Robustness of early warning signals of regime shifts in time-delayed ecological models

Abstract

Various ecological and other complex dynamical systems may exhibit abrupt regime shifts or critical transitions, wherein they reorganize from one stable state to another over relatively short time scales. Because of potential losses to ecosystem services, forecasting such unexpected shifts would be valuable. Using mathematical models of regime shifts, ecologists have proposed various early warning signals of imminent shifts. However, their generality and applicability to real ecosystems remain unclear because these mathematical models are considered too simplistic. Here, we investigate the robustness of recently proposed early warning signals of regime shifts in two well-studied ecological models, but with the inclusion of time-delayed processes. We find that the average variance may either increase or decrease prior to a regime shift and, thus, may not be a robust leading indicator in time-delayed ecological systems. In contrast, changing average skewness, increasing autocorrelation at short time lags, and reddening power spectra of time series of the ecological state variable all show trends consistent with those of models with no time delays. Our results provide insights into the robustness of early warning signals of regime shifts in a broader class of ecological systems.

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References

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

    Article  Google Scholar 

  2. Biggs R, Carpenter SR, Brock WA (2009) Turning back from the brink: detecting an impending regime shift in time to avert it. Proc Natl Acad Sci 106:826–831

    CAS  PubMed Central  PubMed  Article  Google Scholar 

  3. Boerlijst MC, Oudman T, de Roos AM (2013) Catastrophic collapse can occur without early warning: examples of silent catastrophes in structured ecological models. PloS ONE 8:e62033

    CAS  PubMed Central  PubMed  Article  Google Scholar 

  4. Boettiger C, Hastings A (2012a) Early warning signals and the prosecutor’s fallacy. Proc R Soc B Biol Sci 279:4734–4739

    Article  Google Scholar 

  5. Boettiger C, Hastings A (2012b) Quantifying limits to detection of early warning for critical transitions. J R Soc Interface 9:2527–2539

    Article  Google Scholar 

  6. Boettiger C, Hastings A (2013) Tipping points: from patterns to predictions. Nature 493:157–158

    CAS  PubMed  Google Scholar 

  7. Brock W, Carpenter S (2010) Interacting regime shifts in ecosystems: implication for early warnings. Ecol Monogr 80:353–367

    Article  Google Scholar 

  8. Carpenter SR (2005) Eutrophication of aquatic ecosystems: bistability and soil phosphorus. Proc Natl Acad Sci 102:10002–10005

    CAS  PubMed Central  PubMed  Article  Google Scholar 

  9. Carpenter SR, Brock WA (2006) Rising variance: a leading indicator of ecological transition. Ecol Lett 9:308–315

    Google Scholar 

  10. Carpenter S, Cole J, Pace M, Batt R, Brock W, Cline T, Coloso J, Hodgson J, Kitchell J, Seekell D et al (2011) Early warnings of regime shifts: a whole-ecosystem experiment. Science 332:1079–1082

    CAS  PubMed  Article  Google Scholar 

  11. Cushing J (1977) Time delays in single species growth models. J Math Biol 4:257–264

    CAS  PubMed  Article  Google Scholar 

  12. Dai L, Vorselen D, Korolev KS, Gore J (2012) Generic indicators for loss of resilience before a tipping point leading to population collapse. Science 336:1175–1177

    CAS  PubMed  Article  Google Scholar 

  13. Dakos V, Scheffer M, Van Nes EH, Brovkin V, Petoukhov V, Held H (2008) Slowing down as an early warning signal for abrupt climate change. Proc Natl Acad Sci 105:14308–14312

    CAS  PubMed Central  PubMed  Article  Google Scholar 

  14. Dakos V, Kéfi S, Rietkerk M, van Nes EH, Scheffer M (2011) Slowing down in spatially patterned ecosystems at the brink of collapse. Am Nat 177:E153–E166

    PubMed  Article  Google Scholar 

  15. Dakos V, Carpenter SR, Brock WA, Ellison AM, Guttal V, Ives AR, Kéfi S, Livina V, Seekell DA, van Nes EH et al (2012a) Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data. PloS ONE 7:e41010

    CAS  Article  Google Scholar 

  16. Dakos V, van Nes EH, D’Odorico P, Scheffer M (2012b) Robustness of variance and autocorrelation as indicators of critical slowing down. Ecology 93:264–271

    Article  Google Scholar 

  17. Drake JM, Griffen BD (2010) Early warning signals of extinction in deteriorating environments. Nature 467:456–459

    CAS  PubMed  Article  Google Scholar 

  18. Gardiner CW (2003) Handbook of stochastic methods for physics, chemistry and the natural sciences, 3rd edn. Springer, New York

    Google Scholar 

  19. Gopalsamy K (1992) Stability and oscillations in delay differential equations of population dynamics. Springer, Dordrecht

    Google Scholar 

  20. Gurney W, Blythe S, Nisbet R (1980) Nicholson’s blowflies revisited. Nature 287:17–21

    Article  Google Scholar 

  21. Guttal V, Jayaprakash C (2007) Impact of noise on bistable ecological systems. Ecol Model 201:420–428

    Article  Google Scholar 

  22. Guttal V, Jayaprakash C (2008) Changing skewness: an early warning signal of regime shifts in ecological systems. Ecol Lett 11:450–460

    PubMed  Article  Google Scholar 

  23. Guttal V, Jayaprakash C (2009) Spatial variance and spatial skewness: leading indicators of regime shifts in spatial ecological systems. Theor Ecol 2:3–12

    Article  Google Scholar 

  24. Hastings A, Wysham DB (2010) Regime shifts in ecological systems can occur with no warning. Ecol Lett 13:464–472

    PubMed  Article  Google Scholar 

  25. Held H, Kleinen T (2004) Detection of climate system bifurcations by degenerate fingerprinting. Geophys Res Lett 31:L020972

