  We build an evolution model of the central black hole that depends on the
processes of gas accretion, the capture of stars, mergers as well as
electromagnetic torque. In case of gas accretion in the presence of cooling
sources, the flow is momentum-driven, after which the black hole reaches a
saturated mass; subsequently, it grows only by stellar capture and mergers. We
model the evolution of the mass and spin with the initial seed mass and spin in
$\Lambda$CDM cosmology. For stellar capture, we have assumed a power-law
density profile for the stellar cusp in a framework of relativistic loss cone
theory that include the effects of black hole spin, Carter's constant, loss
cone angular momentum, and capture radius. Based on this, the predicted capture
rates of $10^{-5}$--$10^{-6}$ yr$^{-1}$ are closer to the observed range. We
have considered the merger activity to be effective for $z \lesssim 4$, and we
self-consistently include the Blandford-Znajek torque. We calculate these
effects on the black hole growth individually and in combination, for deriving
the evolution. Before saturation, accretion dominates the black hole growth
($\sim 95\%$ of the final mass), and subsequently, stellar capture and mergers
take over with roughly equal contribution. The simulations of the evolution of
the $M_{\bullet} - \sigma$ relation using these effects are consistent with
available observations. We run our model backward in time and retrodict the
parameters at formation. Our model will provide useful inputs for building
demographics of the black holes and in formation scenarios involving stellar
capture.
