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Stabilizing a homoclinic stripe.

Theodore Kolokolnikov1, Michael Ward2, Justin Tzou3

  • 1Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada tkolokol@gmail.com.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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Summary
This summary is machine-generated.

Anisotropy can stabilize two-dimensional stripes in reaction-diffusion systems, preventing pattern break-up. This research explores instabilities and pattern transitions in models like Schnakenberg and Klausmeier.

Keywords:
pattern formationreaction–diffusion systemsstability of patterns

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Area of Science:

  • Reaction-diffusion systems
  • Pattern formation
  • Mathematical modeling

Background:

  • Two-dimensional stripes in reaction-diffusion systems are typically unstable, breaking into spots.
  • Stabilizing these stripes is crucial for understanding pattern formation.

Purpose of the Study:

  • To investigate methods for stabilizing one-dimensional homoclinic spikes in two-dimensional reaction-diffusion systems.
  • To analyze the effects of anisotropy and model parameters on stripe stability.

Main Methods:

  • Theoretical analysis of instability thresholds (zigzag and break-up).
  • Numerical simulations for the Schnakenberg and Klausmeier models.
  • Derivation of instability boundaries in parameter space.

Main Results:

  • Anisotropy in the fast-diffusing variable can stabilize infinite stripes in the Schnakenberg model.
  • Identified distinct zones for stable stripes, bending instability, and break-up instability.
  • Break-up instability leads to a 'spotted-stripe' pattern; transition from spots to stripes examined in the Klausmeier model.

Conclusions:

  • Anisotropy offers a mechanism to stabilize reaction-diffusion stripes.
  • Understanding instability thresholds is key to predicting pattern evolution.
  • The findings have implications for vegetation pattern dynamics and dissipative structures.