Meeting Review

Catastrophic Thresholds: Perspectives, Definitions, and Applications

Introduction

Theconceptofthresholdbehaviorofecosystemvariablesandparametersinspaceandtimereceivedearlytheoretical treatment in the 1970s by the “Upstart” C. S. Holling in a landmark paper on ecological resilience (Holling 1973). Recent treatments have been eerily mirrored in recent bestsellers, Connie Willis’s science fiction novel Bellwether (Willis 1998) and Malcom Gladwell’s Tipping Point: How Little Things Can Make a Big Difference (Gladwell 2002), and concepts presented by Peters et al. (2004) in the paper, “Cross-scale interactions, nonlinearities,andforecastingcatastrophicevents.”InWillis’sBellwethertheprotagonistinvestigatestheorigin,spread, and end of fads, such as pet rocks, mood rings, and miniature golf. Gladwell’s Tipping Point discussesthemechanisms involved in how new ideas, warnings (e.g., Paul Revere’s “the British are Coming!!”), disease (HIV-AIDS), and fads (Hush Puppies) are rapidly spread across society. Similarly, Peters et al. (2004) analyzed how natural phenomena such as disease, wildfire, erosion, and major dust storms spread across the environment. These publications,despitetheirdisparateaudiences,commonlydiscussthresholdsortippingpointsandconceptualmodelsthatcanbeusedtobetterunderstandthesephenomena.

A threshold can be defined by such synonyms as a border, a regime shift, an ecotone, a discontinuity, a phase transition, a point of criticality, and the tipping point between two or more dynamic regimes or states (Mayer and Rietkerk 2004). Adjectives that have been used with “threshold” include transition (Archer 1989), critical, and catastrophic (Scheffer et al. 2001). It is a phenomenon that can be observed in both time and space, but whose observationdependsonscaleandcriteria,e.g.,ecosystemorlandscape (Beisner et al. 2003).

Thresholds can be seen in settings as diverse as artwork, the economy, and in social movements/society.

Bi-stability and thresholds are illustrated in M. C. Escher ‘s 1956 wood engraving “Swans,” where flying black swans rapidly change into white swans in a chiral Möbius strip ‹http://www.mcescher.com/›.Themusicianandecologist Marten Scheffer, author of the forthcoming book Stability: the Mechanisms of Inertia, Chaos and Collapse (Scheffer 2007) showed bi-stability in the economy with the 1929, 19 October 1987, and the 1997 stock market crashes. With reference to social systems, this ESA conference was held near thesitewherethe“Upstart”andnow“Icon”theReverendDr.MartinLutherKing,Jr.wasassassinated.Hisaccomplishmentsanddeathweremajor tipping points in social justice and equality for all Americans. (Also see Gunderson and Holling [2002] and Turchin [2003, 2006] for quantitative treatment of social and ecological phenomena in general.) Further examples ofthresholdbehaviorcanbefoundintheThresholdsBibliographythatisarchivedbytheResilienceAlliance(RA), amultidisciplinaryresearchgroupthatexploresthedynamicsofcomplexadaptivesystems‹http://www.resalliance.org›.

A symposium organized and chaired by Robert A. Washington-Allen and Lucinda F. Salo for the 2006 ESA Annual Meeting at Memphis, Tennessee: “Icons and Upstarts in Ecology,” with support from the Rangeland Ecology Section of ESA.

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Symposium talks

The speakers in this symposium consider thresholds in space and time in both terrestrial and aquatic ecosystems (Table 1). Forecologicalsystems,R.Washington-Allendiscussedthresholdsinrangelands,andillustratedthesewithexamplesofchangesinlandscape-scalevegetationproduction,structure,andpatterninrelationtowetanddry climatic periods, and livestock and wildlife grazing management (Washington-Allen et al. 2006, Zimmermann et al. 2006). This research used power law analysis of vegetation patch dynamics (Olsen et al. 2005) and time-series analysis (the autocorrelation function) (Perry et at. 2000) to quantitatively detect shifts in dynamic regimes (states).

M. Scheffer showed similar evidence for bi-stability in Saharan vegetation cover, the 1980s collapse of

Caribbean coral reefs, North Sea plankton and benthos, State collapses (Sudan), and shallow aquatic systems. He went on to show other dramatic events (dinosaur extinction), but questioned whether these events reallypresented evidence of regimes shifts? Scheffer suggested that observing positive feedback (climate–vegetation feedbacks) is a necessary but not sufficient condition. He suggested observing system behavior using time series analysis for a sudden jump in time, multimodality, dual relationships (use of two functions instead of a single regression), divergence, triggered shifts (but this may be transient), and hysteresis (Scheffer and Carpenter 2003). These behaviors may also be necessary but not sufficient. He also suggested the use of early-warning signals for detection such as increased variance near a threshold (O’Neill et al. 1989), a “red shift” (spectral analysis), or a “critical slowdown” (broadening correlations near the threshold), but in terms of time and space, none of these patternsmaybeconclusive.Attheendofthedaythereisaneedforacombinationofmodelsandlaboratoryandfield experiments to get at the whole picture.

