 Delayed carbon-dioxide ventilation feedback

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.