Turing Patterns in Deserts

  • Jonathan A. Sherratt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)

Abstract

Self-organised patterns of vegetation are a characteristic feature of many semi-arid regions. In particular, banded vegetation is typical on hillsides. Mathematical modelling is widely used to study these banded patterns, because there are no laboratory replicates. I will describe the development of spatial patterns in an established model for banded vegetation via a Turing bifurcation. I will discuss numerical simulations of the phenomenon, and I will summarise nonlinear analysis on the existence and form of spatial patterns as a function of the model parameter that corresponds to mean annual rainfall.

Keywords

Travel Wave Solution Arid Environment Vegetation Pattern Arid Ecosystem Migration Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Valentin, C., d’Herbès, J.M., Poesen, J.: Soil and water components of banded vegetation patterns. Catena 37, 1–24 (1999)CrossRefGoogle Scholar
  2. 2.
    Rietkerk, M., Dekker, S.C., de Ruiter, P.C., van de Koppel, J.: Self–organized patchiness and catastrophic shifts in ecosystems. Science 305, 1926–1929 (2004)CrossRefGoogle Scholar
  3. 3.
    Dunkerley, D.L., Brown, K.J.: Oblique vegetation banding in the Australian arid zone: implications for theories of pattern evolution and maintenance. J. Arid Environ. 52, 163–181 (2002)CrossRefGoogle Scholar
  4. 4.
    Berg, S.S., Dunkerley, D.L.: Patterned mulga near Alice Springs, central Australia, and the potential threat of firewood collection on this vegetation community. J. Arid Environ. 59, 313–350 (2004)CrossRefGoogle Scholar
  5. 5.
    Montaña, C.: The colonization of bare areas in two–phase mosaics of an arid ecosystem. J. Ecol. 80, 315–327 (1992)CrossRefGoogle Scholar
  6. 6.
    McDonald, A.K., Kinucan, R.J., Loomis, L.E.: Ecohydrological interactions within banded vegetation in the northeastern Chihuahuan Desert, USA. Ecohydrology 2, 66–71 (2009)CrossRefGoogle Scholar
  7. 7.
    MacFadyen, W.: Vegetation patterns in the semi–desert plains of British Somaliland. Geographical J. 115, 199–211 (1950)CrossRefGoogle Scholar
  8. 8.
    Valentin, C., d’Herbès, J.M.: Niger tiger bush as a natural water harvesting system. Catena 37, 231–256 (1999)CrossRefGoogle Scholar
  9. 9.
    Couteron, P., Mahamane, A., Ouedraogo, P., Seghieri, J.: Differences between banded thickets (tiger bush) at two sites in West Africa. J. Veg. Sci. 11, 321–328 (2000)CrossRefGoogle Scholar
  10. 10.
    Borgogno, F., D’Odorico, P., Laio, F., Ridolfi, L.: Mathematical models of vegetation pattern formation in ecohydrology. Rev. Geophys. 47, art. no. RG1005 (2009)Google Scholar
  11. 11.
    Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826–1828 (1999)CrossRefGoogle Scholar
  12. 12.
    HilleRisLambers, R., Rietkerk, M., van de Bosch, F., Prins, H.H.T., de Kroon, H.: Vegetation pattern formation in semi–arid grazing systems. Ecology 82, 50–61 (2001)CrossRefGoogle Scholar
  13. 13.
    Rietkerk, M., Boerlijst, M.C., van Langevelde, F., HilleRisLambers, R., van de Koppel, J., Prins, H.H.T., de Roos, A.: Self–organisation of vegetation in arid ecosystems, Am. Am. Nat. 160, 524–530 (2002)CrossRefGoogle Scholar
  14. 14.
    Gilad, E., von Hardenberg, J., Provenzale, A., Shachak, M., Meron, E.: A mathematical model of plants as ecosystem engineers, J. J. Theor. Biol. 244, 680–691 (2007)CrossRefGoogle Scholar
