  A hyperaccreting stellar-mass black hole has been proposed as the candidate
central engine of gamma-ray bursts (GRBs). The rich observations of GRBs by
\textit{Fermi} and \textit{Swift} make it possible to constrain the central
engine model by comparing the model predications against data. This paper is
dedicated to studying the temporal evolution of central engine parameters for
both prompt emission and afterglow phases. We consider two jet launching
mechanisms, i.e., $\nu\bar{\nu}$ annihilations and the Blandford-Znajek (BZ)
processe, and obtain analytical solutions to these two models. We then
investigate the black hole central engine parameters, such as the jet power,
the dimensionless entropy $\eta$, and the central engine parameter $\mu_0=\eta
(1+\sigma_0)$ (where $\sigma_0$ is the initial magnetization of the engine) at
the base of the jet. The black hole may be spun up by accretion, or spun down
by the BZ process, leaving imprints in GRB lightcurves. Usually, a BZ jet is
more powerful and is likely responsible for the late time central engine
activities. However, an initially non-spinning black hole central engine may
first launch a thermal "fireball" via neutrino annihilations, and then launch a
Poynting-flux-dominated jet via the BZ process. Multiple flares, giant bumps,
and plateaus in GRB afterglows can be well produced as the result of late time
accretion onto the black hole.
