  While supermassive black holes are known to co-evolve with their host galaxy,
the precise nature and origin of this co-evolution is not clear. We here
explore the possible connection between star formation and black hole growth in
the circumnuclear disk (CND) to probe this connection in the vicinity close to
the black hole. We adopt here the circumnuclear disk model developed by
Kawakatu & Wada (2008) and Wutschik et al. (2013), and explore both the
dependence on the star formation recipe as well as the role of the
gravitational field, which can be dominated by the central black hole, the CND
itself or the host galaxy. A specific emphasis is put on the turbulence
regulated star formation model by Krumholz et al. (2005) to explore the impact
of a realistic star formation recipe. It is shown that this model helps to
introduce realistic fluctuations in the black hole and star formation rate,
without overestimating them. Consistent with previous works, we show that the
final black hole masses are rather insensitive to the masses of the initial
seeds, even for seed masses of up to 10^6 M_sol. In addition, we apply our
model to the formation of high-redshift quasars, as well as to the nearby
system NGC 6951, where a tentative comparison is made in spite of the presence
of a bar in the galaxy. We show that our model can reproduce the high black
hole masses of the high-redshift quasars within a sufficiently short time,
provided a high mass supply rate from the host galaxy. In addition, it
reproduces several of the properties observed in NGC 6951. With respect to the
latter system, our analysis suggests that supernova feedback may be important
to create the observed fluctuations in the star formation history as a result
of negative feedback effects.
