Divergeasdasdnce and Sensivity Phillips

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J onathanD. Phillips

Divergence, Semitivity, andNoneguilibrium in Ecosystems

Contemporay theoretical debate in ecology and biogeographyis often focused onequilibrium vs. nonequilibrium behavior in ecosystems and on the nature and sourceof ecosystem dynamcs. I t is suggested that these debates be recast in termsof the wayecosystems develop and respond to disturbances, rather than in terms of conceptsoften imported rom mathematics, physics, and other ields. Using nonlinear dynam-cal systems theoy, tisshown that key theoretical implications can be cast in terms ofgeoecologtcally signijcant phenomenologies suchas divergent evolution, sensitivity toinitial conditions and small disturbances, historical contin ency, and path dqen-cal and biogeographical the0y can be problematized from within eography andbalanceof nature, or chaos. Complexity, sensitivity, vartabilit ,nonsteady states,works have manvestations that are evident in observable ecological phenomena, inaddition to the0y andmodels.

dence. Examples show these phenomena are widely observed n ecosystems. Ecologi-ecology rather than fizzy, abstract concepts such as equilibrium, seP-organization,and other concepts often associated with nonequilibrium or comp exity-the0yframe-

1. INTRODUCTIONA fundamental debate in biogeography and ecology is whether, or the extent towhich, communities and ecosystems follow a developmental athway leading towardastable, steady-state equilibrium condition. The absence of, i viation from, or varia-tion in such monotonic developmental pathwaysis likewise a focus of debate, partic-ularlyonthe roles and relative importanceof external disturbances, intrinsic complexdynamics, and historical or path dependencies. The purpose of this aper isnot toprovide a comprehensive review or critique of these debates. Rather, &e goal isto at-tempt to redirect the focus to observable manifestationsof (non)equilbrium, in)sta-bility, and other henomena in ecosystems. Rather than an abstract debate over

search should address specific testable phenomena such as divergent vs. convergentwhether grasslanti , for example, are equilibrium or nonequilibrium systems, re-Critiques and challenges, temperedby encouragements, from several anonymous referees helped im-provethis paper.

J onathun D. Phillips is a professor in theDepartmentof Geography, Universityo Kentucky(j [email protected]).Geographical Analysis,Vol. 36,No.4(October2004)TheOhioState UniversitySubmitted: March 1,2003.Revised version accepted: November 17,2003.


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370 / GeographicalAnalysisevolution and sensitivity to disturbances. The intent is to show that such an approachdoes not involve abandoning metatheory in favor of pure empiricism, because thereare direct links between thebehaviors of nonlinear dynamical systems and measur-able, observable, geoecological phenomena.

Recent decades have seen shifts, or at least broadening, of viewpoints on equilib-rium in bio eography and ecology, as documented by Per (ZOOZ), amon others.events, has become common (or even dominant; PerryZOOZ), the debate is still sig-nificant. First, classicial equilibrium concepts are still quite prominent in appliedecology, geography, and resource management. The whole exercise of delineatingecoregions, for example, is implicitly or explicitly based on the concept of equilib-rium, climax communities, at least at a hi h level of generalization. The concept andpractice of ecosystem restoration, for anotaer example, is often linked to the idea of anatural,equilibrium ecosystem which can be maintained in a steady state. Second,classical equilibrium concepts are not necessarily at odds with modem ecologicalthought. Disturbance, for instance, can be treatedas a riodic factor necessary tomaintain equilibrium, rather thanas a source of non- or g-equilibrium. Third, basicequilibriumvs. non-equilibrium debatesarestill prominent in some fields, such asthe study of semi-arid grazing systems (e.g. Illius and OConnor 1999;Sullivan andRohde2002).Fourth, acceptance of nonequilibrium viewpoints is uite variable,ognize that equilibrium or nonequilibrium may both be possible, and common. Fi-nally, independentlyof thepoints above, there is substantial controversy over theextent to which nonequilibrium behavior is linked to external factors versus intrinsiccomplex dynamics of ecosystems.