  15. 15.
    Ursino, N.: Modeling banded vegetation patterns in semiarid regions: inter–dependence between biomass growth rate and relevant hydrological processes. Water Resour. Res. 43, W04412 (2007)Google Scholar
  16. 16.
    Ursino, N.: Above and below ground biomass patterns in arid lands. Ecological Modelling 220, 1411–1418 (2009)CrossRefGoogle Scholar
  17. 17.
    Ursino, N., Contarini, S.: Stability of banded vegetation patterns under seasonal rainfall and limited soil moisture storage capacity. Adv. Water Resour. 29, 1556–1564 (2006)CrossRefGoogle Scholar
  18. 18.
    Guttal, V., Jayaprakash, C.: Self–organisation and productivity in semi–arid ecosystems: implications of seasonality in rainfall. J. Theor Biol. 248, 290–500 (2007)CrossRefGoogle Scholar
  19. 19.
    Kletter, A.Y., von Hardenberg, J., Meron, E., Provenzale, A.: Patterned vegetation and rainfall intermittency. J. Theor. Biol. 256, 574–583 (2009)CrossRefGoogle Scholar
  20. 20.
    van de Koppel, J., Rietkerk, M., van Langevelde, F., Kumar, L., Klausmeier, C.A., Fryxell, J.M., Hearne, J.W., van Andel, J., de Ridder, N., Skidmore, M.A., Stroosnijder, L., Prins, H.H.T.: Spatial heterogeneity and irreversible vegetation change in semiarid grazing systems. Am. Nat. 159, 209–218 (2002)CrossRefGoogle Scholar
  21. 21.
    Pueyo, Y., Kefi, S., Alados, C.L., Rietkerk, M.: Dispersal strategies and spatial organization of vegetation in arid ecosystems. Oikos 117, 1522–1532 (2008)CrossRefGoogle Scholar
  22. 22.
    Kefi, S., Rietkerk, M., Katul, G.G.: Vegetation pattern shift as a result of rising atmospheric CO2 in arid ecosystems. Theor. Pop. Biol. 74, 332–344 (2008)MATHCrossRefGoogle Scholar
  23. 23.
    Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59, 263–294 (1997)MATHCrossRefGoogle Scholar
  24. 24.
    Couteron, P., Lejeune, O.: Periodic spotted patterns in semi–arid vegetation explained by a propagation–inhibition model. J. Ecol. 89, 616–628 (2001)CrossRefGoogle Scholar
  25. 25.
    Barbier, N., Couteron, P., Lefever, R., Deblauwe, V., Lejeune, O.: Spatial decoupling of facilitation, competition at the origin of gapped vegetation patterns. Ecology 89, 1521–1531 (2008)CrossRefGoogle Scholar
  26. 26.
    Lefever, R., Barbier, N., Couteron, P., Lejeune, O.: Deeply gapped vegetation patterns: on crown/root allometry, criticality and desertification. J. Theor. Biol. 261, 194–209 (2009)CrossRefGoogle Scholar
  27. 27.
    Deblauwe, V.: Modulation des structures de végétation auto–organisées en milieu aride / Self–organized vegetation pattern modulation in arid climates. PhD thesis, Université Libre de Bruxelles (2010)Google Scholar
  28. 28.
    Tongway, D.J., Ludwig, J.A.: Theories on the origins, maintainance, dynamics, and functioning of banded landscapes. In: Tongway, D.J., Valentin, C., Seghieri, J. (eds.) Banded Vegetation Patterning in Arid and Semi–Arid Environments, pp. 20–31. Springer, New York (2001)CrossRefGoogle Scholar
  29. 29.
    Montaña, C., Seghieri, J., Cornet, A.: Vegetation dynamics: recruitment, regeneration in two–phase mosaics. In: Tongway, D.J., Valentin, C., Seghieri, J. (eds.) Banded Vegetation Patterning in Arid and Semi–Arid Environments, pp. 132–145. Springer, New York (2001)CrossRefGoogle Scholar
  30. 30.
