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Turing pattern formation in fractional activator-inhibitor systems.

B I Henry1, T A M Langlands, S L Wearne

  • 1Department of Applied Mathematics, School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia. B.Henry@unsw.edu.au

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2005
PubMed
Summary

This study explores anomalous subdiffusion in activator-inhibitor systems, revealing how fractional calculus impacts pattern formation. Results show altered Turing instability thresholds and complex spatiotemporal patterns under anomalous diffusion.

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Area of Science:

  • Reaction-diffusion systems
  • Anomalous diffusion
  • Pattern formation

Background:

  • Activator-inhibitor systems model pattern formation in various scientific fields.
  • Standard diffusion models use a single diffusion constant, with Turing patterns forming above a critical ratio (d) of inhibitor to activator diffusion.
  • Anomalous subdiffusion introduces a diffusion exponent, altering diffusion dynamics.

Purpose of the Study:

  • To investigate activator-inhibitor systems with anomalous subdiffusion.
  • To analyze the impact of anomalous diffusion on Turing instabilities and pattern formation.
  • To explore fractional reaction-diffusion systems with Gierer-Meinhardt and Brusselator kinetics.

Main Methods:

  • Developed a reaction-diffusion system with fractional temporal derivatives based on continuous-time random walks.

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  • Performed algebraic stability analysis of the homogeneous steady-state solution.
  • Conducted numerical simulations of the fractional activator-inhibitor equations.

    • Main Results:

    • Identified Turing instability bifurcation curves in the diffusion parameter space for fractional systems.
    • Found that the critical value of d decreases with the anomalous diffusion exponent.
    • Observed the formation of complex spatiotemporal patterns, transitioning from stationary to nonstationary as diffusion becomes more anomalous.

    Conclusions:

    • Anomalous diffusion significantly alters Turing instability conditions and pattern characteristics.
    • Fractional calculus provides a framework for modeling anomalous subdiffusion in pattern formation.
    • Specific diffusion properties (e.g., activator vs. inhibitor) can lead to stable patterns even below standard thresholds.