Three speakers examined scaling laws, length scales, and increased variance using models and field data

collection. C. Johnson and his colleagues are concerned with the definition of characteristic or natural length scales (CLS). Determination of CLS allows (1) the definition of the optimum scale for defining independent or complex interactions between different species, (2) monitoring, and (3) interpreting change in ecological dynamics.DeterminationofCLSisaccomplishedbymeasuringthechangesinthemagnitudeofthepredictionerror as the scale of observation of species abundance (e.g., percent cover) changes, using two error estimates of variance spectra called the error X (a scaled error variance) and the prediction r2.Innovationisintroducedbyshowingthattheuseofspace-for-timesubstitutionwithinalimitedtimeseriesofdata,ratherthanamoreoneroustime-series analysis (which requires thousands of realizations) produces the same results (Habeeb et al. 2005; C. Johnsonetal.,unpublished manuscript).

C.Allen observed that variance of a response variable increased near a threshold. He has observed thatdiscontinuous structures at different spatial scales may result in discontinuities between aggregated species’body mass patterns, reflecting the scales of structure available to animal communities within a landscape. This has led toa texturaldiscontinuityhypothesis,where it ispredicted thatvariability inpopulationabundance isgreater inanimalspeciesneartheedgeofbodymassaggregationsthanit is inspeciesintheinteriorofbodymassaggregations.Allenandhiscolleaguesgloballyexaminedspatialandtemporaldataonfaunapopulationabundances and confirmed this relationship. They further examined this relationship for bird populations in the Everglades, and again confirmed the relationship. This may be a predictor of invasions and nomadism.

B.MilneusedpowerlawanalysisonpreviousstudiesofU.S.basinstoobservethatparticularinputandoutputparameters exhibited 4/7 scaling. Consider that a basin, whose length scale (L) is proportional to its Area (A) to its fractal dimension (D) divided by 2 power or L∝AD/2 receives solar irradiance and precipitation (PPT) as inputs. PPT is lost as runoff (Q), evaporation (E), and transpiration (T) from plants; thus the basin can be considered an exchange surface (Λ). Consider that renormalization (increasing pixel size, thus coarsening resolution) of the

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Table 1. Speakers in the symposium “The Detection of Catastrophic Thresholds: Perspectives, Definitions, and Methods” (ESA Annual Meeting, August 2006), titles of their talks, and the methods they used to detect statesandthresholds.

Speaker Affiliation Title Methodofdetection

RobertA.Washington-AllenandLucindaSalo

UniversityofVirginiaandSageEcosystemScience,Boise,ID

Thedetectionofcatastrophicthresholds:perspectives, definitions, andmethods:anintroduction

Timeseriesanalysis:autocorrelationfunctionSelf-organized criticality /Power law analysis

MartenScheffer

WageningenUniversity,TheNetherlands

Regimeshiftsinaquaticandterrestrialecosystems

TimeseriesanalysisofvariabilityRedshiftCriticalslowdown

DaleandJef-frey Lock-wood

ColoradoStateUniversityandUniversityofWyoming

Grasshopperpopulationecology:catastrophe,criticality,andcritique

CatastrophetheorySelf-organized criticality (1/f noise)Fractalanalysis

Tamzen Stringham OregonState

University

Stateandtransitionmodeling definitions fromaecologicalprocessapproach

Cluster analysis for classification of statesInferentialstatistics

CraigJohn-son

UniversityofTasmania

Timeseriesprobesforthresholddetectionatdifferentscales

Space-for-timesubstitutionPredictedr2anderror-Xvariancespectraanalysis

CraigR.Allen Universityof

Nebraska

Cross-scalestructure,thresholds,andthegenerationofnoveltyindiscontinuouscomplexsystems

Spatialanalysisofvariance

DianaThomas MontclairState

University,NJ

Determiningthresholds:WestNilevirusandotherapplications

Timescalecalculus

BruceT.Milne Universityof

NewMexico

Coupleddynamicsofvegetationandterrain:pervasive 4/7 scaling in landscapes

RenormalizationPercolationtheoryTextureanalysisFractalanalysisPowerlawanalysis

David Briske TexasA&MUniversity,TX

Thedetectionofcatastrophicthresholds:perspectives, definitions, andmethods:paneldiscussionandsynthesis

