  We study complex systems arising, in particular, in population dynamics,
developmental biology, and bacterial metabolic processes, in which each
individual element obeys a relatively simple hysteresis law (a non-ideal
relay). Assuming that hysteresis thresholds fluctuate, we consider the arising
reaction-diffusion system. In this case, the spatial variable corresponds to
the hysteresis threshold. We describe the collective behavior of such a system
in terms of the Preisach operator with time-dependent measure which is a part
of the solution for the whole system. We prove the well-posedness of the system
and discuss the long-term behavior of solutions.
