Pinning of Diffusional Patterns by Non-Uniform Curvature

Abstract

Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational symmetry. Deformations, e.g. by a bulge or indentation, break symmetry and can pin a pattern. We adapt methods of conformal mapping and perturbation theory to examine how curvature inhomogeneities select and pin patterns, and confirm the results numerically. The theory provides an analogy to quantum mechanics in a geometry-dependent potential and yields intuitive implications for cell membranes, tissues, thin films, and noise-induced quasipatterns.

substantial re-write of arXiv:1710.00103

References in corpus (3)

• Quasi-cycles in a spatial predator-prey model
• Giant amplification of noise in fluctuation-induced pattern formation
• Geometry of wave propagation on active deformable surfaces

Cited by in corpus (5)

• Pattern Propagation Driven by Surface Curvature
• Reaction-Diffusion Waves Coupled with Membrane Curvature
• Sensing the shape of a cell with reaction-diffusion and energy minimization
• Weakly nonlinear analysis of Turing pattern dynamics on curved surfaces
• Oscillatory and chaotic pattern dynamics driven by surface curvature