Abstract
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.