  Many animals have patterned fur, feathers, or scales, such as the stripes of
a zebra. Turing models, or reaction-diffusion systems, are a class of
mathematical models of interacting species that have been successfully used to
generate animal-like patterns for many species. When diffusion of the inhibitor
is high enough relative to the activator, a diffusion-driven instability can
spontaneously form patterns. However, it is not just the type of pattern but
also the orientation that matters, and it remains unclear how this is done in
practice. Here, we propose a mechanism by which the curvature of the surface
influence the rate of diffusion, and can recapture the correct orientation of
stripes on models of a zebra and of a cat in numerical simulations. Previous
work has shown how anisotropic diffusion can give stripe forming
reaction-diffusion systems a bias in orientation. From the observation that
zebra stripes run around the direction of highest curvature, that is around the
torso and legs, we apply this result by modifying the diffusion rate in a
direction based on the local curvature. These results show how local geometry
can influence the reaction dynamics to give robust, global-scale patterns.
Overall, this model proposes a coupling between the system geometry and
reaction-diffusion dynamics that can give global control over the patterning by
using only local curvature information. Such a model can give shape and
positioning information in animal development without the need for spatially
dependent morphogen gradients.
