Abstract
An extension of the totally asymmetric exclusion process, which incorporates a dynamically extending lattice, is explored. Although originally inspired as a model for filamentous fungal growth, here the dynamically extending exclusion process (DEEP) is studied in its own right as a nontrivial addition to the class of nonequilibrium exclusion process models. Here we discuss various mean-field approximation schemes and elucidate the steady state behaviour of the model and its associated phase diagram. Of particular note is that the dynamics of the extending lattice leads to a new region in the phase diagram in which a shock discontinuity in the density travels forward with a velocity that is lower than the velocity of the tip of the lattice. Thus in this region the shock recedes from both boundaries.