Skip to main content Accessibility help
Hostname: page-component-6c8bd87754-clkrv Total loading time: 0.287 Render date: 2022-01-18T22:07:27.631Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Patterned vegetation, tipping points, and the rate of climate change

Published online by Cambridge University Press:  23 June 2015

Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, USA
Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada email:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada email:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada email:


When faced with slowly depleting resources (such as decrease in precipitation due to climate change), complex ecological systems are prone to sudden irreversible changes (such as desertification) as the resource level dips below a tipping point of the system. A possible coping mechanism is the formation of spatial patterns, which allows for concentration of sparse resources and the survival of the species within “ecological niches” even below the tipping point of the homogeneous vegetation state. However, if the change in resource availability is too sudden, the system may not have time to transition to the patterned state and will pass through the tipping point instead, leading to extinction. We argue that the deciding factors are the speed of resource depletion and the amount of the background noise (seasonal climate changes) in the system. We illustrate this phenomenon on a model of patterned vegetation. Our analysis underscores the importance of, and the interplay between, the speed of climate change, heterogeneity of the environment, and the amount of seasonal variability.

Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Ashwin, P., Wieczorek, S., Vitolo, R. & Cox, P. (2012) Tipping points in open systems: Bifurcation, noise-induced and rate-dependent examples in the climate system. Phil. Trans. R. Soc. A: Math. Phys. Eng. Sci. 370 (1962), 11661184.CrossRefGoogle ScholarPubMed
[2]Baer, S. M., Erneux, T. & Rinzel, J. (1989) The slow passage through a hopf bifurcation: Delay, memory effects, and resonance. SIAM J. Appl. Math. 49 (1), 5571.CrossRefGoogle Scholar
[3]Bender, C. M. & Orszag, S. A. (1999) Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Vol. 1, Springer.CrossRefGoogle Scholar
[4]Berglund, N. & Gentz, B. (2002) Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Relat. Fields 122 (3), 341388.CrossRefGoogle Scholar
[5]Boer, M. & Puigdefábregas, J. (2005) Effects of spatially structured vegetation patterns on hillslope erosion in a semiarid mediterranean environment: A simulation study. Earth Surf. Process. Landf. 30 (2), 149167.CrossRefGoogle Scholar
[6]Breña-Medina, V. F., Avitabile, D., Champneys, A. R., Grierson, C. & Ward, M. J. (2014) Mathematical modeling of plant root hair initiation: Dynamics of localized patches. SIAM J. Appl. Dyn. Syst. 13 (1), 210248.CrossRefGoogle Scholar
[7]Carpenter, S. R., Cole, J. J., Pace, M. L., Batt, R., Brock, W. A., Cline, T., Coloso, J., Hodgson, J. R., Kitchell, J. F., Seekell, D. al. (2011) Early warnings of regime shifts: a whole-ecosystem experiment. Science 332 (6033), 10791082.CrossRefGoogle ScholarPubMed
[8]Dai, A., Trenberth, K. E. & Qian, T. (2004) A global dataset of palmer drought severity index for 1870-2002: Relationship with soil moisture and effects of surface warming. J. Hydrometeorology 5 (6), 11171130.CrossRefGoogle Scholar
[9]Dakos, V., Scheffer, M., van Nes, E. H., Brovkin, V., Petoukhov, V. & Held, H. (2008) Slowing down as an early warning signal for abrupt climate change. Proc. Natl. Acad. Sci. 105 (38), 1430814312.CrossRefGoogle ScholarPubMed
