Influence of the Turing instability on the motion of domain boundaries

Turing's theory of pattern formation has provided crucial insights into the behavior of various biological, geographical, and chemical systems over the last few decades. Existing studies have focused on moving-boundary Turing systems for which the motion of the boundary is prescribed by an external agent. In this paper, we present an extension of this theory to a class of systems in which the front motion is governed by the physical processes that occur within the domain. Biological systems exhibiting apically dominant growth and corrosion of metals and alloys highlight some of the noteworthy examples of such systems. In this study, we characterize the nature of interaction between the moving front and the Turing-instability for both an activator-inhibitor and an activator-substrate model. Behavioral regimes of periodic, as well as nonperiodic (nonconstant), growth rates are obtained. Furthermore, the trends in the first show striking similarities with the cyclic-boundary-kinetics observed in experimental systems. In general, a stationary, periodic structure is also left behind the moving front. If the periodicity of the boundary kinetics agrees with the allowed range of the stable-periodic solutions, the pattern formed tends to persist. Otherwise, it evolves to a nearby energy-minimum either by peak-splitting, peak-decay, or by settling down to a spatially homogeneous state.