Phase separation in membranes due to matter exchange

We consider a two-dimensional membrane exchanging matter with a reservoir. Matter absorption and desorption in the membrane occurs with rates k_A and k_D respectively.
By only assuming matter conservation in the system membrane-reservoir, we show that a complex patterned composition distribution emerges in the membrane, depending on the values k_A and k_D. The pattern emergence is due to Spatio-temporal "memory" effects.