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Mathematics of Planet Earth pp 177–202Cite as

Multistability in Ecosystems: Concerns and Opportunities for Ecosystem Function in Variable Environments

Part of the Mathematics of Planet Earth book series (MPE,volume 5)

Abstract

Ecosystems are highly nonlinear dissipative systems characterized by multiplicity of stable and unstable states. Two major concerns are associated with multistable ecosystems in variable environments. The first is related to the increased likelihood of extreme climate events at regional scales, such as droughts, floods, and heat waves, that may result in abrupt transitions to malfunctioning ecosystem states. The second concern is related to the dominant role played by humans in shaping and transforming the ecology of the Earth, and to the detrimental effects that such transformations often have. Using mathematical models of dryland ecosystems as a case study, we discuss recent advances that shed new light on these concerns. We first argue that state transitions can be gradual or incomplete rather than abrupt, providing opportunities for prevention and recovery. We further argue that analyzing the unstable states that exist along with the stable ones, identifying their existence ranges and their stable and unstable manifolds, can help to devise human intervention forms that direct ecosystems towards desired functional ecosystem states, without impairing ecosystem function. We conclude by presenting open problems and delineating further research directions.

Keywords

  • Dryland ecosystems
  • Vegetation patterns
  • Multistability
  • Front dynamics
  • Abrupt and gradual state transitions
  • Human intervention

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References

  1. Adeel, Z., Safriel, U., Niemeijer, D., et al.: Ecosystems and human well-being: Desertification synthesis. Technical Report of the Millennium Ecosystem Assessment, World Resources Institute, Washington, D.C. (2005)

    Google Scholar 

  2. Barbier, N., Couteron, P., Lefever, R., et al.: Spatial decoupling of facilitation and competition at the origin of gapped vegetation patterns. Ecology 89, 1521–1531 (2008)

    CrossRef  Google Scholar 

  3. Bel, G., Hagberg, A., Meron, E.: Gradual regime shifts in spatially extended ecosystems. Theor. Ecol. 5, 591–604 (2012)

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  5. Borthagaray, A.I., Fuentes, M.A., Marquet, P.A.: Vegetation pattern formation in a fog-dependent ecosystem. J. Theor. Biol. 265(1), 18–26 (2010)

    MathSciNet  MATH  CrossRef  Google Scholar 

  6. Chen, Y., Kolokolnikov, T., Tzou, J., et al.: Patterned vegetation, tipping points, and the rate of climate change. Eur. J. Appl. Math. 26, 945–958 (2015)

    MathSciNet  MATH  CrossRef  Google Scholar 

  7. Coullet, P., Lega, J., Houchmanzadeh, B., et al.: Breaking chirality in nonequilibrium system. Phys. Rev. Lett. 65, 1352 (1990)

    CrossRef  Google Scholar 

  8. Cramer, M.D., Barger, N.N.: Are Namibian fairy circles the consequence of self-organizing spatial vegetation patterning? PloS One 8(8), e70,876 (2013)

    CrossRef  Google Scholar 

  9. Cramer, M.D., Barger, N.N., Tschinkel, W.R.: Edaphic properties enable facilitative and competitive interactions resulting in fairy circle formation. Ecography 40, 1210–1220 (2017)

    CrossRef  Google Scholar 

  10. Cross, M.C., Greenside, H.: Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press, Cambridge (2009)

    MATH  CrossRef  Google Scholar 

  11. Dawes, J.H.P., Williams, J.L.M.: Localised pattern formation in a model for dryland vegetation. J. Math. Biol. 73, 1–28 (2015)

    MathSciNet  MATH  Google Scholar 

  12. DeAngelis, D.L., Gross, L.J. (eds.): Individual-Based Models and Approaches on Ecology: Concepts and Models. Chapman and Hall, New York (1992)

    Google Scholar 

  13. DeAngelis, D.L., Yurek, S.: Spatially explicit modeling in ecology: A review. Ecosystems 20(2), 284–300 (2017). https://doi.org/10.1007/s10021-016-0066-z

    CrossRef  Google Scholar 

  14. Deblauwe, V., Barbier, N., Couteron, P., et al.: The global biogeography of semi-arid periodic vegetation patterns. Glob. Ecol. Biogeogr. 17, 715–723 (2008)

    CrossRef  Google Scholar 

  15. Dirzo, R., Young, H.S., Galetti, M., et al.: Defaunation in the anthropocene. Science 345, 401–406 (2014)

    CrossRef  Google Scholar 

  16. D’Odorico, P., Bhattachan, A., Davis, K.F., et al.: Global desertification: Drivers and feedbacks. Adv. Water Resour. 51, 326–344 (2013)