[10]D'Odorico, P. & Bhattachan, A. (2012) Hydrologic variability in dryland regions: impacts on ecosystem dynamics and food security. Phil. Trans. R. Soc. B: Biolog. Sci. 367 (1606), 31453157.CrossRefGoogle ScholarPubMed
[11]D'Odorico, P., Laio, F. & Ridolfi, L. (2005) Noise-induced stability in dryland plant ecosystems. Proc. Natl. Acad. Sci. USA 102 (31), 1081910822.CrossRefGoogle ScholarPubMed
[12]Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B. & Wang, X. Auto 97: Continuation and bifurcation software for ordinary differential equations (with homcont).Google Scholar
[13]Field, C. B., Barros, V., Stocker, T. F. & Dahe, Q. (2012) Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation: Special Report of the Intergovernmental Panel on Climate Change, Cambridge University Press.CrossRefGoogle Scholar
[14]Foden, W., Midgley, G. F., Hughes, G., Bond, W. J., Thuiller, W., Timm Hoffman, M., Kaleme, P., Underhill, L. G., Rebelo, A. & Hannah, L. (2007) A changing climate is eroding the geographical range of the namib desert tree aloe through population declines and dispersal lags. Diversity Distributions 13 (5), 645653.CrossRefGoogle Scholar
[15]Gowda, K. & Kuehn, C. (2015) Early-warning signs for pattern-formation in stochastic partial differential equations. Commun. Nonlinear. Sci., 22 (1), 5569.CrossRefGoogle Scholar
[16]Guttal, V. & Jayaprakash, C. (2008) Changing skewness: An early warning signal of regime shifts in ecosystems. Ecology Lett. 11 (5), 450460.CrossRefGoogle ScholarPubMed
[17]Hairer, M., Ryser, M. D. & Weber, H. (2012) Triviality of the 2d stochastic allen-cahn equation. Electron. J. Probab. 17 (39), 114.CrossRefGoogle Scholar
[18]Hasselmann, K. (1982) An ocean model for climate variability studies. Progr. Oceanogr. 11 (2), 6992.CrossRefGoogle Scholar
[19]Hoegh-Guldberg, O., Mumby, P. J., Hooten, A. J., Steneck, R. S., Greenfield, P., Gomez, E., Harvell, C. D., Sale, P. F., Edwards, A. J., Caldeira, al. (2007) Coral reefs under rapid climate change and ocean acidification. Science 318 (5857), 17371742.CrossRefGoogle ScholarPubMed
[20]Hofmann, M. & Schellnhuber, H.-J. (2009) Oceanic acidification affects marine carbon pump and triggers extended marine oxygen holes. Proceedings of the National Academy of Sciences 106 (9), 30173022.CrossRefGoogle ScholarPubMed
[21]Holmes, M. H. (2012) Introduction to Perturbation Methods, Vol. 20, Springer Science & Business Media.Google Scholar
[22]Hughes, T. P., Baird, A. H., Bellwood, D. R., Card, M., Connolly, S. R., Folke, C., Grosberg, R., Hoegh-Guldberg, O., Jackson, J. B. C., Kleypas, al. (2003) Climate change, human impacts, and the resilience of coral reefs. Science 301 (5635), 929933.CrossRefGoogle ScholarPubMed
[23]Jansons, K. M. & Lythe, G. D. (1998) Stochastic calculus: Application to dynamic bifurcations and threshold crossings. J. Stat. Phys. 90 (1–2), 227251.CrossRefGoogle Scholar
[24]Kéfi, S., Rietkerk, M., Alados, C. L., Pueyo, Y., Papanastasis, V. P., ElAich, A. & De Ruiter, P. C. (2007) Spatial vegetation patterns and imminent desertification in mediterranean arid ecosystems. Nature 449 (7159), 213217.CrossRefGoogle ScholarPubMed
[25]Klausmeier, C. A. (1999) Regular and irregular patterns in semiarid vegetation. Science 284 (5421), 18261828.CrossRefGoogle ScholarPubMed
[26]Kletter, A. Y., Von Hardenberg, J., Meron, E. & Provenzale, A. (2009) Patterned vegetation and rainfall intermittency. J. Theor. Biol. 256 (4), 574583.CrossRefGoogle ScholarPubMed
