  By following the Kazantsev theory and taking into account both microscopic
and turbulent diffusion of magnetic fields, we develop a unified treatment of
the kinematic and nonlinear stages of turbulent dynamo, and study the dynamo
process for a full range of magnetic Prandtl number Pm and ionization
fractions. We find a striking similarity between the dependence of dynamo
behavior on Pm in a conducting fluid and R (a function of ionization fraction)
in partially ionized gas. In a weakly ionized medium, the kinematic stage is
largely extended, including not only exponential growth but a new regime of
dynamo characterized by linear-in-time growth of magnetic field strength, and
the resulting magnetic energy is much higher than the kinetic energy carried by
viscous-scale eddies. Unlike the kinematic stage, the subsequent nonlinear
stage is unaffected by microscopic diffusion processes and has a universal
linear-in-time growth of magnetic energy with the growth rate as a constant
fraction $3/38$ of the turbulent energy transfer rate, showing good agreement
with earlier numerical results. Applying the analysis to the first stars and
galaxies, we find that the kinematic stage is able to generate a field strength
only an order of magnitude smaller than the final saturation value. But the
generation of large-scale magnetic fields can only be accounted for by the
relatively inefficient nonlinear stage and requires longer time than the
free-fall time. It suggests that magnetic fields may not have played a
dynamically important role during the formation of the first stars.
