= Delayed negative feedback in red-cell production
{title2=$u'=\frac1{1+u(t-d)^n}-\delta u$}
The decreasing production function and increasing linear loss have one positive constant <equilibrium> $u_*$. It scales as $\delta^{-1/(n+1)}$ for small loss and $\delta^{-1}$ for large loss. With $a=nu_*^{n-1}/(1+u_*^n)^2>0$, the linear <characteristic equation> is $\sigma+\delta+ae^{-\sigma d}=0$. It has no real nonnegative root, and zero is never a root. Thus loss of stability from the zero-delay stable state must occur through a nonzero imaginary conjugate pair.
If $a>\delta$, the first crossing has $\omega=\sqrt{a^2-\delta^2}$ and $d_c=\arccos(-\delta/a)/\omega$. Differentiation of the <characteristic equation> gives $\operatorname{Re}\sigma'(d_c)=\omega^2/[(1+\delta d_c)^2+\omega^2d_c^2]>0$, proving transversality. If $a\leq\delta$, the magnitude inequality $|\sigma+\delta|=a|e^{-\sigma d}|$ rules out a root in the closed right half-<plane>. The model needs an initial history on the whole delay interval. The delay represents a production/maturation lag rather than instantaneous production controlled by the current cell count.
Back to article page