    CrossRef  Google Scholar 

  17. Duraiappah, A.K., Naeem, S.: Ecosystems and human well-being: biodiversity synthesis. Technical Report of the Millennium Ecosystem Assessment, World Resources Institute, Washington, DC. (2005)

    Google Scholar 

  18. Eldridge, D.J., Zaady, E., Shachak, M.: Infiltration through three contrasting biological soil crusts in patterned landscapes in the Negev, Israel. J. Stat. Phys. 148, 723–739 (2012)

    Google Scholar 

  19. Ellis, E.C.: Ecology in an anthropogenic biosphere. Ecol. Monogr. 85, 287–331 (2015)

    CrossRef  Google Scholar 

  20. Fernandez-Oto, C., Tlidi, M., Escaff, D., et al.: Strong interaction between plants induces circular barren patches: fairy circles. Phil. Trans. R. Soc. A 372(2027), 20140009 (2014)

    CrossRef  Google Scholar 

  21. Field, C.B., Barros, V., Stocker, T.F., et al.: Managing the risks of extreme events and disasters to advance climate change adaptation: a special report of the Intergovernmental Panel on Climate Change. Technical Report, Cambridge University Press, Cambridge, UK, and New York, NY (2013)

    Google Scholar 

  22. Getzin, S., Wiegand, K., Wiegand, T., et al.: Adopting a spatially explicit perspective to study the mysterious fairy circles of Namibia. Ecography 38, 1–11 (2015)

    CrossRef  Google Scholar 

  23. Getzin, S., Yizhaq, H., Bell, B., et al.: Discovery of fairy circles in Australia supports self-organization theory. Proc. Natl. Acad. Sci. 113(13), 3551–3556 (2016)

    CrossRef  Google Scholar 

  24. Gilad, E., Von Hardenberg, J., Provenzale, A., et al.: Ecosystem engineers: from pattern formation to habitat creation. Phys. Rev. Lett. 93, 098105 (2004)

    CrossRef  Google Scholar 

  25. Gilad, E., Shachak, M., Meron, E.: Dynamics and spatial organization of plant communities in water limited systems. Theor. Popul. Biol. 72, 214–230 (2007)

    MATH  CrossRef  Google Scholar 

  26. Gilad, E., Von Hardenberg, J., Provenzale, A., et al.: A mathematical model for plants as ecosystem engineers. J. Theor. Biol. 244, 680 (2007)

    MathSciNet  CrossRef  Google Scholar 

  27. Goldstein, R.E., Muraki, D.J., Petrich, D.M.: Interface proliferation and the growth of labyrinths in a reaction-diffusion system. Phys. Rev. E 53, 3933–3957 (1996)

    MathSciNet  CrossRef  Google Scholar 

  28. Gowda, K., Riecke, H., Silber, M.: Transitions between patterned states in vegetation models for semiarid ecosystems. Phys. Rev. E 89, 022,701 (2014)

    CrossRef  Google Scholar 

  29. Grimm, V., Railsback, S.F.: Individual-based Modeling and Ecology. Princeton University Press, Princeton (2005)

    MATH  CrossRef  Google Scholar 

  30. Hagberg, A., Meron, E.: Complex patterns in reaction diffusion systems: a tale of two front instabilities. Chaos 4, 477–484 (1994)

    MATH  CrossRef  Google Scholar 

  31. Hagberg, A., Meron, E.: From labyrinthine patterns to spiral turbulence. Phys. Rev. Lett. 72, 2494–2497 (1994)

    CrossRef  Google Scholar 

  32. Hagberg, A., Meron, E.: Pattern formation in non-gradient reaction diffusion systems: the effects of front bifurcations. Nonlinearity 7, 805–835 (1994)

    MathSciNet  MATH  CrossRef  Google Scholar 

  33. Hagberg, A., Meron, E.: The dynamics of curved fronts: beyond geometry. Phys. Rev. Lett. 78, 1166–1169 (1997)

    CrossRef  Google Scholar 

  34. Hagberg, A., Meron, E., Rubinstein, I., et al.: Controlling domain patterns far from equilibrium. Phys. Rev. Lett. 76, 427–430 (1996)

    CrossRef  Google Scholar 

  35. Hastings, A.: The key to long-term ecological understanding? Trends Ecol. Evol. 19, 39–45 (2004)

    CrossRef  Google Scholar 

  36. van Heijster, P., Doelman, A., Kaper, T.J., et al.: Front interactions in a three-component system. SIAM J. Appl. Dyn. Syst. 9(2), 292–332 (2010)

