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Numerical study of anisotropic diffusion in Turing patterns based on Finsler geometry modeling.

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This study introduces Finsler geometry to model anisotropic Turing patterns in reaction-diffusion systems. A novel internal degree of freedom dynamically generates diffusion, explaining emergent biological patterns and enabling control.

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Area of Science:

  • Theoretical Physics
  • Mathematical Biology
  • Chemical Kinetics

Background:

  • Turing patterns (TPs) are crucial for understanding biological pattern formation.
  • Standard models often assume isotropic or manually defined anisotropic diffusion.
  • The origin of spontaneously emergent anisotropic patterns in nature remains an active research area.

Purpose of the Study:

  • To numerically investigate anisotropic Turing patterns using Finsler geometry (FG) modeling.
  • To explore the role of a dynamically generated, direction-dependent diffusion coefficient.
  • To propose a novel mechanism for the emergence and control of anisotropic biological patterns.

Main Methods:

  • Reaction-diffusion (RD) equations within a Finsler geometry framework.
  • A hybrid numerical technique combining Metropolis Monte Carlo for internal degree of freedom (IDOF) updates and discrete RD equations for steady-state solutions.
  • Dynamical generation of diffusion coefficients influenced by an IDOF.

Main Results:

  • The FG model successfully generates anisotropic Turing patterns.
  • The internal degree of freedom (IDOF) and its interaction with system components are identified as a source of emergent anisotropy.
  • Demonstrated that the IDOF can be linked to cell diffusion and thermal fluctuations for pattern control.

Conclusions:

  • Finsler geometry provides a novel framework for modeling anisotropic diffusion in reaction-diffusion systems.
  • The introduced IDOF offers a potential explanation for the spontaneous emergence of anisotropic patterns observed in organisms like zebras and fish.
  • The IDOF presents a pathway for externally controlling Turing patterns in biological systems.