Boundary-Induced Pattern Formation from Temporal Oscillation: Spatial Map Analysis

TL;DR

This paper investigates how fixed boundary conditions in reaction-diffusion systems can induce spatial pattern formation from uniform states, using a novel spatial map approach to analyze the transformation of temporal oscillations into spatial patterns.

Contribution

It introduces a spatial map method that reproduces emergent patterns from temporal oscillations and explores the relationship between pattern wavelength and oscillation period.

Findings

Spatial map accurately reproduces pattern formation

Pattern wavelength correlates with oscillation period

Method applicable to biological morphogenesis

Abstract

Boundary-induced pattern formation from a spatially uniform state is investigated using one-dimensional reaction-diffusion equations. The temporal oscillation is successively transformed into a spatially periodic pattern, triggered by diffusion from the fixed boundary. We introduced a spatial map, whose temporal sequence, under selection criteria from multiple stationary solutions, can completely reproduce the emergent pattern, by replacing the time with space. The relationship of the pattern wavelength with the period of oscillation is also obtained. The generality of the pattern selection process and algorithm is discussed with possible relevance to biological morphogenesis.