[27]Kolokolnikov, T., Ward, M. J. & Wei, J. (2005) The existence and stability of spike equilibria in the one-dimensional gray–scott model: The low feed-rate regime. Stud. Appl. Math. 115 (1), 2171.CrossRefGoogle Scholar
[28]Kuehn, C. (2011) A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics. Physica D: Nonlinear Phenom. 240 (12), 10201035.CrossRefGoogle Scholar
[29]Kuske, R. (1999) Probability densities for noisy delay bifurcations. J. Stat. Phys. 96 (3–4), 797816.CrossRefGoogle Scholar
[30]Kuske, R. & Baer, S. M. (2002) Asymptotic analysis of noise sensitivity in a neuronal burster. Bull. Math. Biol. 64 (3), 447481.CrossRefGoogle Scholar
[31]Lejeune, O, Tlidi, M. & Couteron, P. (2002) Localized vegetation patches: A self-organized response to resource scarcity. Phys. Rev. E 66 (1), 010901.CrossRefGoogle ScholarPubMed
[32]Lenton, T. M., Held, H., Kriegler, E., Hall, J. W., Lucht, W., Rahmstorf, S. & Joachim Schellnhuber, H. (2008) Tipping elements in the earth's climate system. Proc. Natl. Acad. Sci. 105 (6), 17861793.CrossRefGoogle ScholarPubMed
[33]Lindner, B., Garcia-Ojalvo, J., Neiman, A. & Schimansky-Geier, L. (2004) Effects of noise in excitable systems. Phys. Rep. 392 (6), 321424.CrossRefGoogle Scholar
[34]Mandel, P. & Erneux, T. (1987) The slow passage through a steady bifurcation: Delay and memory effects. J. Stat. Phys. 48 (5–6), 10591070.CrossRefGoogle Scholar
[35]May, R. M. (1977) Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269 (5628), 471477.CrossRefGoogle Scholar
[36]Meron, E., Gilad, E., von Hardenberg, J., Shachak, M. & Zarmi, Y. (2004) Vegetation patterns along a rainfall gradient. Chaos, Solitons Fractals 19 (2), 367376.CrossRefGoogle Scholar
[37]Meron, E., Yizhaq, H. & Gilad, E. (2007) Localized structures in dryland vegetation: Forms and functions. Chaos: Interdiscip. Nonlinear Sci. 17 (3), 037109.CrossRefGoogle ScholarPubMed
[38]Overpeck, J. T. & Cole, J. E. (2006) Abrupt change in earth's climate system. Annu. Rev. Environ. Resour. 31, 131.CrossRefGoogle Scholar
[39]Pearson, P. N. & Palmer, M. R. (2000) Atmospheric carbon dioxide concentrations over the past 60 million years. Nature 406 (6797), 695699.CrossRefGoogle ScholarPubMed
[40]Petit, J.-R., Jouzel, J., Raynaud, D., Barkov, N. I., Barnola, J.-M., Basile, I., Bender, M., Chappellaz, J., Davis, M., Delaygue, al. (1999) Climate and atmospheric history of the past 420,000 years from the vostok ice core, antarctica. Nature 399 (6735), 429436.CrossRefGoogle Scholar
[41]Praetorius, S. K. & Mix, A. C. (2014) Synchronization of north pacific and greenland climates preceded abrupt deglacial warming. Science 345 (6195), 444448.CrossRefGoogle ScholarPubMed
[42]Rietkerk, M., Dekker, S. C., de Ruiter, P. C. & van de Koppel, J. (2004) Self-organized patchiness and catastrophic shifts in ecosystems. Science 305 (5692), 19261929.CrossRefGoogle ScholarPubMed
[43]Ryser, M. D., Nigam, N. & Tupper, P. F. (2012) On the well-posedness of the stochastic allen–cahn equation in two dimensions. Journal of Computational Physics 231 (6), 25372550.CrossRefGoogle Scholar
[44]Scheffer, M., Bascompte, J., Brock, W. A., Brovkin, V., Carpenter, S. R., Dakos, V., Held, H., van Nes, E. H., Rietkerk, M. & Sugihara, G. (2009) Early-warning signals for critical transitions. Nature 461 (7260), 5359.CrossRefGoogle ScholarPubMed
[45]Scheffer, M., Carpenter, S., Foley, J. A., Folke, C. & Walker, B. (2001) Catastrophic shifts in ecosystems. Nature 413 (6856), 591596.CrossRefGoogle ScholarPubMed
[46]Schellnhuber, H. al. (2013) Turn down the heat: climate extremes, regional impacts, and the case for resilience. International Bank for Reconstruction and Development, World Bank, pp. xii + 213 pp.Google Scholar
