Context : The recent although contoversial discovery of two main stellar populations in the Galactic bulge , one metal poor , with a spheroid kinematics and the other metal rich , with a bar-like kinematics , suggests to revise the classical model for bulge formation . Aims : We aim at computing the chemical evolution of the Galactic bulge to explain the existence of the two main stellar populations . We also plan to explore the possible existence of spatial abundance gradients inside the bulge . Methods : We adopt a chemical evolution model which follows the evolution of several chemical species ( from H to Ba ) . We assume that the metal poor population formed first and on a short timescale while the metal rich population formed later and out of the enriched gas . We predict the stellar distribution functions for Fe and Mg , the mean < [ Fe / H ] > and < [ Mg / H ] > and the [ Mg/Fe ] vs. [ Fe/H ] relations in the two stellar populations . We also consider the case in which the metal poor population could be the result of sub-populations formed with different chemical enrichment rates . Results : Our results , when compared with observations , indicate that the old more metal poor stellar population formed very fast ( on a timescale of 0.1-0.3 Gyr ) by means of an intense burst of star formation and an initial mass function flatter than in the solar vicinity . The metal rich population formed on a longer timescale ( 3 Gyr ) . We predict differences in the mean abundances of the two populations ( -0.52 dex for < [ Fe / H ] > ) which can be interpreted as a metallicity gradients . We also predict possible gradients for Fe , O , Mg , Si , S and Ba between sub-populations inside the metal poor population itself ( e.g . -0.145 dex for < [ Fe / H ] > ) . Finally , by means of a chemo-dynamical model following a dissipational collapse , we predict a gradient inside 500 pc from the Galactic center of -0.26 dexkpc ^ { -1 } in Fe . Conclusions : A stellar population forming by means of a classical gravitational gas collapse is probably mixed with a younger stellar population created perhaps by the bar evolution . The differences among their mean abundances can be interpreted as gradients . On the basis of both chemical and chemo-dynamical models , we also conclude that it is possible that the metal poor population contains abundance gradients itself , and therefore different stellar populations .