    Sherratt, J.A., Lord, G.J.: Nonlinear dynamics, pattern bifurcations in a model for vegetation stripes in semi–arid environments. Theor. Pop. Biol. 71, 1–11 (2007)MATHCrossRefGoogle Scholar
  31. 31.
    Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi–arid environments II: patterns with the largest possible propagation speeds. Proc. R. Soc. Lond. A 467, 3272–3294 (2011)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation semi–arid environments IV: slowly moving patterns and their stability (submitted)Google Scholar
  33. 33.
    Doedel, E.J.: AUTO, a program for the automatic bifurcation analysis of autonomous systems. Cong. Numer. 30, 265–384 (1981)MathSciNetGoogle Scholar
  34. 34.
    Sherratt, J.A.: An analysis of vegetation stripe formation in semi–arid landscapes. J. Math. Biol. 51, 183–197 (2005)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi–arid environments I. Nonlinearity 23, 2657–2675 (2010)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semi–arid environments III: the transition between homoclinic solutions (submitted)Google Scholar
  37. 37.
    Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)CrossRefGoogle Scholar
  38. 38.
    Lejeune, O., Tlidi, M., Couteron, P.: Localized vegetation patches: a self–organized response to resource scarcity. Phys. Rev. E 66, 010901 (2002)CrossRefGoogle Scholar
  39. 39.
    Ben Wu, X., Archer, S.R.: Scale–dependent influence of topography–based hydrologic features on patterns of woody plant encroachment in savanna landscapes. Landscape Ecol. 20, 733–742 (2005)CrossRefGoogle Scholar
  40. 40.
    Belyea, L.R.: Climatic and topographic limits to the abundance of bog pools. Hydrological Processes 21, 675–687 (2007)CrossRefGoogle Scholar
  41. 41.
    Eppinga, M.B., de Ruiter, P.C., Wassen, M.J., Rietkerk, M.: Nutrients and hydrology indicate the driving mechanisms of peatland surface patterning. Am. Nat. 173, 803–818 (2009)CrossRefGoogle Scholar
  42. 42.
    van de Koppel, J., Rietkerk, M., Dankers, N., Herman, P.M.J.: Scale–dependent feedback and regular spatial patterns in young mussel beds. Am. Nat. 165, E66–E77 (2005)CrossRefGoogle Scholar
  43. 43.
    van de Koppel, J., Gascoigne, J.C., Theraulaz, G., Rietkerk, M., Mooij, W.M., Herman, P.M.J.: Experimental evidence for spatial self–organization and its emergent effects in mussel bed ecosystems. Science 322, 739–742 (2008)CrossRefGoogle Scholar
  44. 44.
    Hiemstra, C.A., Liston, G.E., Reiners, W.A.: Observing, modelling, and validating snow redistribution by wind in a Wyoming upper treeline landscape. Ecol. Modelling 197, 35–51 (2006)CrossRefGoogle Scholar
  45. 45.
    Bekker, M.F., Clark, J.T., Jackson, M.W.: Landscape metrics indicate differences in patterns and dominant controls of ribbon forests in the Rocky Mountains, USA. Applied Vegetation Science 12, 237–249 (2009)CrossRefGoogle Scholar
  46. 46.
    Borthagaray, A.I., Fuentes, M.A., Marquet, P.A.: Vegetation pattern formation in a fog–dependent ecosystem. J. Theor. Biol. 265, 18–26 (2010)CrossRefGoogle Scholar
  47. 47.
    Wang, R.H., Liu, Q.X., Sun, G.Q., Jin, Z., van de Koppel, J.: Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds. J. R. Soc. Interface 6, 705–718 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jonathan A. Sherratt
    • 1
  1. 1.Department of Mathematics and Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghUnited Kingdom

Personalised recommendations