  We model the constraints set on the evolution of dusty starburst galaxies by
the current deep extragalactic surveys performed in the far-infrared with $\it
Spitzer$, and at radio wavelengths with the VLA. Our models fit the number
counts in all the available spectral bands well, and also provide a reasonably
close match to the redshift distribution of the $\it Spitzer$ detections. We
find: 1.) dusty starburst galaxies with infrared burst phases triggered by
galactic interactions at redshift $z \sim 1-2$ are good candidates to fit the
$\it Spitzer$ results at $24\mu m$, $70 \mu m$ and $160 \mu m$, assuming
plausible strengths for the PAH features for the infrared luminous sources. An
Arp220-like spectral energy distribution (SED) for Ultraluminous Infrared
Galaxies (ULIGs) of $L_{\rm ir}>10^{12} L_{\odot}$ and one like that of M82 for
Luminous Infrared Galaxies (LIGs) of $L_{\rm ir}\sim 10^{11-12} L_{\odot}$ give
a successful fit to the $\it Spitzer$ $24\mu m$ and $\rm ISOCAM 15\mu m$ number
counts at flux levels of $S_{\nu}< 1 mJy$; 2.) the strong evolution of the
number density of the ULIGs from redshift $z\sim 0$ to $\sim 1$ predicted by
our models is consistent with the current deep $\rm 1.4 GHz$ radio surveys and
accounts for the upturn in the $\rm 1.4 GHz$ differential counts at the sub-mJy
flux level; and 3.) comparing with number counts at near infrared bands, as
well as the background measurements using DIRBE and 2MASS, shows that only a
fraction of the stellar mass in the Universe is included in our models of dusty
starburst mergers at $z \sim 1 - 2$.
