  We estimate the fraction of halos that host supermassive black holes (SMBHs)
forming through the direct collapse (DC) scenario by using cosmological N -body
simulations combined with a semi-analytic model for galaxy evolution. While in
most of earlier studies the occurrence of the DC is limited only in chemically
pristine halos, we here suppose that the DC can occur also in halos with
metallicity below a threshold value $Z_{\rm th} = 0$--$10^{-3}~{\rm
Z}_{\bigodot}$, considering the super-competitive accretion pathway for DC
black hole (DCBH) formation. In addition, we consider for the first time the
effect of Lyman-Werner (LW) radiation from stars within host halos, i.e.,
internal radiation. We find that, with low threshold metallicities of $Z_{\rm
th} \leq 10^{-4}~{\rm Z}_{\bigodot}$, the inclusion of internal radiation
rather reduces the number density of DCBHs from $0.2$--$0.3$ to
$0.03$--$0.06~{\rm Mpc}^{-3}$. This is because star formation is suppressed due
to self-regulation, and the LW flux emitted by neighboring halos is reduced.
Only when $Z_{\rm th}$ is as high as $10^{-3}~{\rm Z}_{\bigodot}$, internal
radiation enhances the number density of DCBHs from $0.4$ to $1~{\rm
Mpc}^{-3}$, thereby decreasing the threshold halo mass above which at least one
DCBH forms from $2\times 10^{9}$ to $9\times 10^{8}~{\rm M}_{\bigodot}$. We
also find that halos with $M_{\rm halo} \gtrsim 10^{11}$--$10^{12}~{\rm
M}_{\bigodot}$ can host more than one DCBH at $z = 0$. This indicates that the
DC scenario alone can explain the observed number of SMBH-hosting galaxies.