Equilibrium is defined in various, and often imprecise, ways in geo raphy and

Even thougaa nonequilibrium view, emphasizing the roleo7isturbance an% chance

ranging from those who view nonequilibriumas the rule and norm, to9, se who rec-

ecology. Heretheterm is used to refer to a steady-state, whereb small Puctuationsaround a constant mean condition. For example, stead-state equilibriummatter implies that the rate of litter additions is roughly balanced by theso that soil organic matter remains a roximately constant.Asteady-state grassland community, for another example, wouf)d)be haracterized by anoverall species composition that does not vary much, even as individual plants andpatches come and go.The link between observable ecosystem properties and histories and theory is ac-complished here via nonlinear dynamical systems(NDS)theory, This should not beconstruedas an implicit claim of primacy forNDS theory. However,NDS theory andmethods are critical or relevant to many of the concepts of nonequilibrium ecosys-tems, and at the very least thiswork should serveas an illustration of howcomplexsystems concepts can be translated into or interpreted in the context of real-worldecosystem patterns, processes, and histories.Thepurpose of this paper is not to pro-vide a comprehensive overview of nonlinear dynamics in geography or ecology.Rather, the goal is to identify complex nonlinear phenomenologies of ecosystems and

and other sciences that areto a laboratory or simula-and even conceptsmodification.

ver ent; originally similar ve etation plots, for example, become progressively moredifBrent over time. The coni tions under which the development of ecosystems andcommunities is divergent or convergent are of fundamental ecological importance


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J onathan D. Phillips / 371and interest. In studies to assess or predict ecolo 'cal responses to climate or otherpoint out that understandin the stability of communities may be more important

Having acknowledged that there are no claims for comprehensiveness, note thatthe examples are selective and biased toward subjects reflecting the interests and ex-perience of the author, and they should not be interpreted as suggesting chaos ismore common or significant in these ecosystems or subfields as opposed to otherareasof biogeography and ecology.

environmental change, for example, Mitchell anf sillag (2001) and Hobbs (1994)than predicting levels of pro%uctivity or other ecological particulars.

2.BACK GROUND: CHAOS AND I NSTABILI TYThe starting oint is an ecological system withn components that at least poten-tially affect, anBare affected by, each other, Thus the analysis of ecological systems

can be approached based on dynamical equation systems and interaction matrices. Asthese methods are described in some detail in geographical and ecological contextselsewhere (Logofet 1993;Pahl-Wostl 1995; Phillips 1992; 1999a),this section willfocus on a few key points linking nonlinear dynamical systems theory and methodswith observed geographical and ecological patterns and behaviors.The stability of a system to small perturbations can be determined based on the in-teraction (Jacobian) matrix evenif only qualitative information (positive, negative, orzero entries) is available. Dynamical instability in thissense is identical to determinis-tic chaos. Chaotic systems are often said to have sensitivedependenceon initial con-ditions because even minuscule differences in startingvalues lead to divergent resultslater. This terminology has been unfortunately misleading for paleoecology,as it su-perficially implies the ability to estimate or infer initial conditions from the presentstate of a system. Ecologically, sensitivity to initial conditions s best understoodas fn-dependence of initial conditions; even minor, initiallyecologically insignificant varia-tions in starting conditions could lead to a very different system state. Chaoticsystems arealso sensitive to small disturbances. Sensitivity to initial conditions is usu-ally stressed in the chaos theory literature because of the predominance of numericalmodeling. In field studies, as opposed to numerical models and laboratorye eri-time periods, disturbances are likely. Sensitivity to perturbations is therefore at leastas importantas sensitivity to initial conditions when the goal is to understand or re-ments, initial conditions are unknowable. And because ecologists often deal withTongdict system res onses to disturbance, or to reconstruct system states or behaviorPomclues in the p2orecord.In a stable, nonchaotic system, development is convergent-that is, variations ininitial conditions are reduced, on average, over time. Unstable, chaotic systems aredivergent,as variations in initial conditions (on average) are exa erated and increaseover time. Observations and measurements of convergence or %vergence, indepen-dent of external controls and forcings, can therefore be linked directly to dynamicalstability. There is also a direct link between Kolmogorov (K-) entropy and chaos,whereby finite positive K-entropy is associated with instability and chaos. K-entropy,in essence, measures the change in statistical (Shannon) entropy-a well-known andwidely used tool in spatial analysis-as a system evolves (Oono 1978; Culling 1988).3. ECOLOGICAL MANIFESTATIONS

Based on the discussion above, ecological signatures or manifestations of instabilican be identified. Because they are based on ecological phenomena, they can bes t u xied independently of chaos theory andNDS formalisms, though the latter are oftenvital to understanding, interpreting, and modeling these phenomena (Pahl-Wostl


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372 / Geographical Analysis1995; Philli s 1999a; Stone and Ezrati 1996).This section should not be understoodhistorical contingency, an characterized y multiple equilibria, as there are exam-ples of convergence, gradual change, time-independence, and single stable equilib-ria. Even where the phenomena listed below do exist, they may be unrelated tononlinear dynamics.3.1.Divergent Evolution

