= Solution
The decreasing production term represents <negative feedback>: low red-cell abundance stimulates production, whereas high abundance suppresses it. The linear loss term models removal at constant per-cell rate. Set
$$
\boxed{u=E/\theta,\quad\tau=F_0t/\theta,\quad\delta=\gamma\theta/F_0.}
$$
Then $u'=f(u)-\delta u$, with $f(u)=1/(1+u^n)$. On the physical half-line $u\geq0$, $f$ decreases from one to zero while $\delta u$ increases from zero to infinity. Their unique intersection $u_*>0$ is the <steady state>. Its linear <derivative> is $f'(u_*)-\delta<0$, so it is asymptotically stable.
For small $\delta$, the intersection lies at large $u$, where $f\sim u^{-n}$; for large $\delta$ it lies at small $u$, where $f\sim1$. Hence
$$
\boxed{u_*\sim\delta^{-1/(n+1)}\quad(\delta\downarrow0),\qquad u_*\sim\delta^{-1}\quad(\delta\to\infty).}
$$
Implicit differentiation of $f(u_*)=\delta u_*$ gives $du_*/d\delta=u_*/[f'(u_*)-\delta]<0$. The plot joins these two decreasing asymptotic regimes.
\Image[/past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2008-rbc-equilibrium.png]
{title=Negative production feedback gives one stable red-cell equilibrium and two limiting concentration scalings}
The delay $t_0$ represents the time between feedback sensing and the appearance of newly produced mature cells. In <dimensionless> time it is $d=F_0t_0/\theta$, so the equation becomes
$$
\boxed{u'(\tau)=\frac1{1+u(\tau-d)^n}-\delta u(\tau).}
$$
It requires a nonnegative initial history on $[-d,0]$, not just one initial value. Its constant <equilibrium> is unchanged. Put $a=-f'(u_*)=nu_*^{n-1}/(1+u_*^n)^2>0$. A small perturbation obeys $v'=-\delta v-av(\tau-d)$, whose <characteristic equation> is
$$
\boxed{\sigma+\delta+ae^{-\sigma d}=0.}
$$
There is no real root with $\sigma\geq0$, because all three terms would have positive sum; zero is never a root either. Starting from the stable zero-delay case, loss of <linear stability> can therefore occur only when a <complex conjugate> pair crosses the imaginary axis at a nonzero frequency, the linear signature of a <Hopf bifurcation>.
For completeness, substituting $\sigma=i\omega$ gives $\delta+a\cos(\omega d)=0$ and $\omega=a\sin(\omega d)$. Such a crossing requires $a>\delta$, with
$$
\boxed{\omega=\sqrt{a^2-\delta^2},\qquad d_c=\frac{\arccos(-\delta/a)}{\sqrt{a^2-\delta^2}}.}
$$
The first crossing is transversal: $\operatorname{Re}(d\sigma/dd)=\omega^2/[(1+\delta d)^2+\omega^2d^2]>0$. If $a\leq\delta$, the magnitude relation rules out roots in the closed right half-<plane> for every delay. A nondegenerate nonlinear Hopf conclusion additionally uses the usual bifurcation hypotheses. This is <delayed negative feedback in red-cell production>.
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