  Recent observations have indicated that a large fraction of the low to
intermediate luminosity AGN population lives in disk-dominated hosts, while the
more luminous quasars live in bulge-dominated hosts, in conflict with some
previous model predictions. We therefore build and compare a semi-empirical
model for AGN fueling which accounts for both merger and non-merger
'triggering.' In particular, we show that the 'stochastic accretion' model - in
which fueling in disk galaxies is essentially a random process arising whenever
dense gas clouds reach the nucleus - provides a good match to the present
observations at low/intermediate luminosities. However it falls short of the
high-luminosity population. We combine this with models for major
merger-induced AGN fueling, which lead to rarer but more luminous events, and
predict the resulting abundance of disk-dominated and bulge-dominated AGN host
galaxies as a function of luminosity and redshift. We compile and compare
observational constraints from z~0-2. The models and observations generically
show a transition from disk to bulge dominance in hosts near the Seyfert-quasar
transition, at all redshifts. 'Stochastic' fueling dominates AGN by number
(dominant at low luminosity), and dominates BH growth below the knee in the
present-day BH mass function (<10^7 M_sun). However it accounts for just ~10%
of BH mass growth at masses >10^8 M_sun. In total, fueling in disky hosts
accounts for ~30% of the total AGN luminosity density/BH mass density. The
combined model also accurately predicts the AGN luminosity function and
clustering/bias as a function of luminosity and redshift; however, we argue
that these are not sensitive probes of BH fueling mechanisms.
