  Recent experiments on high density excitons in Cu$_2$O provide evidence for
degenerate quantum statistics and Bose-Einstein condensation of this nearly
ideal gas. We model the time dependence of this bosonic system including
exciton decay mechanisms, energy exchange with phonons, and interconversion
between ortho (triplet-state) and para (singlet-state) excitons, using
parameters for the excitonic decay, the coupling to acoustic and low-lying
optical phonons, Auger recombination, and ortho-para interconversion derived
from experiment. The single adjustable parameter in our model is the
optical-phonon cooling rate for Auger and laser-produced hot excitons. We show
that the orthoexcitons move along the phase boundary without crossing it (i.e.,
exhibit a ``quantum saturation''), as a consequence of the balance of entropy
changes due to cooling of excitons by phonons and heating by the non-radiative
Auger two-exciton recombination process. The Auger annihilation rate for
para-para collisions is much smaller than that for ortho-para and ortho-ortho
collisions, explaining why, under the given experimental conditions, the
paraexcitons condense while the orthoexcitons fail to do so.
