Advection and Autocatalysis as Organizing Principles for Banded Vegetation Patterns

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Notes

  1. 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. 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. 3.

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

References

  1. Borgogno, F., D’Odorico, P., Laio, F., Ridolfi, L.: Mathematical models of vegetation pattern formation in ecohydrology. Rev. Geophys. 47(1), RG1005 (2009)

    Article  Google Scholar 

  2. Bricmont, J., Kupiainen, A., Lin, G.: Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun. Pure Appl. Math. 47(6), 893–922 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  3. 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)

  4. Coullet, P., Risler, E., Vandenberghe, N.: Spatial unfolding of elementary bifurcations. J. Stat. Phys. 101(1), 521–541 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  5. Doedel, E.J., Oldeman, B.E.: AUTO07p software for continuation and bifurcation problems in ordinary differential equations. http://indy.cs.concordia.ca/auto/ (2007)

  6. Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains. Mem. Am. Math. Soc. 199(934), viii+105 (2009)

  7. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  8. 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)

  9. Goh, R.N., Mesuro, S., Scheel, A.: Coherent structures in reaction–diffusion models for precipitation. SIAM J. Appl. Dyn. Syst. 10(1), 360–402 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  10. Gowda, K., Riecke, H., Silber, M.: Transitions between patterned states in vegetation models for semiarid ecosystems. Phys. Rev. E 89, 022701 (2014)

    Article  Google Scholar 

  11. Gowda, K., Chen, Y., Iams, S., Silber, M.: Assessing the robustness of spatial pattern sequences in a dryland vegetation model. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2187), 20150893 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  12. Gowda, K., Iams, S., Silber, M.: Dynamics and resilience of vegetation bands in the Horn of Africa. ArXiv e-prints (2017)

  13. Haragus, M., Scheel, A.: Almost planar waves in anisotropic media. Commun. Partial Differ. Equ. 31(4–6), 791–815 (2006a)

    MathSciNet  MATH  Article  Google Scholar 

  14. Haragus, M., Scheel, A.: Corner defects in almost planar interface propagation. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(3), 283–329 (2006b)

    MathSciNet  MATH  Article  Google Scholar 

  15. HilleRisLambers, R., Rietkerk, M., van den Bosch, F., Prins, H.H.T., de Kroon, H.: Vegetation pattern formation in semi-arid grazing systems. Ecology 82(1), 50–61 (2001)

    Article  Google Scholar 

  16. Holzer, M., Scheel, A.: Criteria for pointwise growth and their role in invasion processes. J. Nonlinear Sci. 24(4), 661–709 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  17. Jimbo, S., Morita, Y.: Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation. J. Differ. Equ. 255(7), 1657–1683 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  18. Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284(5421), 1826–1828 (1999)

    Article  Google Scholar 

  19. 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)

  20. Kuwamura, M., Morita, Y.: Perturbations and dynamics of reaction–diffusion systems with mass conservation. Phys. Rev. E 92, 012908 (2015)

    MathSciNet  Article  Google Scholar 

  21. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn., volume 112 of Applied Mathematical Sciences. Springer, New York (1998)

  22. Lejeune, O., Tlidi, M., Lefever, R.: Vegetation spots and stripes: dissipative structures in arid landscapes. Int. J. Quantum Chem. 98(2), 261–271 (2004)

    Article  Google Scholar 

  23. Meron, E.: Pattern-formation approach to modelling spatially extended ecosystems. Ecol. Model. 234(Suppl C), 70–82 (2012)

    Article  Google Scholar 

  24. Meron, E.: Nonlinear Physics of Ecosystems. CRC Press, London (2015)

    Google Scholar 

  25. Meron, E., Gilad, E., von Hardenberg, J., Shachak, M., Zarmi, Y.: Vegetation patterns along a rainfall gradient. Chaos Solitons Fractals 19(2), 367–376 (2004)

    MATH  Article  Google Scholar 

  26. Mori, Y., Jilkine, A., Edelstein-Keshet, L.: Wave-pinning and cell polarity from a bistable reaction–diffusion system. Biophys. J. 94(9), 3684–3697 (2008)

    Article  Google Scholar 

  27. Pogan, A., Scheel, A.: Instability of spikes in the presence of conservation laws. Z. Angew. Math. Phys. 61(6), 979–998 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  28. Pogan, A., Scheel, A.: Traveling fronts bifurcating from stable layers in the presence of conservation laws. Discrete Contin. Dyn. Syst. 37(5), 2619–2651 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  29. Pogan, A., Scheel, A., Zumbrun, K.: Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns. Differ. Integral Equ. 26(3–4), 389–438 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Rademacher, J.D.M., Scheel, A.: The saddle-node of nearly homogeneous wave trains in reaction–diffusion systems. J. Dyn. Differ. Equ. 19(2), 479–496 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. Sandstede, B., Scheel, A.: Defects in oscillatory media: toward a classification. SIAM J. Appl. Dyn. Syst. 3(1), 1–68 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  33. Scheel, A., Stevens, A.: Wavenumber selection in coupled transport equations. J. Math. Biol. 75(5), 1047–1073 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  34. Sewalt, L., Doelman, A.: Spatially periodic multipulse patterns in a generalized Klausmeier–Gray–Scott model. SIAM J. Appl. Dyn. Syst. 16(2), 1113–1163 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  35. Shashkov, M.V.: On bifurcations of separatrix contours with two saddles. Int. J. Bifurc. Chaos Appl. Sci. Eng. 2(4), 911–915 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  36. Sherratt, J.A.: Using wavelength and slope to infer the historical origin of semiarid vegetation bands. Proc. Nat. Acad. Sci. 112(14), 4202–4207 (2015)

    Article  Google Scholar 

  37. Sherratt, J.A.: When does colonisation of a semi-arid hillslope generate vegetation patterns? J. Math. Biol. 73(1), 199–226 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  38. 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)

  39. Siero, E., Doelman, A., Eppinga, M.B., Rademacher, J.D.M., Rietkerk, M., Siteur, K.: Striped pattern selection by advective reaction–diffusion systems: resilience of banded vegetation on slopes. Chaos Interdiscipl. J. Nonlinear Sci. 25(3), 036411 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  40. van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386(2), 29–222 (2003)

    MATH  Article  Google Scholar 

  41. Verschueren, N., Champneys, A.: A model for cell polarization without mass conservation. SIAM J. Appl. Dyn. Syst. 16(4), 1797–1830 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  42. von Hardenberg, J., Kletter, A.Y., Yizhaq, H., Nathan, J., Meron, E.: Periodic versus scale-free patterns in dryland vegetation. Proc. R. Soc. Lond. B Biol. Sci. 277, 1771–1776 (2010)

    Article  Google Scholar 

  43. Wuyts, B., Champneys, A.R., House, J.I.: Amazonian forest-savanna bistability and human impact. Nat. Commun. 8, 15519 (2017)

    Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Arnd Scheel.

Additional information

Communicated by Michael Ward.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Conservation laws
  • Traveling waves
  • Heteroclinic bifurcation
  • Undercompressive shocks

Mathematics Subject Classification

  • 34K18
  • 92D40
  • 35C07