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.
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