Nighttime Chapman ozone

With photolysis switched off, constant molecular oxygen and collider reservoirs give $\dot{x}=-x(a+by)$ and $\dot{y}=x(a-by)$ for atomic oxygen $x$, ozone $y$, $a=k_1[{\mathrm{O}}_2][M]$ and $b=k_3$. Dividing the equations and integrating givesFor $y_0<a/b$, ozone grows toward a finite-reservoir endpoint below $a/b$. Its value at sunrise depends on the dark-interval duration; $a/b$ is an approximate quasi-steady balance while atomic oxygen remains available.