  We propose and study a one-dimensional (1D) model consisting of two lanes
with open boundaries. One of the lanes executes diffusive and the other lane
driven unidirectional or asymmetric exclusion dynamics, which are mutually
coupled through particle exchanges in the bulk. We elucidate the generic
nonuniform steady states in this model. We show that in a parameter regime,
where hopping along the TASEP lane, diffusion along the SEP lane and the
exchange of particles between the TASEP and SEP lanes compete, the SEP
diffusivity $D$ appears as a tuning parameter for both the SEP and TASEP
densities for a given exchange rate in the nonequilibrium steady states of this
model. Indeed, $D$ can be tuned to achieve phase coexistence in the asymmetric
exclusion dynamics together with spatially smoothly varying density in the
diffusive dynamics in the steady state. We obtain phase diagrams of the model
by using mean field theories, and corroborate and complement the results by
stochastic Monte Carlo simulations. This model reduces to an isolated open
totally asymmetric exclusion process (TASEP) and an open TASEP with bulk
particle nonconserving Langmuir kinetics (LK), respectively, in the limits of
vanishing and diverging particle diffusivity in the lane executing diffusive
dynamics. Thus this model works as an overarching general model, connecting
both pure TASEPs and TASEPs with LK in different asymptotic limits. We further
define phases in the SEP and obtain phase diagrams, and show their
correspondence with the TASEP phases. In addition to its significance as a 1D
driven, diffusive model, this model also serves as a simple reduced model for
cell biological transport by molecular motors undergoing diffusive and directed
motion inside eukaryotic cells.
