Flow- and Diffusion Distributed Structures with noise at the inlet

Flow and Diffusion Distributed Structures (FDS) are stationary spatially periodic patterns that can be observed in reaction-diffusion-advection systems. These structures arise when the flow rate exceeds a certain bifurcation point provided that concentrations of interacting species at the inlet differ from steady state values and the concentrations are held constant. Normally, theoretical studies of these patterns are developed without concerning a noise. In this paper we consider FDS for a more realistic conditions and assume that the inlet concentrations are perturbed by a small noise. When the flow rate is small, the FDS is linearly sensitive to noise at the inlet. Even weak fluctuations destroy the stationary pattern and an oscillatory solution appears instead. For higher flow rates the instability becomes nonlinear: the pattern remains unaltered for a weak noise and undergoes the destruction when the noise amplitude passes a certain threshold. We represent a detailed description of these effects and examine two scenarios for the stabilization.