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Pattern formation from spatially heterogeneous reaction-diffusion systems.

Robert A Van Gorder1

  • 1Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|November 8, 2021
PubMed
Summary

Spatial heterogeneity in reaction-diffusion systems can generate complex, irregular Turing patterns. This study extends Turing instability analysis to understand how spatial variations influence pattern formation, revealing new insights into morphogenesis.

Keywords:
Turing instabilityTuring patternsreaction–diffusion systemsspatial heterogeneity

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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Pattern Formation

Background:

  • Turing instability and patterns are fundamental to understanding diffusion-driven pattern formation.
  • Classical Turing systems with spatial homogeneity yield organized, repeating patterns.
  • Irregular patterns are of increasing interest, with spatial heterogeneity proposed as a key factor.

Purpose of the Study:

  • To investigate pattern formation in reaction-diffusion systems with spatial heterogeneity.
  • To extend classical Turing instability analysis to heterogeneous systems.
  • To understand the influence of spatial heterogeneity on the evolution and characteristics of Turing patterns.

Main Methods:

  • Extended Turing instability analysis to track linear Turing mode evolution and nascent patterns.
  • Calculation of nonlinear mode coefficients to assess the long-time evolution of patterns.
  • Analytical and numerical techniques applied to reaction-diffusion systems with various forms of spatial heterogeneity.

Main Results:

  • Developed a generalized instability criterion applicable to spatially heterogeneous systems.
  • Demonstrated that heterogeneity can lead to the simultaneous interaction of multiple Turing modes with different wavelengths.
  • Observed that heterogeneous systems produce patterns with significant spatial variation, unlike homogeneous systems.

Conclusions:

  • Spatial heterogeneity is a significant factor in generating complex and irregular Turing patterns.
  • The extended analysis provides a more general framework for studying pattern formation in non-uniform environments.
  • Various mathematical and physical examples illustrate how heterogeneity can modify classical Turing patterns, offering insights into morphogenesis.