  Using a large set of high resolution numerical simulations incorporating
non-equilibrium molecular hydrogen chemistry and a constant source of external
radiation, we study gas collapse in previously photo-ionized mini-galaxies with
virial temperatures less than 10^4 K in the early universe (redshifts z=10-20).
We confirm that the mechanism of positive feedback of ionizing radiation on
star formation in mini-galaxies proposed by Ricotti, Gnedin, & Shull (2002) can
be efficient despite a significant flux of metagalactic photo-dissociating
radiation. We derive critical fluxes for the Lyman-Werner background radiation
sufficient to prevent the collapse of gas in mini-galaxies as a function of the
virial mass of the halo and redshift. In our model, the formation of
mini-galaxies in defunct HII regions is most efficient at large redshifts
(z>15) and/or for large local gas overdensity delta>10. We show that
non-equilibrium chemistry plays an important dynamical role not only during the
initial evolutionary phase leading to the gas becoming gravitationally unstable
inside the mini-halo, but also at the advanced stages of the core collapse,
resulting in efficient gas accretion in the core region. We speculate on a
possible connection between our objects and metal-poor globular clusters and
dwarf spheroidal galaxies.
