  The conversion factor $\alpha_{\rm CO}$ from the observable CO(1-0)
luminosity to the mass of molecular gas is known to vary between isolated
galaxies and some mergers, but the underlying reasons are not clearly
understood. Thus, the value(s) of $\alpha_{\rm CO}$ to be adopted remain highly
uncertain. To provide better constraints, we apply the large velocity gradient
method to a series of hydrodynamical simulations of galaxies and derive the
evolution of $\alpha_{\rm CO}$. We report significant variations of
$\alpha_{\rm CO}$, and identify three distinct regimes: disk galaxies,
starbursts and post-burst phases. We show that estimating the star formation
rate over 20 Myr smooths out some of these differences, but still maintains a
distinction between disks and starbursts. We find a tighter correlation of
$\alpha_{\rm CO}$ with the gas depletion time than with star formation rate,
yet with deviations induced by the transitions to and from the starburst
episodes. We conclude that $\alpha_{\rm CO}$ fluctuates because of both the
feedback energy and the velocity dispersion. Identifying the phase of an
interaction by classical means (e.g. morphology, luminosity) could then help
selecting the relevant conversion factor to be used and get more accurate
estimates of the molecular masses of galaxies.