    MathSciNet  MATH  CrossRef  Google Scholar 

  37. Hilker, F.M., Lewis, M.A., Seno, H., et al.: Pathogens can slow down or reverse invasion fronts of their hosts. Biol. Invasions 7(5), 817–832 (2005)

    CrossRef  Google Scholar 

  38. Homburg, A.J., Sandstede, B.: Homoclinic and heteroclinic bifurcations in vector fields. In: Handbook of Dynamical Systems, vol. 3, pp. 379–524. Elsevier, Amsterdam (2010)

    Google Scholar 

  39. Ikeda, H., Mimura, M., Nishiura, Y.: Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems. Nonlinear Anal. Theory Methods Appl. 13, 507–526 (1989)

    MathSciNet  MATH  CrossRef  Google Scholar 

  40. Juergens, N.: The biological underpinnings of Namib Desert fairy circles. Science 339(6127), 1618–1621 (2013)

    CrossRef  Google Scholar 

  41. Kéfi, S., Vishwesha, G., Brock, W.A.: Early warning signals of ecological transitions: methods for spatial patterns. Plos One 9, e92097 (2014)

    CrossRef  Google Scholar 

  42. Kinast, S., Zelnik, Y.R., Bel, G., et al.: Interplay between turing mechanisms can increase pattern diversity. Phys. Rev. Lett. 112, 078701 (2014)

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  44. Kletter, A.Y., von Hardenberg, J., Meron, E.: Ostwald ripening in dryland vegetation. Commun. Pure Appl. Anal. 11, 261–273 (2012)

    MathSciNet  MATH  CrossRef  Google Scholar 

  45. Knobloch, E.: Spatially localized structures in dissipative systems: open problems. Nonlinearity 21, T45 (2008)

    MathSciNet  MATH  CrossRef  Google Scholar 

  46. Knobloch, E.: Spatial localization in dissipative systems. Ann. Rev. Condens. Matter Phys. 6(1), 325–359 (2015)

    CrossRef  Google Scholar 

  47. Kyriazopoulos, P., Jonathan, N., Meron, E.: Species coexistence by front pinning. Ecol. Complex. 20, 271–281 (2014)

    CrossRef  Google Scholar 

  48. Lefever, R., Lejeune, O.: On the origin of tiger bush. Bull. Math. Biol. 59, 263–294 (1997)

    MATH  CrossRef  Google Scholar 

  49. Lejeune, O., Couteron, P., Lefever, R.: Short range co-operativity competing with long range inhibition explains vegetation patterns. Acta Oecol. 20(3), 171–183 (1999)

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  51. Maestre, F.T., Eldridge, D.J., Soliveres, S., et al.: Structure and functioning of dryland ecosystems in a changing world. Annu. Rev. Ecol. Evol. Syst. 47(1), 215–237 (2016)

    CrossRef  Google Scholar 

  52. Marten, G.G.: Human Ecology - Basic Concepts for Sustainable Development. Earthscan Publications, London (2001)

    Google Scholar 

  53. Marts, B., Hagberg, A., Meron, E., et al.: Bloch-front turbulence in a periodically forced Belousov-Zhabotinsky reaction. Phys. Rev. Lett. 93(108305), 1–4 (2004)

    Google Scholar 

  54. Mau, Y., Hagberg, A., Meron, E.: Spatial periodic forcing can displace patterns it is intended to control. Phys. Rev. Lett. 109, 034102 (2012)

    CrossRef  Google Scholar 

  55. Mau, Y., Haim, L., Hagberg, A., et al.: Competing resonances in spatially forced pattern-forming systems. Phys. Rev. E 88, 032,917 (2013)

    CrossRef  Google Scholar 

  56. Mau, Y., Haim, L., Meron, E.: Reversing desertification as a spatial resonance problem. Phys. Rev. E 91, 012,903 (2015)

    CrossRef  Google Scholar 

  57. Meron, E.: Modeling dryland landscapes. Math. Model. Nat. Phenom. 6, 163–187 (2011)

    MathSciNet  CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  59. Meron, E.: Nonlinear Physics of Ecosystems. CRC Press, Taylor & Francis Group, Boca Raton (2015)

    Google Scholar 

  60. Meron, E.: Pattern formation – a missing link in the study of ecosystem response to environmental changes. Math. Biosci. 271, 1–18 (2016)

