Turing Instabilities in Reaction–Diffusion Systems with Temporally or Spatially Varying Parameters

Abstract

In Chap. 10 we considered the Turing instability in systems where the kinetic parameters and the transport coefficients are constant in space and time. While the vast majority of theoretical work on Turing patterns deals with such systems, there are good reasons from applications in biology and ecology to account for the effect of spatial or temporal variations on the threshold of the Turing instability. Chemical or biological systems are rarely completely uniform. Pattern formation in the Drosophila egg, for example, occurs in the presence of maternally imposed gradients of gene products [106]. Experimental studies of Turing patterns in the CIMA and CDIMA reactions use continuously fed unstirred reactors (CFURs), see Chap. 12, which unavoidably exhibit gradients in the concentrations of the feed reactants. The problem of determining diffusion-driven instabilities in reacting systems with spatially or temporally varying parameters is in general a rather difficult one. The tools of the linear 10 cannot be extended to such systems, since they do not posses a uniform steady state in most cases. Reaction–diffusion systems with weak heterogeneities can be studied with perturbation techniques [19, 50, 92, 40, 341, 342]. Lengyel and coworkers [249] used an approximation of the reaction–diffusion equation to study the effect of the gradients in CFURs on the position and the possible three-dimensional character of the Turing structures.