Boundary-Induced Pattern Formation from Uniform Temporal Oscillation

TL;DR

This paper investigates how fixing a boundary in a reaction-diffusion system can induce various pattern formations from uniform oscillations, revealing three distinct phases influenced by diffusion ratios and analyzing their mechanisms.

Contribution

It introduces a novel boundary-induced pattern formation mechanism in reaction-diffusion systems and develops a spatial map approach to analyze these patterns.

Findings

Transformation of oscillations into spatial patterns

Emergence of traveling waves from boundaries

Identification of aperiodic spatiotemporal dynamics

Abstract

Pattern dynamics triggered by fixing a boundary is investigated. By considering a reaction-diffusion equation that has a unique spatially-uniform and limit cycle attractor under a periodic or Neumann boundary condition, and then by choosing a fixed boundary condition, we found three novel phases depending on the ratio of diffusion constants of activator to inhibitor: transformation of temporally periodic oscillation into a spatially-periodic fixed pattern, travelling wave emitted from the boundary, and aperiodic spatiotemporal dynamics. The transformation into a fixed, periodic pattern is analyzed by crossing of local nullclines at each spatial point, shifted by diffusion terms. A spatial map, then, is introduced, whose temporal sequence can reproduce the spatially periodic pattern, by replacing the time with space. The generality of the boundary-induced pattern formation as well as its relevance to biological morphogenesis is discussed.