Robust pattern formation in reaction-diffusion systems in the presence of heterogeneity

The emergence of spatial patterns in biological systems plays a key role in many important functions. Reaction-diffusion systems have been used to explain the emergence of biological patterns such as stripes, hexagons, fronts, and spirals through interacting chemical species. These systems must be robust to microscopic spatial fluctuations in reactant properties in order to form stable patterns. Our aim is to build such robustness by modifying local interaction rules such that small changes in reactant properties do not affect the scale of the formed pattern. Further, we aim for larger changes in reactant properties to result in discrete jumps in the pattern scale, corresponding to the formation of a discrete new module of the underlying system. The Gierer-Meinhardt (GM) model is a canonical example of a two-species reaction-diffusion system that produces periodic spatial patterns. We augment the GM model with additional species not central to pattern formation but which stabilize the formed patterns and provide the desired robustness properties. To demonstrate these results in our augmented model, we show that adding a continuous gradient in the intrinsic activator and inhibitor length scales leads to discrete steps in the emergent scale of the formed pattern.