    MathSciNet  MATH  CrossRef  Google Scholar 

  61. Mimura, M., Tohma, M.: Dynamic coexistence in a three-species competition–diffusion system. Ecol. Complex. 21, 215–232 (2015)

    CrossRef  Google Scholar 

  62. Petraitis, P.: Multiple Stable States in Natural Ecosystems. Oxford University Press, Oxford (2013)

    CrossRef  Google Scholar 

  63. Pismen, L.: Patterns and Interfaces in Dissipative Dynamics. Springer Series in Synergetics. Springer, Berlin (2006)

    Google Scholar 

  64. Pomeau, Y.: Front motion, metastability and subcritical bifurcations in hydrodynamics. Phys. D 23, 3 (1986)

    CrossRef  Google Scholar 

  65. Ravi, S., Wang, L., Kaseke, K.F., et al.: Ecohydrological interactions within “fairy circles” in the Namib Desert: revisiting the self-organization hypothesis. J. Geophys. Res. Biogeosci. 122(2), 405–414 (2017)

    CrossRef  Google Scholar 

  66. Reynolds, J.F., Smith, D.M.S., Lambin, E.F., et al.: Global desertification: building a science for dryland development. Science 316(5826), 847–851 (2007)

    CrossRef  Google Scholar 

  67. Rietkerk, M., van de Koppel, J.: Regular pattern formation in real ecosystems. Trends Ecol. Evol. 23(3), 169–175 (2008)

    CrossRef  Google Scholar 

  68. Rietkerk, M., Boerlijst, M.C., van Langevelde, F., et al.: Self-organization of vegetation in arid ecosystems. Am. Nat. 160, 524–530 (2002)

    CrossRef  Google Scholar 

  69. 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)

    CrossRef  Google Scholar 

  70. Scheffer, M., Bascompte, J., Brock, W.A., et al.: Early-warning signals for critical transitions. Nature 461, 387–393 (2009)

    CrossRef  Google Scholar 

  71. Scheffer, M., Carpenter, S.R.: Catastrophic regime shifts in ecosystems: linking theory to observation. Trends Ecol. Evol. 18, 648–656 (2003)

    CrossRef  Google Scholar 

  72. Scheffer, M., Carpenter, S., Foley, J.A., et al.: Catastrophic shifts in ecosystems. Nature 413, 591–596 (2001)

    CrossRef  Google Scholar 

  73. Sherratt, J.A.: An analysis of vegetation stripe formation in semi-arid landscapes. J. Math. Biol. 51(2), 183–197 (2005). https://doi.org/10.1007/s00285-005-0319-5

    MathSciNet  MATH  CrossRef  Google Scholar 

  74. Sherratt, J.A.: Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments, V: the transition from patterns to desert. SIAM J. Appl. Math. 73, 1347–1367 (2013)

    MATH  Google Scholar 

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

    MathSciNet  MATH  CrossRef  Google Scholar 

  76. Sherratt, J.A., Synodinos, A.D.: Vegetation patterns and desertification waves in semi-arid environments: mathematical models based on local facilitation in plants. Discrete Contin. Dynam. Systems B 17(8), 2815–2827 (2012)

    MathSciNet  MATH  CrossRef  Google Scholar 

  77. Shnerb, N.M., Sarah, P., Lavee, H., et al.: Reactive glass and vegetation patterns. Phys. Rev. Lett. 90, 0381011 (2003)

    CrossRef  Google Scholar 

  78. Siero, E., Doelman, A., Eppinga, M.B., et al.: Striped pattern selection by advective reaction-diffusion systems: resilience of banded vegetation on slopes. Chaos 25(3), 036411 (2015)

    MathSciNet  MATH  CrossRef  Google Scholar 

  79. Siteur, K., Siero, E., Eppinga, M.B., et al.: Beyond Turing: The response of patterned ecosystems to environmental change. Ecol. Complex. 20(0), 81–96 (2014)

    CrossRef  Google Scholar 

  80. Siteur, K., Eppinga, M.B., Doelman, A., et al.: Ecosystems off track: rate-induced critical transitions in ecological models. Oikos 125, 1689–1699 (2016)

    CrossRef  Google Scholar 

  81. Stone, L., Weisburd, R.S.J.: Positive feedback in aquatic ecosystems. Trends Ecol. Evol. 7, 263–267 (2016)

    CrossRef  Google Scholar 

  82. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2001)

