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Diffusive instabilities in heterogeneous systems

We investigate the behavior of a system composed of two small identical water droplets loaded with an activator-inhibitor system possessing only a single stable steady state and coupled through a third, “signaling” species able to diffuse through the interdroplet medium. Depending on whether the third species is coupled to the activator or the inhibitor, the system can exhibit wave instability, which is equivalent to out-of-phase oscillations, or Turing instability, which is equivalent to the birth of two new stationary steady states. For a large interdroplet gap, equivalent to a large time delay, droplets can undergo synchronous, in-phase oscillations. Our analysis can be extended to a system of many coupled in-line droplets, as we illustrate.