Progressive differentiation of communities, ecosystems, and landscapes is diagnos-tic of instability and chaos if the divergence occurs in the absence of, or is dispropor-tionately large relative to, variability in environmental controls and forcings. Thelatter is an important consideration,as nonchaotic divergence may also occur, associ-ated with obvious, measurable variations in environmental controls or external forc-ings. Divergence is most readily visualized in a spatial context, whereby initiallysimilar locations become increasingly different, on average, over time. An examplewould be increasin patchiness of species composition or soil nutrients. Systemsgent. Initial variations or minor perturbations are gradually obscured as the steadystate is approached or restored.Convergence and divergence are two fundamentally different forms of evolutionthat are directly linked to equilibrium and nonequilibrium. In the case of conver-gence, whatever the starting points or the variations in initial conditions, the ecosys-tem inexorably roceeds toward some articular end state (Clementsian succession is

B t:o suggest ti t ecological s stems are alwa sdivergent, subject to abrupt change and

characterized by sta% e equilibrium, or progression toward such a state, are conver-

7classic exam e). The latter may bei fined specifically (an oak-hicko climax for-est, for exampPP), or generally (steady state or a critical threshold state, or instance).Disturbances may interru t or delay but do not stop this progression.Divergence indicates gat the landscape does not move toward a predetermineddestination, and in fact becomes more diverse over time. Initial differences become,on average, magnified. The effects of perturbations or disturbances, internal or exter-nal, tend to persist and grow over time. Where this occurs in the absence of, or inde-pendently of, external controls or forcings, it is indicative of chaos. Further,divergence that is disproportionately arge compared to the initial trigger is evidenceof instability and chaos.There are several examples from the literature,shown in Table1.Many of the ex-amples are not grounded in NDS theory. I argue that divergence that is demonstrablylarge and long-lived relative to the variations or perturbations that initiated it must re-flect dynamical instability and chaos. Acceptance of this argument, however, is notnecessary for recognition that this key implication of NDS theory can be directlylinked to observable, geoecologically significant phenomena.A few studies address divergenceas sensitivity to initial conditions within theoreti-cal and methodological frameworks explicitly linked to nonlinear dynamical systemstheory and landscape sensitivity. These are summarized n Table2.3.2.Abrupt Changes and Sensitivity

Chaotic systems are characterized by sensitivity to small perturbations, implyingthat small disturbances have disproportionately ar e and long-lived effects. WhentheseoftenTaracterized as a flip or switch) in the state of an ecological system. Thres -olds can,of course, be transgressed due to gradual ongoing changesor large forcingevents. Abrupt chan es triggered by dis ro ortionately small (even undetectable)nonequilibrium in ecosystems. Any ecosystem, stable or unstable, may changeabruptly in response to large disturbances (for example, volcanic eruptions, tsunamis,

as of responses occur near a threshold, taere may be an abrupt chan edisturbances are attri Jutable to chaos. TR f ls 'stinction is critical for equilibrium vs.


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TABLE 1Empirical Examples of Divergence inEcological Systems.Dylanrl land degrdti on.

Plant-soil-microclimate nteractions.

Coastal marshes.

Positive vegetation feedbacks.

Self-reinforcing tree effects.

Grazing effects.

Climate-vegetation feedbacks.

Evolution of biodiversity.

Increasing variability over time in the form of a progressively morecomplex attern of vegetation and soil nutrient resources. Initiallymore uniform distribution of vegetation, soil nutrients and carbon,and surface erosion diverges into an increasin ly atchy distrihu-tion (A braham, Parsons, and Wainwright199%; hlesinger etal .1990; 1996).Microscale heterogeneil increases due to pro ressive modifica-tions by ve etation; patc y vegetationbare soifmosaics develofrom ini tiay more uniform patterns (Beatty1987;Hardy and 11-bert 1995;Holtmeier and Brol ll992; Wilson and A pew 1992).Increasin ly complex mosaicof marsh plant communities, saltpans, andgopen water due to complex interactions between ve etation, substrate, erosioddeposition, and h droperiod (Boston1%3-Hackney et al . 1996;Nyrnan etal. 1993; T994;Pethick 1974).Verti-cal accretion on tidal flats and subsequent salt marsh developmentis sensitive to the distributions and grazin predation of benthicmicroalgae in the initial stages of mudfiat5evelopment (C ola197Q).Mutual adjustments of vegetation and environmental factors suchas microclimate, hydrology, fire regimes, and soils produce diver-gence in theform of sh ened ecological boundaries and localcommunity switches (%son and Agnew 1992).In forest communities, individual trees create nutrient-rich mi-crosites that encourage future tree establishment; treespreferen-tially reoccupy the same microsites over multi leforesty t i o n s , with pedologic effects increasingTocal soil variabilityPhillips and Marion2004;Van L ear, Kapeluck, and Carroll 2000).Initial recruitment in floodplain forests controlled by variations ingeomorphic processes persists and is enhanced in the compositionof older forests (Robertson and Augspurger 1999).Alternative stable vegetation states within a single landsca e andvegetation discontinuities, arise after grazing due to compfei inter-actions between animal im acts, ve etation, microclimate, soilmoisture, and soil erosion &allin f988; 1989;Hobbs 1994;Lay-cock 1991;Friedel 1991;Tausch,%i and, and Burkhardt 1993;Walker 1993;Westoby, Walker, and ioy-bleir 1989).OverQuaternadevelopment ofya root mne is characterized by instabil ities thatlead to differential res onse and divergence into tro ical forest, sa-vanna, semiarid, andLsert environments from simifar initial con-ditions (Lapenis and Shabalova1994).Isoto e trends acrossthePermo-Triassicbounda do not su portmod& of biodiversity collapse and recovery baseyon a singufarevent. Rather, the transitions seem to reflect multiple ecosystemstable states and abrupt responses to small changes durinforcingsof a complex nonlinear biotic system (de Wit et af%%!)?

