Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / A / 2 / iv

At equilibrium $v=u/(u+K)$ and adding $u^{\prime }+ϵv^{\prime }$ gives $0=\eta u-\lambda v$. The positive equilibrium isThis condition also gives $\eta <1$. The nullclines are $v=u/(u+K)$ and $v=(1-\eta )u/(u+K-\lambda )$. The second lies above the first for $0<u<u_e$ and below it for $u>u_e$. Below the first nullcline $v^{\prime }>0$; above it $v^{\prime }<0$. Below the second nullcline $u^{\prime }<0$; above it $u^{\prime }>0$.

At the origin the Jacobian has trace $\eta -1-K/ϵ<0$ and determinant $(\lambda -\eta K)/ϵ>0$. Its off-diagonal entries are positive, making the eigenvalues real, so this is an attracting node. At the positive equilibrium the slow reduced derivative is $\eta -\lambda K/(u_e+K{)}^2=\eta v_e>0$, while the fast derivative is $-(u_e+K)/ϵ<0$; equivalently the full Jacobian determinant is negative. Thus the positive point is a saddle equilibrium. Its stable manifold separates extinction from unbounded substrate growth.

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.