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Turing pattern dynamics in an activator-inhibitor system with superdiffusion.

Lai Zhang1, Canrong Tian2

  • 1Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187, Umeå, Sweden.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 24, 2015
PubMed
Summary
This summary is machine-generated.

Anomalous superdiffusion in activator-inhibitor systems influences Turing pattern formation. A smaller ratio of anomalous diffusion exponents between inhibitor and activator promotes pattern emergence compared to normal diffusion.

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Area of Science:

  • Chemical kinetics
  • Non-equilibrium thermodynamics
  • Mathematical biology

Background:

  • Activator-inhibitor systems are fundamental models for pattern formation.
  • Anomalous diffusion, specifically superdiffusion, describes non-standard particle movement.
  • Understanding diffusion's role is crucial for predicting pattern emergence.

Purpose of the Study:

  • To investigate the impact of anomalous superdiffusion on pattern formation in activator-inhibitor systems.
  • To analyze how the superdiffusive exponent affects Turing pattern characteristics.
  • To explore the interplay between diffusion rates and exponents in pattern emergence.

Main Methods:

  • Introduction of a fractional operator to model anomalous superdiffusion.
  • Linear stability analysis to determine wave numbers of Turing patterns.
  • Weakly nonlinear analysis to derive amplitude equations.
  • Numerical simulations to validate analytical predictions.

Main Results:

  • Turing pattern wave number increases with the superdiffusive exponent.
  • Amplitude equation analysis predicts parameter regimes for hexagons, stripes, and their coexistence.
  • Critical diffusion rate ratio for pattern formation increases with the superdiffusive exponent.
  • Numerical simulations confirm analytical findings.

Conclusions:

  • Anomalous diffusion significantly impacts pattern formation and selection in activator-inhibitor systems.
  • A lower ratio of anomalous diffusion exponents between inhibitor and activator favors Turing pattern emergence.
  • The findings offer insights into pattern formation mechanisms beyond Fickian diffusion.