Plasma simulation models have been developed which replicate the fundamental features of the convective and oscillating instability in the nonlinear weak beam regime. The oscillating instability is characterized in this regime by a randomly fluctuating electric field which is caused by the presence of a fresh, strong beam in a region of space in which the electric field is large. Nonlinear behavior of the background plasma need not be present during the production of the oscillating instability. The convective instability develops into a quasilinear quiescent state during its nonlinear evolution because the nonlinear behavior of the system is confined to a diffused beam.