a . __ E3 $1 ELSEVIER 24June 1996 Physics Letters A 216 (1996) 262-268 PHYSICS LETTERS A Control of activator-inhibitor systems by differential transport Arkady B. Rovinsky, Satoshi Nakata ‘, Vladimir Z. Yakhnin, Michael Menzinger * zyxwvutsrqponm Department of Chemistry, University of Toronto, kmto, Ont., Cunadu M5S IA1 Received I7 August 1995; revised manuscript received 7 March 1996; accepted for publication 18 March 1996 Communicated by CR. Doering zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM Abstract When an activator-inhibitor system switches from a spatially uniform to a patterned state by a differential transport insta- bility - the differential flow instability, or the diffusive or Turing instability - the values of variables, such as concentrations or reaction rates, including their averages, may drastically change. Analysis shows that such control of productivity by differential transport is a general property of nonlinear activator-inhibitor systems. By simulations and experiments using the Belousov-Zhabotinsky reaction, we show that average concentrations of the key species may be significantly enhanced when the reaction is run in the presence of a differential flow. 1. Introduction Nonlinear systems that are excitable or multistable may be switched between states whose state variables differ significantly. A bistable chemical system (e.g. a continuous flow of combustable mixture) possesses two states, one “ignited” and the other “extinguished”, in which the concentrations and reaction rates may be different by orders of magnitude. This ability to switch a system between two or more different states without changing its internal parameters provides a valuable control tool. We show here that a class of dynamical systems which are neither excitable nor multistablecan still be controlled externally over a wide range. These are the systems that are susceptible to differential transport instabilities (the Turing instability and the differential flow instability). * Corresponding author. ’ Pcrmnnent address: Nara Universty of Fducation. Takabatake- cho. Nara 630, Japan. The homogeneous, stable state of a reaction- transport system, characterized by activator-inhibitor kinetics (e.g. existence of an autocatalyst) [ I ,2], may become unstable through the differential trans- port of its key species. In the case of differential diffusion between activator and inhibitor, this may lead to Turing patterns [ 3,4]. A differential flow may give rise to travelling waves - consequences of the differential flow instability DIFI [ 5,6]. The analysis presented here shows that the average values of the variables may change drastically compared to the ref- erence homogeneous state when the differential flow patterns are formed. While the variable values vary along the pattern, the space or time averages of some variables may be above or below the correspond- ing steady state values. The effect is caused by a non-trivial interplay of activator-inhibitor dynamics, differential flow, and diffusion. The Turing instability leads to a similar, although less pronounced, change of the system variables. The paper is organized as follows: to begin with we 0375.9601/96/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00270-S