[47]Shardlow, T. (2000) Stochastic perturbations of the allen–cahn equation. Electron. J. Differ. Equ. 2000 (47), 119.Google Scholar
[48]Sherratt, J. A. (2013) History-dependent patterns of whole ecosystems. Ecological Complexity 14, 820.CrossRefGoogle Scholar
[49]Sherratt, J. A. & Lord, G. J. (2007) Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. Theor. Population Biol. 71 (1), 111.CrossRefGoogle Scholar
[50]Shisanya, C. A., Recha, C., Anyamba, al. (2011) Rainfall variability and its impact on normalized difference vegetation index in arid and semi-arid lands of kenya. Int. J. Geosci. 2 (01), 36.CrossRefGoogle Scholar
[51]Siteur, K., Siero, E., Eppinga, M. B., Rademacher, J. D. M., Doelman, A. & Rietkerk, M. (2014) Beyond turing: The response of patterned ecosystems to environmental change. Ecological Complexity 20, 8196.CrossRefGoogle Scholar
[52]Solomon, al. (2007) Intergovernmental panel on climate change, group I. Intergovernmental Panel on Climate Change, Group I., Cambridge University Press.Google Scholar
[53]Stocks, N. G., Mannella, R. & McClintock, P. V. E. (1989) Influence of random fluctuations on delayed bifurcations: The case of additive white noise. Phys. Rev. A 40 (9), 5361.CrossRefGoogle ScholarPubMed
[54]Sutera, A. (1981) On stochastic perturbation and long-term climate behaviour. Q. J. R. Meteorol. Soc. 107 (451), 137151.CrossRefGoogle Scholar
[55]Michael, J., Thompson, T. & Sieber, J. (2011) Climate tipping as a noisy bifurcation: A predictive technique. IMA J. Appl. Math. 76 (1), 2746.Google Scholar
[56]Tierney, J. E. & deMenocal, P. B. (2013) Abrupt shifts in horn of africa hydroclimate since the last glacial maximum. Science 342 (6160), 843846.CrossRefGoogle ScholarPubMed
[57]Torrent, M. C. & San Miguel, M. (1988) Stochastic-dynamics characterization of delayed laser threshold instability with swept control parameter. Phys. Rev. A 38 (1), 245.CrossRefGoogle ScholarPubMed
[58]Travis, J. M. J. (2003) Climate change and habitat destruction: A deadly anthropogenic cocktail. Proc. R. Soc. Lond. Ser. B: Biol. Sci. 270 (1514), 467473.CrossRefGoogle ScholarPubMed
[59]Trenberth, K. E. (2011) Changes in precipitation with climate change. Clim. Res. 47 (1), 123.CrossRefGoogle Scholar
[60]Turley, C., Blackford, J. C., Widdicombe, S., Lowe, D., Nightingale, P. D. & Rees, A. P. (2006) Reviewing the impact of increased atmospheric co2 on oceanic ph and the marine ecosystem. Avoiding Dangerous Clim. Change 8, 6570.Google Scholar
[61]Tzou, J. C., Ward, M. J. & Kolokolnikov, T. (2015) Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction–diffusion systems. Physica D: Nonlinear Phenom. 290, 2443.CrossRefGoogle Scholar
[62]van der Stelt, S., Doelman, A., Hek, G. & Rademacher, J. D. M. (2013) Rise and fall of periodic patterns for a generalized klausmeier–gray–scott model. J. Nonlinear Sci. 23 (1), 3995.CrossRefGoogle Scholar
[63]Von Hardenberg, J., Meron, E., Shachak, M. & Zarmi, Y. (2001) Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87 (19), 198101.CrossRefGoogle ScholarPubMed
[64]Zeghlache, H., Mandel, P. & Van den Broeck, C. (1989) Influence of noise on delayed bifurcations. Phys. Rev. A 40 (1), 286.CrossRefGoogle ScholarPubMed
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Patterned vegetation, tipping points, and the rate of climate change
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Patterned vegetation, tipping points, and the rate of climate change
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Patterned vegetation, tipping points, and the rate of climate change
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *