  Coupling diffusion process of signaling molecules with nonlinear interactions
of intracellular processes and cellular growth/transformation leads to a system
of reaction-diffusion equations coupled with ordinary differential equations
(diffusion-ODE models), which differ from the usual reaction-diffusion systems.
One of the mechanisms of pattern formation in such systems is based on the
existence of multiple steady states and hysteresis in the ODE subsystem.
Diffusion tries to average different states and is the cause of spatio-temporal
patterns. In this paper we provide a systematic description of stationary
solutions of such systems, having the form of transition or boundary layers.
The solutions are discontinuous in the case of non-diffusing variables whose
quasi-stationary dynamics exhibit hysteresis. The considered model is motivated
by biological applications and elucidates a possible mechanism of formation of
patterns with sharp transitions.
