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Turing Pattern Formation in a Semiarid Vegetation Model with Fractional-in-Space Diffusion.

Canrong Tian1

  • 1Department of Basic Sciences, Yancheng Institute of Technology, Yancheng, 224003, China. tiancanrong@163.com.

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Summary
This summary is machine-generated.

This study introduces anomalous diffusion to vegetation models, revealing that spatial patterns stabilize with increased superdiffusion. The findings align with real-world vegetation patterns in arid environments.

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Amplitude equationsPattern formationSuperdiffusionTuring instability

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Area of Science:

  • Mathematical modeling
  • Ecology
  • Physics

Background:

  • Reaction-diffusion systems model ecological patterns.
  • Anomalous diffusion describes non-standard particle movement.
  • Water's diffusion in semiarid environments is crucial for vegetation.

Purpose of the Study:

  • To incorporate anomalous diffusion into vegetation pattern models.
  • To analyze the impact of superdiffusion on Turing patterns.
  • To understand the stability and characteristics of emergent spatial patterns.

Main Methods:

  • Introduced fractional Laplacian for anomalous diffusion.
  • Performed linear and weakly nonlinear stability analyses.
  • Derived and analyzed amplitude equations.
  • Conducted numerical simulations.

Main Results:

  • Turing pattern wavenumber increases with the superdiffusive exponent.
  • Supercritical Turing bifurcation ensures asymptotic stability of spatial patterns.
  • Numerical simulations confirmed a bistable regime (hexagons and stripes).

Conclusions:

  • Fractional Laplacian effectively models anomalous diffusion in vegetation systems.
  • Superdiffusion enhances pattern stability and influences characteristic length.
  • Model predictions are consistent with observed vegetation patterns in semiarid regions.