  A model for the Universe is proposed whose constituents are: (a) a dark
energy field modeled by a fermionic field non-minimally coupled with the
gravitational field, (b) a matter field which consists of pressureless baryonic
and dark matter fields and (c) a field which represents the radiation and the
neutrinos. The coupled system of Dirac's equations and Einstein field equations
is solved numerically by considering a spatially flat homogeneous and isotropic
Universe. It is shown that the proposed model can reproduce the expected
red-shift behaviors of the deceleration parameter, of the density parameters of
each constituent and of the luminosity distance. Furthermore, for small values
of the red-shift the constant which couples the fermionic and gravitational
fields has a remarkable influence on the density and deceleration parameters.