N/A

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Columbia River Basin and other energy-limited areas scale as the 4/7 power of cell length, that actual ETvarieswith PPT to the 4/7 power in U.S. basins, and that plant cover is proportional to L to the 7/4 power. Consider thatgradientpercolation,anerosiveforcegradientfromlandtoshoretosea,produced−4/7 scaling, suggesting a universal statistical signature in the geometry of the terrain. Can this scaling be seen in basin boundaries? B. Milne tested 29 basins and found that they supported the hypothesis of 4/7 scaling. Thus basin geometry will affect interceptionofsolarradiationthatpowersevaporationandtranspirationthroughoutthebasin,ortheexchangesurface (Λ) is approximately equal to PPT × ETandtoA1+4/7 = (L2)11/7 = Lπ. So where do the 4/7 and the π scaling constraints that produced this nonlinear coupling come from? The 4/7 scaling is from the gradient, and π is from the cross-sectional area of sunbeams (which are usually circular), i.e., the solar irradiance (energy for transpiration and evaporation) input as scattered hemispherical radiance fills the basin within the fractal perimeter, subject to π. This is true because the fractal dimension of the basin perimeter evolves to satisfy the equality entailed by the solar disk. B. Milne concluded that ecohydrological theory is based on a foundation of coupled fluxes of water and energy. Universal to this foundation are basin boundaries (formed by erosion) that exhibit 4/7 scaling and a fractal geometrical limit on the interception of sunbeams. Thus the sustainability of ecosystem services of food, fiber, and watercanbeassessedfromtheviewpointofscalingtheory.

Two speakers examined the application of mathematical models to field data. The Lockwoods and D. Thomas are interested in the dynamics of grasshopper and mosquito population outbreaks, respectively, and the development of nonlinear tools to provide explanation and forecasting for management strategies. The Lockwoods have used both catastrophe theory (CT) and self-organized criticality (SOC) to examine outbreaks, and Thomas has used time-scale calculus (TSC, Hilger 1990). Catastrophe theory is a mathematical framework from topology that can be used to explain both discontinuous and continuous behavior in systems (Thom 1972, Gilmore 1981). Self-organized criticality is theemergenceofcomplexbehavior fromlocal interactionbetweenneighbors inasystem that is self-reinforcing or self-tuning, i.e., positive feedback at the local scale maintains a poised state betweenorderanddisorder.Self-organized criticalitycanbedetectedbyexaminationofscale-invarianceusingeither power law analysis or 1/f noise (Bak et al. 1987). Regime shifts are detected by lack of scale-invariance in apreviouslyself-organized system (Olsen et al. 2005). The Lockwoods fit the cusp catastrophe geometry to the state variable (grasshopper population size) and two driving parameters (bimonthly temperature and precipitation) inalinearregressionmannerusingthesoftwareGEMCATII (Olivia et al. 1987 ‹http://astro.temple.edu/~oliva/cat-theory.htm›). They were able to temporally and spatially predict grasshopper outbreaks over a 28-year period of reconstruction. They also found self-similarity by fractal analysis and 1/fnoiseinthefrequencydistributionof areas infested by grasshoppers or scale-invariance, a feature diagnostic of a self-organized criticality (SOC). Theyalso found, to their surprise, thatCTcuspbifurcationcanproduceSOCdynamics.Consider that this isconvergence between two different mathematical models where CT emphasizes strong exogenous controls and SOC strong endogenous processes. This brings into question whether mosquito population outbreaks and crashes are internally or externally controlled? It may be reasonable to consider that due to processes that drive evolution, the control parameters are not exogenous if they exert selective pressure on the state variables (adaptation being a feedback), as appears to be the case for El Niños (Gibbs and Grant 1987).

D. Thomas developed a simple time-scale calculus (TSC) model of mosquito, birds, and human population

dynamics to successfully direct the pesticide-spraying plan for New York City to battle the transmission of West Nile encephalitis (Thomas and Urena 2001). In the spring, mosquitoes contract West Nile virus from migrant birds,transmitittotheiroffspring,andinfectbothhumansandbirdsthroughoutthesummer.Infectedhumansandbirds become sick, die, or generally recover from the infection. Infected birds migrate to South America, and over-winteringmosquitolarvaeincubatethevirusinNorthAmerica.Thesetransmissiondynamicsareatime-varyingprocess that presents a special mathematical challenge where both discrete (seasonal breaks or sudden jumps) and continuous (day-by-day) changes occur. Generally, discrete events are modeled using difference equations, and continuous behavior is modeled using differential equations (calculus). However, time-scale calculus is a

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discovery of the relationship between difference and differential equations, and thus like catastrophe theory can be used to model systems that exhibit both continuous and discrete behavior (Hilger 1990, Bohner and Peterson 2001).

