= Slow and fast limits of a delayed stem-cell model
{title2=$u'=g(u(t-1))-g(u)+\epsilon[\mu g(u(t-1))-u]$}
For resting-cell recruitment $g(u)=bu/(1+u^n)$, a positive constant <equilibrium> exists precisely when $\mu b>1$, with $u_*=(\mu b-1)^{1/n}$. On slow time $T=\epsilon t$, expand the delayed value as $u(T-\epsilon)=u-\epsilon u_T+O(\epsilon^2)$. Substitution gives $(1+g'(u))u_T=\mu g(u)-u$ to leading order. This reduction requires smooth slow motion and $1+g'(u)$ bounded away from zero.
On the original fast time, only the explicit order-$\epsilon$ terms are dropped, leaving $u'=g(u(t-1))-g(u(t))$. The unit delay remains order one and cannot be Taylor-expanded on that scale. For the biological two-compartment derivation, a fixed cell-cycle time $\tau$ and proliferating death hazard $\gamma_d$ give survival $e^{-\gamma_d\tau}$ and daughter-cell gain twice the surviving exit rate. Scaling gives $b=\beta_0\tau$, $\epsilon=\delta\tau$ and $1+\epsilon\mu=2e^{-\gamma_d\tau}$. Thus small resting-cell loss over one cycle creates the slow-fast structure.
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