  A one-component bistable reaction-diffusion system with asymmetric nonlocal
coup ling is derived as limiting case of a two-component activator-inhibitor
reaction -diffusion model with differential advection. The effects of
asymmetric nonlocal couplings in such a bistable reaction-diffusi on system are
then compared to the previously studied case of a system with symm etric
nonlocal coupling. We carry out a linear stability analysis of the spatially
homogeneous steady sta tes of the model and numerical simulations of the model
to show how the asymmetr ic nonlocal coupling controls and alters the steady
states and the front dynamic s in the system. In a second step, a third fast
reaction-diffusion equation is included which ind uces the formation of more
complex patterns. A linear stability analysis predicts traveling waves for
asymmetric nonlocal coupling in contrast to a stationary Turing patterns for a
system with symmetric nonlocal coupling. These findings are verified by direct
numerical integration of the full equations with nonlocal coupling.