    MATH  Google Scholar 

  83. Tarnita, C., Bonachela, J.A., Sheffer, E., et al.: A theoretical foundation for multi-scale regular vegetation patterns. Nature 541, 398–401 (2017)

    CrossRef  Google Scholar 

  84. Tongway, D.J., Valentin, C., Seghieri, J., et al. (eds.): Banded Vegetation Patterning in Arid and Semiarid Environments: Ecological Processes and Consequences for Management. Ecological Studies, vol. 149. Springer, Basel (2001)

    Google Scholar 

  85. Tschinkel, W.: The life cycle and life span of Namibian fairy circles. PloS One 7(6), e38056 (2012)

    CrossRef  Google Scholar 

  86. Valentine, C., d’Herbes, J., Poesen, J.: Soil and water components of banded vegetation patterns. Catena 37, 1–24 (1999)

    CrossRef  Google Scholar 

  87. van der Stelt, S., Doelman, A., Hek, G.M., et al.: Rise and fall of periodic patterns for a generalized Klausmeier–Gray–Scott model. J. Nonlinear Sci. 23, 39–95 (2013)

    MathSciNet  MATH  CrossRef  Google Scholar 

  88. Vohland, K., Barry, B.: A review of in situ rainwater harvesting (RWH) practices modifying landscape functions in African drylands. Agric. Ecosyst. Environ. 131, 119–127 (2009)

    CrossRef  Google Scholar 

  89. von Hardenberg, J., Meron, E., Shachak, M., et al.: Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 89, 198101 (2001)

    CrossRef  Google Scholar 

  90. Wiegand, T., Kissling, W.D., Cipriotti, P.A., et al.: Extending point pattern analysis for objects of finite size and irregular shape. J. Ecol. 94(4), 825–837 (2006)

    CrossRef  Google Scholar 

  91. Zelnik, Y.R., Meron, E.: Regime shifts by front dynamics. Ecol. Indic. 94, 544–552 (2018). https://doi.org/10.1016/j.ecolind.2017.10.068

    CrossRef  Google Scholar 

  92. Zelnik, Y.R., Kinast, S., Yizhaq, H., et al.: Regime shifts in models of dryland vegetation. Philos. Trans. R. Soc. A 371, 20120358 (2013)

    MathSciNet  MATH  CrossRef  Google Scholar 

  93. Zelnik, Y.R., Meron, E., Bel, G.: Gradual regime shifts in fairy circles. Proc. Natl. Acad. Sci. 112, 12,327–12,331 (2015)

    Google Scholar 

  94. Zelnik, Y.R., Meron, E., Bel, G.: Localized states qualitatively change the response of ecosystems to varying conditions and local disturbances. Ecol. Complex. 25, 26–34 (2016)

    CrossRef  Google Scholar 

  95. Zelnik, Y.R., Uecker, H., Feudel, U., et al.: Desertification by front propagation? J. Theor. Biol. 418, 27–35 (2017)

    MathSciNet  CrossRef  Google Scholar 

  96. Zelnik, Y.R., Gandhi, P., Knobloch, E., et al.: Implications of tristability in pattern-forming ecosystems. Chaos Interdiscip. J. Nonlinear Sci. 28(3), 033609 (2018)

    MathSciNet  CrossRef  Google Scholar 

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Acknowledgements

Some of the results described here have been reported in earlier publications with additional colleagues, including Golan Bel, Stephan Getzin, Aric Hagberg, Lev Haim, Omer Tzuk, and Hezi Yizhaq. We gratefully acknowledge their contributions. The research leading to the results described in this chapter received funding from the Israel Science Foundation Grant 305/13.

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Authors and Affiliations

  1. Blaustein Institutes for Desert Research and Physics Department, Ben-Gurion University of the Negev, Beersheba, Israel

    Ehud Meron

  2. Department of Soil and Water Sciences, Robert H. Smith Faculty of Agriculture, Food and Environment, The Hebrew University of Jerusalem, Rehovot, Israel

    Yair Mau

  3. Centre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS, Moulis, France

    Yuval R. Zelnik

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  1. Ehud Meron
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  1. Mathematics and Statistics, Georgetown University, Washington, DC, USA

    Hans G. Kaper

  2. DIMACS Center, Rutgers University, Piscataway, NJ, USA

    Fred S. Roberts

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Meron, E., Mau, Y., Zelnik, Y.R. (2019). Multistability in Ecosystems: Concerns and Opportunities for Ecosystem Function in Variable Environments. In: Kaper, H., Roberts, F. (eds) Mathematics of Planet Earth. Mathematics of Planet Earth, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-22044-0_7

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