T.Stringhamappliestheconceptsofmultiplestablestatesandregimeshiftstothereal-worldchallengeofdescribing vegetation dynamics on rangelands. Previously, the ecological basis of land management modelsassumed linearchanges inplantcommunitycomposition thatwereprimarilyendogenouslydriven.However,federallandmanagementagencies,includingtheUSDA’sForestServiceandNRCS,andtheU.S.DepartmentoftheInterior’sBLM,haverecentlyadoptedthenonlinearStatesandTransitionmodelthatassumesstatesareexogenouslydrivenbyecosystemprocesses.Theseagenciesarecurrentlypreparingecologicalsiteordynamicregimedescriptionsthatillustratehomogenousvegetationandsoildynamicsatlocalmanagementscalesinthewestern United States. T. Stringham defines a “state” as a resilient complex of soil and vegetation that is connected byecologicalprocessessuchashydrology.Withinastate,plantcommunitydynamicscalled“phases”occur.A“communitypathway”isamechanismthatresultsinchangefromonephasetoanother.Similarlya“stressor”isa mechanism that degrades the soil–vegetation complex’s ecological processes enough to result in a “transition” orchangefromonestatetoanother.Sitesthatresistchangeexhibitresilience.Stringhamnotedthattheecologicalsite descriptions are being developed by interview with expert stakeholders who aid in identifying reference or control ecological sites. Plot-level vegetation and soil data are collected, classified into states by cluster analysis, and inferential statistics are used to detect significant change in ecosystem processes relative to controls; this is indicativeofathresholdbeingcrossed.

Conclusion

David Briske tied the symposium together by synthesizing the concepts presented that further contributed to our growing knowledge of complex systems in ecology. The talks used nonlinear concepts such as dynamic regimes, soil–vegetation complex dynamics, scaling laws, catastrophe theory, self-organization, time-scale calculus,andcross-scaleinteractions,i.e.,thetexturaldiscontinuityhypothesis,providedinsightsonconstraintson basin geometry, characteristic scale length of communities, management of disease and insect outbreaks, body massaggregationsindiscontinuouslandscapes,andrangelandmonitoringandassessment.

This symposiumdevelopedprotocols fordetecting states and thresholds, including theuseof catastrophe

theory, self-organized criticality, and time scale calculus, analysis of the increase in variance (including predicted r2 and error-X variance spectra analysis), red shifts, and critical slowdown time series analyses, multivariate and inferential statistical analyses, and power law analyses. Speakers defined characteristics of nonlinear ecology anddescribedapplicationsfornaturalresourcemanagementthathavegrownoutofnonlinearconcepts.Theseincludestateandtransitionsmodels,whichgrewoutofalternativestablestates,theconceptofthresholds,whichdevelopedfromregimeshifts,andtheideaofrangelandhealth,whichisapracticalapplicationofecosystemself-organization. These applications have helped natural resource managers and other decision-makers deal with “realworld”problemsofrangelandstewardship,managementofWestNilevirus,andcontrolofgrasshopperinfestations.

Notsurprisingly,thisgatheringraisedevenmorequestions. How do we proceed from here? The time scale calculusstudyhasmovedtowardanincreaseinmodelcomplexitytoaccountforbothtemperatureandcontinuedseasonal dynamics. In general, processes contributing to the observed dynamics of insect outbreaks and rangeland vegetation dynamics remain inconclusive. For example: What is the relationship of self-organized criticality to catastrophe theory? What is the relationship of time scale calculus to catastrophe theory and the cyclic regime shifts typical of insect outbreaks? Next steps must include more fully integrating concepts of nonlinear ecology, such as thresholds, into mainstream ecology and developing a more comprehensive nonlinear framework that recognizes the importance of scale in ecological systems.

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RobertA.Washington-Allen(contact for information)CenterforRegionalandEnvironmentalStudiesDepartmentofEnvironmentalSciencesUniversity of Virginia, Clark HallCharlottesville, VA 22904-4123 (434) 243-2331, E-mail:washingtonra@mac.com

and

LucindaF.SaloSageEcosystemScienceBoise ID 83707-3315(208) 850-3313E-mail:csalo11@hotmail.com

the EsA Family of scientific journals.