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In this delayed carbon-dioxide ventilation feedback model, $p$ is a representative carbon-dioxide partial pressure, $K$ an effective storage capacitance, $M>0$ metabolic production, and $pV$ removal by pulmonary ventilation. The delay represents transport and sensing between the gas-exchange compartment and the ventilatory response. The Hill equation response rises sigmoidally with pressure, reaches half its maximum $\alpha$ at $p=\theta$, and saturates at $\alpha$; $n>1$ permits a steep threshold-like response.

For positive parameters, $pV(p)$ increases strictly from zero to infinity because its derivative is $V+p{V}^{\prime }>0$. Its graph therefore intersects the horizontal line $M$ exactly once: $\boxed{M=p^{\ast }V(p^{\ast }),p^{\ast }>0}$. The figure shows this argument and the first imaginary-root crossing.

For a small perturbation $z=p-p^{\ast }$, linearization givesThe sigmoidal delayed ventilation threshold follows from the characteristic equation of a delay differential equation, $\lambda +A+B{e}^{-\lambda \tau }=0$. At an imaginary root $\lambda =i\omega$, separate real and imaginary parts:For $B>A>0$, the first positive crossing hasAt zero delay the scalar equation is stable. No root can cross the imaginary axis before this first crossing, and zero itself is not a root. To check the crossing direction, implicit differentiation givesAll subsequent imaginary crossings have the same direction. Thus the equilibrium is linearly stable below ${\tau }_c$ and unstable above it. This establishes the oscillatory linear threshold; the full nonlinear Hopf bifurcation additionally requires the usual nondegeneracy conditions. If $B\le A$, no finite imaginary-root crossing occurs and the equilibrium is stable for every finite delay. For the stated response, $B/A=n{\theta }^n/({\theta }^n+(p^{\ast }{)}^n)$, showing explicitly when the gain can exceed instantaneous damping.

If $A\ll B$, then ${\omega }_c\approx B$ and ${\omega }_c{\tau }_c\approx \pi /2$. Consequently the small-amplitude period near onset is $\boxed{2\pi /{\omega }_c\approx 4{\tau }_c}$. The comparison is to the critical delay near the first instability, rather than an arbitrary delay far beyond onset. A longer circulatory transit time can move the physiological system across this delayed-feedback threshold, providing a mechanism for periodic breathing associated with impaired cardiac output.

In delayed carbon-dioxide ventilation feedback, $P$ is the effective carbon-dioxide partial pressure of the lumped body compartment. Its stored carbon-dioxide amount is proportional to $P$, with effective capacitance $K$, so $KdP/dT$ is its accumulation rate. Metabolism produces carbon dioxide at rate $M$. Ventilation removes it at a rate proportional to partial pressure times ventilation, $P\dot{V}$, with gas-conversion constants absorbed into the model coefficients. The equation is therefore a one-compartment carbon-dioxide balance: production minus respiratory removal equals accumulation. It idealizes mixing and treats metabolic production as constant over the time scale of interest.