  We studied simple random-walk models with asymmetric time delays. Stochastic
simulations were performed for hyperbolic-tangent fitness functions and to
obtain analytical results we approximated them by step functions. A novel
behavior has been observed. Namely, the mean position of a walker depends on
time delays. This is a joint effect of both stochasticity and time delays
present in the system. We also observed that by shifting appropriately fitness
functions we may reverse the effect of time delays - the mean position of the
walker changes the sign.
