  We investigated the evolution of the background expansion and the growth rate
of the matter density fluctuations in the scalar field $\phi$CDM Ratra-Peebles
model. We constrained the model parameter $\alpha$ and the matter density
parameter $\Omega_{\rm m}$ using the recent measurements of the growth rate of
the matter density fluctuations and baryon acoustic oscillation peak position.
In addition, we studied a number of the $\phi$CDM scalar field models in order
to determine whether or not it would be possible to discriminate them from the
$\Lambda$CDM model using predicted data for the future Dark Energy
Spectroscopic Instrument (DESI) observations. For this purpose, we carried out
the statistical Bayesian analysis, and computed Bayes coefficients, as well as
Akaike and Bayesian information criteria. We found that projected DESI results
could provide compelling when comparing most $\phi$CDM models with the
$\Lambda$CDM model. We also conducted the MCMC analysis and obtained the
constraints on the parameters of the scalar field models, comparing the
observational data for: the universe expansion rate, the angular diameter
distance and the growth rate function, with the corresponding data generated
for the $\Lambda$CDM model. We investigated how well the
Chevallier-Polarsky-Linder (CPL) parametrization approximates the various
scalar field models. We determined the location of scalar field model in the
phase space of the CPL parameters.
