 Chapman family lifetime versus relaxation time

Under fast Chapman cycle partitioning, an odd oxygen family abundance $X$ can satisfy $\dot{X}=P-k{X}^2$. At equilibrium $P=k{X}_e^2$, its turnover lifetime is $X_e/P=1/(k{X}_e)$, while linearizing gives $\delta \dot{X}=-2k{X}_e\delta X$ and relaxation time $1/(2k{X}_e)$. A chemical lifetime is therefore not automatically an eigenmode relaxation time. The often-used expression $1/(2k_3[\mathrm{O}])$ also assumes ozone dominates the family abundance.