Molecular motors are involved in key transport processes inside
actin-based cellular protrusions. The motors carry cargo proteins to
the protrusion tip which participate in regulating the actin
polymerization, and play a key role in facilitating the growth and
formation of such protrusions. It is observed that the motors
accumulate at the tips of cellular protrusions, and in addition form
aggregates that are found to drift towards the protrusion base at
the rate of actin treadmilling. We present a one-dimensional driven
lattice model, where motors become inactive after delivering their
cargo at the tip, or by loosing their cargo to a cargo-less
neighbor. The results suggest that the experimental observations may
be explained by the formation of traffic jams that form at the tip.
The model is solved using a novel application of mean-field and
shock analysis. We find a new class of shocks that undergo
intermittent collapses, and on average do not obey the
Rankine-Hugoniot relation.