  In order to relate the observed evolution of the galaxy stellar mass function
and the luminosity function of active galactic nuclei (AGN), we explore a
co-evolution scenario in which AGN are associated only with the very last
phases of the star-forming life of a galaxy. We derive analytically the
connections between the parameters of the observed quasar luminosity functions
and galaxy mass functions. The $(m_{\rm bh}/m_{*})_{Qing}$ associated with
quenching is given by the ratio of the global black hole accretion rate density
(BHARD) and star-formation rate density (SFRD) at the epoch in question.
Observational data on the SFRD and BHARD suggests $(m_{\rm bh}/m_{*})_{Qing}
\propto (1+z)^{1.5}$ below redshift 2. This evolution reproduces the observed
mass-luminosity plane of SDSS quasars, and also reproduces the local $m_{\rm
bh}/m_{*}$ relation in passive galaxies. The characteristic Eddington ratio,
$\lambda^*$, is derived from both the BHARD/SFRD ratio and the evolving $L^*$
of the AGN population. This increases up to $z \sim 2$ as $\lambda^* \propto
(1+z)^{2.5}$ but at higher redshifts, $\lambda^*$ stabilizes at the physically
interesting Eddington limit, $\lambda^* \sim 1$. The new model may be thought
of as an opposite extreme to our earlier co-evolution scenario in Caplar et al.
2015. The main observable difference between the two co-evolution scenarios,
presented here and in Caplar et al. 2015, is in the active fraction of low mass
star-forming galaxies. We compare the predictions with the data from deep
multi-wavelength surveys and find that the "quenching" scenario developed in
the current paper is much to be preferred.
