  A simple model for the nonlinear collective transport of interacting
particles in a random medium with strong disorder is introduced and analyzed. A
finite threshold for the driving force divides the behavior into two regimes
characterized by the presence or absence of a steady-state particle current.
Below this threshold, transient motion is found in response to an increase in
the force, while above threshold the flow approaches a steady state with motion
only on a network of channels which is sparse near threshold. Some of the
critical behavior near threshold is analyzed via mean field theory, and
analytic results on the statistics of the moving phase are derived. Many of the
results should apply, at least qualitatively, to the motion of magnetic bubble
arrays and to the driven motion of vortices in thin film superconductors when
the randomness is strong enough to destroy the tendencies to lattice order even
on short length scales. Various history dependent phenomena are also discussed.
