Super Multi-Armed and Segmented Spiral Pattern in a Reaction-Diffusion Model

Jian Gao,Changgui Gu

doi:10.1109/access.2019.2943421

Abstract

The super multi-armed and segmented (SMAS) spiral pattern has been observed in nature, such as sunflower inflorescence, spiral aloe, pine cone, ball cactus and Roman broccoli, which is characterized by several segmented spiral arms sharing the same spiral tip. The mechanism for the emergency of the SMAS spiral pattern has not been found. In this article, we observed the emergence of the SMAS spiral pattern by the simulations of a reaction-diffusion model. Additionally, our theoretical analysis found that the instability of concentrations in spiral arms leads to the emergence of this pattern. This study provides an alternative explanation for the formation of this type of pattern in nature and sheds light on the dynamics of pattern formation.

Highlights

Pattern formation in reaction-diffusion (RD) systems has been suggested to be an underlying mechanism that gives rise to complex patterns in biology [1]–[7]
Segmented spiral patterns result from an interaction between front rippling via a transverse instability and front symmetry breaking by a fast-diffusing inhibitor far from the Hopf-Turing bifurcation [9], and multi-armed spiral patterns are formed due to attraction of single spirals if these spirals rotate in the same direction and their tips are less than one wavelength apart [10], [11], [16]–[18], [21]
The superstructure contains five super arms which are all segmented, and these super arms share the same spiral tip with the original spiral. This superstructure is an super multi-armed and segmented (SMAS) spiral pattern, and we call this type of spiral wave the SMAS spiral wave

Summary

INTRODUCTION

Pattern formation in reaction-diffusion (RD) systems has been suggested to be an underlying mechanism that gives rise to complex patterns in biology [1]–[7]. Segmented spiral patterns result from an interaction between front rippling via a transverse instability and front symmetry breaking by a fast-diffusing inhibitor far from the Hopf-Turing bifurcation [9], and multi-armed spiral patterns are formed due to attraction of single spirals if these spirals rotate in the same direction and their tips are less than one wavelength apart [10], [11], [16]–[18], [21] In addition to these two patterns, other spiral patterns have been observed in nature, e.g., sunflower inflorescence, spiral aloe, pine cone, ball cactus and Roman broccoli [24], [25].

MODEL AND METHOD

THE COMPOSITION OF THE SUPERSTRUCTURE

NUMERICAL RESULTS OBTAINED WITH SMALLER SPACE STEPS AND TIME STEPS

ANALYTICAL RESULTS

DISCUSSION AND CONCLUSION