timescales feedback between precipitation and

TABLE 2Examplesof Studies Showing Sensitivity to Initial Conditions in Ecological Systems.Coastal wetland$. Unstable, chaotic interactions between deposition, marsh surface eleva-tion, hydroperiod, and vegetation produce increasingly complex marshshorelines and s atial atterns of marsh and open water over time(Phillips1989; 1592; 1b9b; Rankey2003).

thresholds of grazin ressure, control subsequent successional path-ways (Miles et al . &?).Statistical models that best fit tree-rin width data indicate determinis-tic chaos and sensitivity to initial condftions that lead to rogressivedif-ferentiation in forest stand condition (Van Deusen 199Of:

Grasslandlwoodland transitions. Historical contingency and sensitivity to intitial conditions, linked toRed Spruce decline.


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374 / Geographical Analysisurbanization),or to gradual changes that go on long enough or whose cumulativeef-fects are sufficient.Very little of theextensive literature on disturbance ecology has been interpretedin light of the magnitude and longevityof responses relative to the size and durationof the disturbance, but it is likely such an interpretation would yield numerous addi-tional examples of sensitivity to small perturbations. Some selected exam les are pre-sented in Table3.Again, many of the studies cited were not conducteBusing NDStheory and methods, and some might argue that the results therefore do not neces-sarily constitute proof or evidence of chaos and instability. Here I sidestep that argu-ment to simply int out that an important im lication of NDS theory-that manybe linked to well-documentedcaseswhere ecosystems ndeed display such sensitivity,3.3.HistoricalContingency and Path Dependence

There is an ever-increasing mass of evidence that historical contin ency and pathdependence are critical in ecology and biogeography (Currie and Na8elhoffer 2002;Hendry and McGlade 1995; Foster 2000; Mailly, K immins, and Busing 2000; Nys-trom and Folke 2001; Parker and Pickett 1998;Peterson 2002; Polakow and Dunne2001; Sinton et al. 2000; Tausch,Wi and, and Burkhardt1993;Whittaker,Willis,andField 2001). Instability and chaos cfctate a degreeof historical contingency in thatsuch systems have a "memory" of disturbances or initial variations. Historical contin-ency and path dependence can also be due to inheritanceof persistent or relictformsor features, or to simple conditionality (e.g., whether or not asite is farmed,

systems may berghly sensitive to small perturf tions and to initial conditions-can

TABLE 3Examples of Studies Showing Sensitivity to Small Perturbations and Disturbances in Ecological Systems.Lake twphic status.Coastal wetlands.

Seminrtd ecosystems.

Tern irmelowland orest.Vegetation-wtnd interactions.

Vegetation change.Forest composition.

Multiple equlNbrtum states.

