Turing instabilities with nearly equal diffusion coefficients

Abstract

We show that if a Turing instability occurs in a reaction–diffusion system with a nearly scalar diffusion matrix, then the parameters of the corresponding well-mixed system are necessarily such that the well-mixed system has at least two eigenvalues near zero. Conversely, if the corresponding well-mixed system is sufficiently close to a coalescence point of Hopf and saddle-node bifurcations (two eigenvalues are zero at such a point), and if the spatial domain is sufficiently large, then there exists a nearly scalar diffusion matrix such that a Turing instability occurs. These results imply that information on bifurcation loci from experiments in continuous-flow stirred tank reactor suffices to locate regions of parameter space where Turing instabilities are likely to occur; no knowledge of the reaction mechanism and rate constants is needed. In order to illustrate these results, we have analyzed a six-step model of the Belousov–Zhabotinskii reaction due to Showalter, Noyes, and Bar-Eli. In this model, the sixth step describes the reduction of the metal ion catalyst by the organic substrate. Depending on the rate constant of this step, we find that Turing instabilities can occur when the diffusion coefficient of either Ce4+ or of HOBr is very near, but slightly larger than that of Br−.