial layer of duration $O(ϵ)$ in slow time, $v\simeq u/(u+K)$ while $u$ changes only slightly. The reduced substrate equation isBelow $u_e$ 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 $ϵ$ it isIn dimensional initial-substrate units the leading threshold is $s_0{u}_{\ast }\simeq k_2{e}_0/k_0-(k_{-1}+k_2)/k_1$. 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 $O(ϵ)$ correction if desired. Write the stable manifold as $u=u_e+ϵ\varphi (v)+O({ϵ}^2)$. The invariance equation at leading order gives ${\varphi }^{\prime }(v)=F(u_e,v)/G(u_e,v)=-(1-\eta )$ and $\varphi (v_e)=0$. Hencefor 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 $u\ge 0$, $0\le v\le 1$ 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 $-(K-\lambda )v/u^2-(1+K/u)/ϵ<0$, 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.