Many lake ecoas abrupt switcxs between clear and tugi doroligotrophideutrophicconditions(Lauand Lane2002).Effects of local disturbances persist and row dispro ortionately l argeand long-lived relativetomagnitude andgduration otthe intia perturba-tion (Orson andHowes1992; Orson, Simpson, and Good 1992;Graceand Guntspergen 1999).Minor vegetation change leads to disproportionately arge eomorphicresponse, with semipermanent soil and vegetation results fibrahams,Parsons, and Wainwright1995; Nicolau etal . 1996; Puigdefabregas et al.1996; Parsons, Abrahams, and Wainwright1996).Steady-state biogeochemicalcycl es unstable and sensitive to minorchanges (Brinkmann1989).Small vegetation disturbances in dunes lead to unstable self-accelerationand formation of unvegetated blowouts (Barth 1982; Gares and Nord-strom 1995); wind-sha d forests linked to perturbations in wind velocityfields (Robertson1 9 xVegetation change indata set of 15,000 experimental lotsisunpre-dictable, due manly to growth and ersistence of locidisturbancesorinfluence of smd ocal variations (&& 1998).Chance events (gap disturbances) contribute more, and niche partition-ing less, than expected to themaintenance of t ree species richness in for-est gaps (Brokaw and Busing2000).Complex forest structures arise fromcontingent interactions amon climate, landforms, stand condition, anddisturbance (Sinton etal. 2 d ) .Persistence and exa erationof local disturbances important in creatingmultiple successionygthways ande uil ibrium states (Cattelino et al.1979;DeAn elis and aterhouse 198q; Hobbs, 1994; Tausch, Wigand,and Burkhar%t1993).

tems sensitive to small erturbations often manifested


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J onathan D. Phillips / 375burned, grazed, etc.). Where historical contingency can be directly linked to progres-sive exaggeration of initial variations (independent of external factors), or to the dis-proportionately large and persistent effects of perturbations, then chaotic dynamicsare indicated.3.4. Multiple Equilibrla

Many ecological systems have multiple possible equilibria, as opposed to a singlepreordained equilibrium condition, such as a climatic climax community. In somecases this is due to the instability of equilibria, which makes it unlikely that any givenequilibrium state can persist for long in light of the small disturbances that inevitablyoccur. Multiple stable states have been much discussed in ecology, and these discus-sions do not necessarily relate to NDS principles. However, d amical instability andThornes's (1985) NDS model found that the interaction of vegetation cover and soilerosion was dynamically unstable. Any disturbance to the system would tip it to oneof twostable states, associated with erodedho vegetation or full vegetation coverhoerosion conditions. Subsequent work has confirmed the applicability of Thornes'smodel (Abrahams, Parsons, and Wainwri ht 1995; Parsons, Abrahams, and Wain-nomena have been identified in lake ecosystems (Lau and Lane2002).

chaos can be responsible for the presence of multiple stabr"states. For example,

wright 1996; Nicolau et al. 1996; PuigdeBbregas and Sanchez 1996). Similar phe-4.TESTING AND VALIDATION

Analtytical techniques for testing and validating NDS-based hypotheses in ecologyand geography are treated at length elsewhere (Cushing et al. 2003; Logofet 1994;Pahl-Wostl 1995; Phillips 1999% 1999b;2000). Where time series are of suitablelength and quality, a variety of methods exist for detectin complex nonlinear dynam-ics, and for distinguishing deterministic from stochasticP rcings. These are reviewedinthecontext of h drology by Sivakumar(2004); hesame approaches can be appliedto biogeographicJand ecological time series. Experimental nvestigations of chaos inecology are outlined and illustrated by Cushing et al. (2003).This section will focuson other means for interpreting potential field evidence of complex nonlinear dy-namics, and for distinguishing chaos and instability from other potential causes ofspatial differentiation, abrupt switches, path dependence, and multiple equilibria.4.1. Chronosequences

A chronosequence represents an ergodic space-for-time substitution. The idealchronosequence consists of a series of sites or environments where all the factorscontrolling ecosystems except time or age are invariant,so that the sequence can beassumed to represent a temporal progression. The chronosequence a proach hasbeen most often applied in edology and soil geography (Huggett 1998 but also inwhere factors other than age or time are negligibly variable, and even en there isthe problem of the longer, and thus different, histories of the older members of thechronosequence. Nonetheless, given the long time scales and historical investigationsthat often characterize physical geography and ecolo chronosequences may offerinsight into temporal trends that is not otherwise avaiPble.A good chronosequence, by definition, means that there are no si nificant varia-tions in geology, climate, disturbance regime, or other factors that inPuence ecosys-tem development. Thus the hypothesis of increasing variability over time associatedwith divergence can be directly tested,as any increase in variabili must occur due tothe unstable, chaotic magnification of the effects of minor initi2ariations or smallperturbations. Any relevant metric of the spatial variability or richness of the phe-

zlant ecology (Foster andTPman 2000). It is fiendishly difficultto identi a situation


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376 / Geographical Analysisnomenon of interest can be computed for each stage of the chronosequence to deter-mine whether variability s increasing. Generally,

where DI is a divergence index, andV is some measure of variability measured at twotime incrementst and t - . DI >0 indicates divergent and DI


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J onathanD. Phillips / 377whereS is richness and c andb are empirical parameters. Speciedarea curves canalso be derived for each elementary area,

