  We study the dynamics of an asymmetric simple exclusion process with open
boundaries and local interactions using a pair approximation which generalizes
the 2-node cluster mean field theory and the Markov chain approach to kinetics
and shares with these approaches the property of reproducing exact results for
the bulk current-density relation and the steady state phase diagrams. We find
that the relaxation rate exhibits a dynamical transition, with no static
counterpart, analogous to that found without interactions. Remarkably, for some
values of the model's parameters, we find 2 dynamical transitions in the same
low density phase. We study the dynamics of relaxation to the steady state on
both sides of these transitions and make an attempt at providing a physical
interpretation for this phenomenon. Results from numerical approaches and a
modified Domain Wall Theory confirm the picture provided by the pair
approximation.
