  We analyse the redshift dependence of space number density of quasars
assuming that they are the short-lived active stages of the massive galaxies
and arise immediately after the collapse of homogeneous central part of
protogalaxy clouds. Obtained dependence fits the observational data
ChaMP+CDF+ROSAT (Silverman et al. 2005) very well for protogalaxy clouds of
mass $M\approx 8\cdot 10^{11}$ $h^{-1}M_{\odot}$ and ellipticity $e<0.4$. The
lifetime of bright X-ray AGNs or QSOs with $L_X>10^{44.5}$ erg$\cdot s^{-1}$ in
the range of energies $0.3-8$ keV is $\tau_{QSO}\sim 6\cdot10^6$ years when the
mass of supermassive black hole is $M_{SMBH}\sim 10^{9}$ $M_{\odot}$ and the
values of other quasar parameters are reasonable. The analysis and all
calculations were carried out in the framework of $\Lambda$CDM-model with
parameters determined from 5-years WMAP, SNIa and large scale structure data
(Komatsu et al. 2009). It is concluded, that the halo model of galaxy formation
in the $\Lambda$CDM cosmological model matches well observational data on AGNs
and QSOs number density coming from current optical and X-ray surveys.
