  We present a new self-consistent semi-analytic model of the first stars and
galaxies to explore the high-redshift ($z{>}15$) Population III (PopIII) and
metal-enriched star formation histories. Our model includes the detailed merger
history of dark matter halos generated with Monte Carlo merger trees. We
calibrate the minimum halo mass for PopIII star formation from recent
hydrodynamical cosmological simulations that simultaneously include the
baryon-dark matter streaming velocity, Lyman-Werner (LW) feedback, and
molecular hydrogen self-shielding. We find an overall increase in the resulting
star formation rate density (SFRD) compared to calibrations based on previous
simulations (e.g., the PopIII SFRD is over an order of magnitude higher at
$z=35-15$). We evaluate the effect of the halo-to-halo scatter in this critical
mass and find that it increases the PopIII stellar mass density by a factor of
${\sim}1.5$ at $z{>}15$. Additionally, we assess the impact of various
semi-analytic/analytic prescriptions for halo assembly and star formation
previously adopted in the literature. For example, we find that models assuming
smooth halo growth computed via abundance matching predict SFRDs similar to the
merger tree model for our fiducial model parameters, but that they may
underestimate the PopIII SFRD in cases of strong LW feedback. Finally, we
simulate sub-volumes of the Universe with our model both to quantify the
reduction in total star formation in numerical simulations due to a lack of
density fluctuations on spatial scales larger than the simulation box, and to
determine spatial fluctuations in SFRD due to the diversity in halo abundances
and merger histories.
