Haematopoiesis

Formation of the cellular constituents of blood; normal adult production is chiefly associated with bone marrow.

Production of red blood cells from dividing precursors, distinguished from division of mature mammalian red blood cells.

The decreasing production function and increasing linear loss have one positive constant equilibrium $u_{\ast }$. It scales as ${\delta }^{-1/(n+1)}$ for small loss and ${\delta }^{-1}$ for large loss. With $a=n{u}_{\ast }^{n-1}/(1+u_{\ast }^n{)}^2>0$, the linear characteristic equation is $\sigma +\delta +a{e}^{-\sigma d}=0$. 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 $\mathrm{Re}{\sigma }^{\prime }(d_c)={\omega }^2/[(1+\delta d_c{)}^2+{\omega }^2{d}_c^2]>0$, proving transversality. If $a\le \delta$, 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.