  Recent numerical and theoretical studies have demonstrated that the modes at
threshold of a random laser are in direct correspondence with the resonances of
the same system without gain, a feature which is well known in a conventional
laser but which was not known until recently for a random laser. This paper
presents numerical results, which extend such studies to the multimode regime
that takes place when the pumping rate is progressively increased above
threshold. Behavior that is already known in standard lasers, such as mode
competition and nonlinear wave-mixing, are shown to also take place in random
lasers thus reinforcing their recent modal description. However, due to the
complexity of the laser modes and to the openness of such lasers, which
requires large external pumping to compensate for strong loss, one observes
that these effects are more pronounced than in a conventional laser.