    Article  Google Scholar 

  26. Holling CS (1973) Resilience and stability of ecological systems. Annu Rev Ecol Evol Syst 4:1–23

    Article  Google Scholar 

  27. Horsthemke W, Lefever R (1984) Noise-induced transitions. Springer, New York

    Google Scholar 

  28. Hutchinson GE (1948) Circular causal systems in ecology. Ann N Y Acad Sci 50:221–246

    CAS  PubMed  Article  Google Scholar 

  29. Kéfi S, Dakos V, Scheffer M, van Nes EH, Rietkerk M (2012) Early warning signals also precede non-catastrophic transitions. Oikos 122:641–648

    Article  Google Scholar 

  30. Kleinen T, Held H, Petschel-Held G (2003) The potential role of spectral properties in detecting thresholds in the Earth system: application to the thermohaline circulation. Ocean Dyn 53:53–63

    Article  Google Scholar 

  31. Kuang Y (1993) Delay differential equations: with applications in population dynamics, vol 191. Academic, San Diego

    Google Scholar 

  32. Lenton T, Livina V, Dakos V, van Nes E, Scheffer M (2012) Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness. Philos Trans R Soc A Math Phys Eng Sci 370:1185–1204

    CAS  Article  Google Scholar 

  33. Ludwig D, Jones DD, Holling CS (1978) Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J Anim Ecol 47:315–332

    Article  Google Scholar 

  34. Ma SK (1976) Modern theory of critical phenomena. Benjamin/Cummings, Reading

    Google Scholar 

  35. May RM (1973) Time-delay versus stability in population models with two and three trophic levels. Ecology 54:315–325

    Article  Google Scholar 

  36. May RM (1977) Thresholds and breakpoints in ecosystems with a multiplicity of stable states. Nature 269:471–477

    Article  Google Scholar 

  37. Noy-Meir I (1975) Stability of grazing systems: an application of predator-prey graphs. J Ecol 63:459–482

    Article  Google Scholar 

  38. Risken H (1984) The Fokker-Planck equation: methods of solution and applications. Springer, New York

    Google Scholar 

  39. Scheffer M, Carpenter SR (2003) Catastrophic regime shifts in ecosystems: linking theory and observation. Trends Ecol Evol 18:648–656

    Article  Google Scholar 

  40. Scheffer M, Carpenter SR, Foley JA, Folke C, Walker B (2001) Catastrophic shifts in ecosystems. Nature 413:591–596

    CAS  PubMed  Article  Google Scholar 

  41. Scheffer M, Bascompte J, Brock W, Brovkin V, Carpenter S, Dakos V, Held H, van Nes E, Rietkerk M, Sugihara G (2009) Early-warning signals for critical transitions. Nature 461:53–59

    CAS  PubMed  Article  Google Scholar 

  42. Scheffer M, Carpenter SR, Lenton TM, Bascompte J, Brock W, Dakos V, van de Koppel J, van de Leemput IA, Levin SA, van Nes EH et al (2012) Anticipating critical transitions. Science 338:344–348

    CAS  PubMed  Article  Google Scholar 

  43. Steele JH, Henderson EW (1984) Modelling long-term fluctuations in fish stocks. Science 224:985–987

    CAS  PubMed  Article  Google Scholar 

  44. Strogatz S (1994) Nonlinear dynamics and chaos. Westview, Boulder

    Google Scholar 

  45. Turchin P, Taylor AD (1992) Complex dynamics in ecological time series. Ecology 73:289–305

    Article  Google Scholar 

  46. van de Koppel J, Rietkerk M, Weissing FJ (1997) Catastrophic vegetation shifts and soil degradation in terrestrial grazing systems. Trends Ecol Evol 12:352–356

    PubMed  Article  Google Scholar 

  47. van Nes EH, Scheffer M (2007) Slow recovery from perurbations as a generic indicator of a nearby catastrophic regime shift. Am Nat 169:738–747

    PubMed  Article  Google Scholar 

  48. Veraat AJ, Faassen EJ, Dakos V, van Nes EH, Lurling M, Scheffer M (2012) Recovery rates reflect distance to a tipping point in a living system. Nature 481:357–359

    Google Scholar 

  49. Wang R, Dearing JA, Langdon PG, Zhang E, Yang X, Dakos V, Scheffer M (2012) Flickering gives early warning signals of a critical transition to a eutrophic lake state. Nature 492:419–422

    PubMed  Article  Google Scholar 

  50. Wissel C (1984) A universal law of the characteristic return time near thresholds. Oecologia 65:101–107

    Article  Google Scholar 

  51. Zeng C, Wang H (2012) Noise and large time delay: Accelerated catastrophic regime shifts in ecosystems. Ecol Model 233:52–58

    Article  Google Scholar 

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Acknowledgments

V.G. is supported by a Ramalingaswamy Fellowship from the Department of Biotechnology, Government of India and the Ministry of Environment and Forests, Government of India.

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Correspondence to Vishwesha Guttal.

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All authors contributed equally to this manuscript and the names are listed in alphabetical order.

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Guttal, V., Jayaprakash, C. & Tabbaa, O.P. Robustness of early warning signals of regime shifts in time-delayed ecological models. Theor Ecol 6, 271–283 (2013). https://doi.org/10.1007/s12080-013-0194-4

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Keywords

  • Regime shifts
  • Critical transitions
  • Alternative stable states
  • Resilience
  • Delay differential equations
  • Early warning signals
  • Variance
  • Skewness
  • Autocorrelation
  • Power spectra