Sl =c1A,hi (3)Then

where the overbars indicate mean values andm =C S,/S to adjust for species whichare counted in more than one elementary area. Because-F I A , h imn =cA b (4)

the ratiogI/b ndicates the relative importance of intrinsic variability attributable tocomplex nonlinear dynamics to that associated with environmental heterogeneity(that is, the habitat difference between theA J .This method is described more com-pletely by Phillips(2001).For example, Phillips and Marion (2004)hypothesized that short-range soil vari-ability in forestsoilsin Arkansas was due to instabilities associated with the effects ofindividual trees on soil morphology. One test of thz hypothesis was a richnesdareaanalysis of 16elementary area plots, where the b,/bratio of 1.15indicated thatwithin-plot variation of soil taxa, consistent with the hypothesis, was greater than be-tween-plot4.4.Landscape and Ecosystem Entropy

Asnoted earlier, there is a direct relationship between Kolmogorov (K-) entropy,representing the change in Shannon entropy as a dynamical system evolves, andchaos. K-entropy is given by

wherep, represents the proportion of pixels, cells, sample sites, area, etc., in categoryi . The categories could represent vegetation communities, classifications of spectraldata, soil types, etc. Hk >0 indicates chaos,as finite positive K-entropy is related tothe Lyapunov exponents by

Recall that a positive Lyapunov exponent is required for deterministic chaos. Theap-plicationof entrop in a chronosequence or historical contextas described above ispreted to determine the nature and direction of entropy changes, and thus whetherthere is positive K-entropy. This ap lication of K-entropy in an interpretive context isillustrated for tidal creek networksfly Rankey(2003), or soils and weathering profilesby Phillips(ZOOO) ,and for soil landscapes by Ibanez et al. (1994)and Phillips, Gares,and Slattery(1999).For exam le, if it is clear that that vegetation typeA is encroach-ing into typesB andC,then i e ffects of these changes in ( p, pb, p, ) can be exam-ined to seeif entropyis increasing or decreasing.

straightforward.Ac rditionally, in some cases ecosystems and landscapes may be inter-


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378 / Geographical Analysis4.5. Magnitudeo Perturbations

Chaotic systems are characterized by sensitivity to small perturbations, wherebysmall disturbances have disproportionately arge and long-lived effects. Thus in somecases a test of whether abrupt changes, historical contingency, or multiple statesmight be attributable to dynamical instability and chaos can be based on whether aperturbation (or variation in initial conditions) s small elative to the magnitude ofthe resulting change.This can be based on the magnitude of the perturbation in terms of longevity, areaof impact, energy (or ower, force, shear stress), or mass. For example, a perturbationmagnitude index couPd be based on

PMZ, =A$,/&, (7)whereAtpandAt, represent the duration of the perturbation and the response, re-spectively. Similarly,

wherepandA, are the footprint or area of the erturbation and response. AnalagousA small perturbation would be associatedwithPMZ


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J onathanD. Phillips / 379ing considerations of nonequilibrium systems, multiple stable states, historical con-tingency, path dependence, and divergent development.The presence and possibility of complex nonlinear dynamics is also important inthat it may complicate (or inform) efforts to interpret historical evidence such as pa-leoecological data. Due to dynamical instabilities and chaos, small perturbations orshort-lived disturbances rather than larger environmental change may be responsiblefor changes or switches. Due to multiple equilibria, there may not be a one-to-onecorrespondence between system states and environmental controls.In some cases complex nonlinear dynamics can provide explanations of phenom-ena not otherwise explained. In climatology there are numerous examples wherenonlinear d amical instabilities provide the only plausible, or the most widely ac-proxy data. Stone and Ezrati (1996) ve a number of ecological exam les.plaining, and describing complex ecological systems. Chaos is sometimes nterpretedto mean that prediction is impossible, and that for practical purposes chaotic systemsare indistinguishable rom genuinely random systems. If an irregular pattern is shownto be chaotic, this actually poses several advantages. Chaos indeed inhibits someformsof redictability and limits the rangeof forecasts. However, chaos by definitionimplies tfepresence of underlying deterministic dynamics. Finding the underlyinggenerator (particularly in terms of processes or environmental controls) of pseudo-randomness may be an important step to redictability. While chaotic systems pro-duce complicated patterns that are unpreb:ctable in detail over long timeddistancesor many interations, these atterns are often perfectly predictable a few iterationspatterns have well-behaved statistical moments. By definition, chaos occurs withindeterministically defined boundaries: unlike true randomness, all outcomes are notequally possible, and the entire hase space is not potentially filled.These properties have yet to ewell exploited in bio eography and ecosystem sci-ence, but examples are available from other field-base sciences. For example, non-linear models performed better than linear models in forecasting surf zone sedimentsuspension( affe and Rubin 1996).The recurrence patterns of lake-volume fluctua-classical linear methods of time series analysis (Sangoyami, Lall, and Abarbanel1996).The methods used in these studies capitalized on two important characteristicsof chaotic sequences: the ability to predict deterministicall a few iterations ahead,

cepted, expr"nations for phenomena such as abrupt shifts observed in paleoclimaticNonlinear formalisms and methof may also provide tools for unb"erstanding, ex-

into the future. Further, wEl le the details are irregular and unpredictable, chaotic

