Delayed carbon-dioxide ventilation feedback ($K\dot{P}=M-PG[P(t-\tau )-P_0{]}_{+}$)

A one-compartment carbon-dioxide balance couples metabolic production with pressure-dependent ventilatory removal. A positive-part response models a nonnegative threshold-linear respiratory drive, and a transport/response delay can destabilize negative feedback. With $T=t/\tau$, $P=P_0+[M/(G{P}_0)]p$, the scaled law is $p^{\prime }=\alpha [1-(1+\mu p)[p(T-1){]}_{+}]$, where $\alpha =\tau G{P}_0/K$ and $\mu =M/(G{P}_0^2)$. Its positive equilibrium is $2/[1+\sqrt{1+4\mu }]$; $1/(1+\mu )$ is only its first-order small-$\mu$ approximation.

For $K\dot{p}=M-pV(p(t-\tau ))$ and a positive steady state, linearization gives $\dot{z}=-Az-Bz(t-\tau )$, where $A=V(p^{\ast })/K$ and $B=p^{\ast }{V}^{\prime }(p^{\ast })/K$. Splitting the characteristic equation of a delay differential equation at an imaginary root yields ${\omega }^2=B^2-A^2$ and the displayed first critical delay. The root crosses into the unstable half-plane with real derivative ${\omega }^2/[(1+A{\tau }_c{)}^2+{\omega }^2{\tau }_c^2]>0$. If $A\ll B$, the period near onset is approximately $4{\tau }_c$. For a Hill equation response, $B/A=n{\theta }^n/({\theta }^n+(p^{\ast }{)}^n)$ identifies whether delayed gain can exceed instantaneous damping.

A longer circulation time increases the delay between gas exchange and its sensed feedback. The American Thoracic Society's documented case relates a change in breathing pattern during one night to a longer estimated circulation time; the simple scalar model isolates a possible feedback mechanism rather than representing every determinant of that observation.^[ref]