  We present the results of fully 3-D hydrodynamic simulations of the
gravitational collapse of isolated, turbulent molecular cloud cores. Starting
from initial states of hydrostatic equilibrium, we follow the collapse of both
singular and nonsingular logatropic cores until the central protostar has
accreted > 90% of the total available mass. We find that, in the collapse of a
singular core with access to a finite mass reservoir, the mass of the central
protostar increases as M_acc proportional to t^4 until it has accreted about
35% of the total available mass. For nonsingular cores of fiducial masses 1,
2.5, and 5 M_solar, we find that protostellar accretion proceeds slowly prior
to the formation of a singular density profile. Immediately thereafter, the
accretion rate in each case increases to about 10^{-6} M_solar/yr, for cores
with central temperature T_c= 10 K and truncation pressure P_s = 1.3E5 k_B
K/cm^3. It remains at that level until half the available mass has been
accreted. After this point, the accretion rate falls steadily as the remaining
material is accreted onto the growing protostellar core. We suggest that this
general behaviour of the protostellar accretion rate may be indicative of
evolution from the Class 0 to the Class I protostellar phase.
