Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns
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.
KeywordsConservation laws Traveling waves Heteroclinic bifurcation Undercompressive shocks
Mathematics Subject Classification34K18 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.
- Chow, S.N., Hale, J.K.: Methods of bifurcation theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 251. Springer, New York (1982)Google Scholar
- Doedel, E.J., Oldeman, B.E.: AUTO07p software for continuation and bifurcation problems in ordinary differential equations. http://indy.cs.concordia.ca/auto/ (2007)
- Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains. Mem. Am. Math. Soc. 199(934), viii+105 (2009)Google Scholar
- Goh, R.N., Mesuro, S., Scheel, A.: Spatial wavenumber selection in recurrent precipitation. In: Precipitation Patterns in Reaction–Diffusion Systems, pp. 73–92. Research Signpost (2010)Google Scholar
- Gowda, K., Iams, S., Silber, M.: Dynamics and resilience of vegetation bands in the Horn of Africa. ArXiv e-prints (2017)Google Scholar
- Kotzagiannidis, M., Peterson, J., Redford, J., Scheel, A., Wu, Q.: Stable pattern selection through invasion fronts in closed two-species reaction–diffusion systems. In: Far-from-Equilibrium Dynamics, RIMS Kôkyûroku Bessatsu, B31, pp. 79–92. Res. Inst. Math. Sci. (RIMS), Kyoto (2012)Google Scholar
- Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn., volume 112 of Applied Mathematical Sciences. Springer, New York (1998)Google Scholar
- Rietkerk, M., Boerlijst, M., van Langevelde, F., HilleRisLambers, R., de Koppel, J., Kumar, L., Prins, H.T., de Roos, A.: Self-organization of vegetation in arid ecosystems. Am. Nat. 160(4), 524–530 (2002)Google Scholar
- Shilnikov, L.P., Shilnikov, A.L., Turaev, D., Chua, L.O.: Methods of Qualitative Theory in Nonlinear Dynamics. Part II, volume 5 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. World Scientific, River Edge(2001)Google Scholar