Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns

  • Richard Samuelson
  • Zachary Singer
  • Jasper Weinburd
  • Arnd Scheel


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.


Conservation laws Traveling waves Heteroclinic bifurcation Undercompressive shocks 

Mathematics Subject Classification

34K18 92D40 35C07 



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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Trinity CollegeHartfordUSA
  2. 2.Department of Mathematical Sciences, Wean Hall 6113Carnegie Mellon UniversityPittsburghUSA
  3. 3.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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