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Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns

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Abstract

We motivate and analyze a simple model for the formation of banded vegetation patterns. The model incorporates a minimal number of ingredients for vegetation growth in semiarid landscapes. It allows for comprehensive analysis and sheds new light onto phenomena such as the migration of vegetation bands and the interplay between their upper and lower edges. The key ingredient is the formulation as a closed reaction–diffusion system, thus introducing a conservation law that both allows for analysis and provides ready intuition and understanding through analogies with characteristic speeds of propagation and shock waves.

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Notes

  1. Compare also with (Sandstede and Scheel 2004, (1.8)) where such conditions were derived in reaction–diffusion systems when the underlying conserved quantity is the phase of an oscillation rather than an explicit variable.

  2. We are not aware of observations of such “strip”-like vegetation gaps—gaps appear to be mostly patches, localized in all spatial directions.

  3. This dichotomy is central to the proof and the reader may verify the statement by computing the vector field on \(\partial \Sigma \).

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Acknowledgements

This work was supported through Grant NSF DMS—1311740. Most of the analysis was carried out during an NSF-funded REU project on Complex Systems at the University of Minnesota in Summer 2017. The authors gratefully acknowledge conversations with Arjen Doelman and Punit Gandhi, who pointed to many of the references included here and provided many helpful comments and suggestions on an early version of the manuscript.

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Correspondence to Arnd Scheel.

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Communicated by Michael Ward.

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Samuelson, R., Singer, Z., Weinburd, J. et al. Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns. J Nonlinear Sci 29, 255–285 (2019). https://doi.org/10.1007/s00332-018-9486-6

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