Plasma density gratings induced in current carrying plasmas are investigated through the particle in cell method. Simulation results show that the electron phase space holes can be formed at the saturation time which confirms the counter-streaming in these plasmas. It is found that the generation of grating-like pattern strongly depends on the saturation time and velocity of electrons and positrons. Moreover, in the presence of delta distribution function the grating-like pattern is obvious, compared to the Maxwellian distribution. Finally, using the fluid model, it is demonstrated that increasing the velocity of electrons and positrons causes to increase the maximum amount of growth rate of instability which is in good agreement with those obtained using the simulation model.