%tions were s ,own to be better modeled and forecast by nonlinear models than byand the role of histo in conditioning subsequent valuesoJa time series.Neither ispossible inasystem2 t is treated as linear and random.5.2. Concretization

If ecosystems are in or progressing inexorably towards stable steady-state equilib-rium, they will be characterized by convergent evolution, nsensitivity to minor varia-tions in initial conditions, and the abilityto recover from and negate the effects ofsmall disturbances. If ecosystems are not in or progressing toward a stable steadystate, they are likely to be characterized by divergent evolution, sensitivity to initialconditions, and the unstable exaggerationof the effectsof small perturbations. His-torical contingency and ath dependence may occur in equilibrium or nonequilib-and small perturbations.Thus at least some implicationsof nonequilibrium and equilibrium in ecosystemscan be directly linked both to theoretical formalisms(inthiscasenonlinear dynamicalsystems theory) and to observable, geographically and ecologically meaningful he-nomenologies. These phenomenologies relate directly to issuesof the role ande p cts

rium systems, but in the P tter case should be linked to sensitivity to initial conditions


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380 / GeographicalAnalysisof disturbances, spatial heterogeneity, and evolution and historical development ofecosystems.It seems, then, that b shifting the terms of debate from equilibrium, chaos, self-organization, andsofoA to ecological1 meaningful phenomenologies suchas diver-can be firmly rounded in geographical and historical observables without losing con-nections to de%ates grounded in metatheories such as NDS theory.gence and disturbance responses, theBbates can be concretized. The controversies6. SUMMARY AND RESEARCH AGENDA

Sullivan and Rohde(2002,p. 1,595)write that adebate in ecology rages over thesources and types of dynamic behavior driving ecological systems. This study sug-gests that in many cases intrinsic properties of nonlinear ecosystems drive the dy-namic behavior, and that these are manifested n measurable, observable, ecologicallymeaningful phenomena in real-world ecosystems. Perry (2002,p. 354)writes, It hasbeen realized thats ace can fundamentally alter the dynamics and outcomesof eco-logical processes. &s has been largely achieved through theoretical, model-basedstudies; the challenge remains to empirically test and explore the large bodyof spatialecological theory. In the case of NDS, the implications of theory can be directlytranslated into ecological phenomenologies that are empirically testable. It is likelythat this is, or will,alsobe the case with other threads of spatial ecological theory,This suggests that there should be more links between models and observationaldata. This can be accomplished not only by theorists linking their work to field stud-ies and experiments, but also b the examination of historical contingency, multipleto develop hypotheses regarding ecosystem dynamics and spatial ecology that aretestable on the basisof observations in nature.Chaos, instability, and other forms of nonequilibrium do not preclude predictabil-ity but do provide a new context for predictabili We need to know the range, accu-phenomena such as chaos. Where nonequilibrium exists, this further implies a criticalneed to assess the extent to which unpredictability and uncertainty are inherent insystem structures and dynamics,as opposed to being associated with stochastic forc-ings, large degreesof freedom, and information inadequacies. The new context forpredictability may well be one wherewerecognize that we can perhaps never accu-rately forecast the time and spatial coordinates of pest invasions or ecotone shifts, butcan move toward accurate and recise identificationof synoptic situations, where

This research agenda should be problematizedfromwithin ecology and geographyrather than NDS orcom lexity theory. The fundamental manifestations of chaos, forexample, inecology canfle linked to observable ecological phenemona. Ecosystemsthemselves, as opposed to models, should be the primary source of ideas on ecosys-tem dynamics.

equilibria, etc., in the contextoP DS and other theories. This encompasses the need

racy, precision, and uncertainty of predictions, w%ch are all profoundly influenced by

such events are moreor less likeP , and better identification of trigger mechanisms.

