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The formation of ionizing channels and the propagation of cathode- and anode-directed streamers in nitrogen gas have been examined theoretically by a self-consistent, fully two-dimensional simulation, assuming a cylindrically symmetric geometry. The dynamical model of the electron, considering the photoionization effect, is described by equilibrium (a single-moment equation) and non-equilibrium (a three-moment equation) models. A more accurate flux-corrected transport (MAFCT) technique, which provides a solution for the steep and varying gradient problem, is used to solve the electron fluid equations. This is the first investigation of the gas pre-breakdown based on a non-equilibrium two-dimensional fluid model, which is more accurate than the equilibrium fluid model. The differences in the simulation results of the three- and single-moment fluid models for streamers are found. Moreover, a comparison with a Monte Carlo simulation result (a kinetic model) is used to test the validity of fluid models, and the comparison illustrates that the non-equilibrium fluid model provides a more accurate description from the equilibrium fluid model.