LITERATURE CITE DAbrahams, A .D., A .J . Parsons, and Wainwripht. (1995) . Effects of Vegetation Change on I nterrill

Runoff and Erosion: Walnut Gulck, Arizona. In Biogeomo hology: Terrestrial and Freshwater Sys-tems, edited by C . R . Hupp, W. Osterkamp, and A. Howard, %-48 Amsterdam: Elsevier.Ballin ,R. C. J r. (1988).The Climatic Impact of a Sonoran Vegetation Discontinuity.CltmaHc Change13,b-lO9Balling,R . C.,Jr. (1989).The Im actof Summer Rainfall on the Temperature Gradient alongthe UnitedStates-Mexico Border.]ournafof Applied Meteorokgy 28,304-8.Barrett,L . R. (2001).A Strand Plsn Soil Development-SequenceinNorthern M ichigan, U.S.A .Catena44,163-86.


13/15

J onathan D. Phillips / 381Barth, H.K. (1982). AcceleratedErosion of Fossil Dunes in the Gourma Region(M&) as a ManifestationBeatty,S.W. (1987) . Origin and Role of Soil Variability n Southern California Chaparral.Physical Geog-of Desertification.Catena suppl. 1,211-19.raphy 8, l - 17.Boston, K. G. (1983). The Development of Salt Pans on Tidal JMarshes, with Particular Reference toBrinkmann, W. L. F. (1989) . System Propulsion of an Amazonia Lowland Forest:An Outline.Geo]uur-Brokaw,N., andR. T. Busin , (2000). Niche versus Chance andTreeDiversity in Forest Gaps.Trends inCattelino, P. J ., I. R. Noble, R. 0.Sla er, andS. R. Kessell. (1979).Predicting the Multiple Pathways ofColes,S.M. (1979). Benthic Microalgal Populations on Intertidal Sediments andTheir Roleas Precursorsto Salt Marsh Develo ment. InEcological Processes in Coastal Environments,edited byR. L. JeffreysandA. J . Davy,25- 4l Oxford: Blackwell.Culling, W. E. H. (1988) .Dimension and Entropy in the Soil-Covered Landscape.Earth SurfaceProcesses6Landforms 13, 619- 48.Currie, W. S. , andK. J . Nadelhoffer. (2002) . TheIm rint of Land-Use History: Patterns of Carbon andNitrogen in Downed Woody Debris at the Harvard gorest.Ecosystems5, 446- 60.Cushing, . M., R. F. Constantino,B. Dennis,B. Desharnais, andS. M. Henson.(2003).Chaos in E cology:Experimental Nonlinear Dynamics. San Diego: Academc Press.DeAn elis,D. L., andJ .C.Waterhouse. 1987) . Equilibriumand Nonequilibrium Concepts n EcologicalM o$els.Ecological Monographs 57, 1-21.de Wit, M. J ,, . G. Ghosh, S. de Vfiers, N. Rakotosofolo, .Alexander,A. Tripathi, and C. h y 2002).

wana {equences: Clues to Extinction Patterns and Delayed Ecosystem Recovery. ]ournal o GeologyFoster, D.R. (2000). From Bobolinksto Bears: Inte ectingGeographical History into Ecolo cal Studies,Environmental nterpretation, and Conservation &mning.]ournal o Biogeography27, 2$-30.Foster,B. L.,and D. Timan. (zo00).Dynamc and Static Views of Succession: Testing the Descriptive

Powerof the Chronosequence Approach.Phnt Ecology 146, l - 10.Friedel,M. (1991). Range Condition Assessment and the Concept of Thresholds:A Viewpoint.]ournalo Range Management 44, 422-26.Cares, P. A., andK. F. Nordstrom.(1995). A Cyclic Model of Foredune Blowout Evolutionfora LeewardCoast: Island Beach, New Jersey.Annalso the Assodation of Amerfcan Geographers85, l - 20.Gracheva,R. G.,V. 0.Targulian, and I.V. Zamotaev.(2001).Time-Dependent Factors of Soil and Weath-ering Mantle Diversity in the Humidlko ics and Subtropics:A Concept of Soil Self-development andDenudation.Quaternay InternationalA,3-10.Gustafson, E. J . (1998).Quantifying Landscape Spatial Pattern: What Is the State of the Art?EcosystemsHackne C. T.,S.Brad L. Stemmy,M. Boris, C. Dennis, T. Hancock,M. OByron, C. Tilton, and E.Bar-bee. fi996).Does Ztertidal Vegetation Indicate SpecificSoil and Hydrologic Conditions?WetlandsHaines-Youn ,R., andM. Choppin , (1996) . Quantifjdnq Landscape Structure:A Reviewof LandscapeIndices anf Their Application to gorested Landscapes. Progress in Physical Geography 20, 418-45.Hardy, J . P., andM. R. Albert. (1995).Snow-Induced Thermal Variations around a Single ConiferTree.Hwdroloetcal Processes 9. 923-34.

South-eastern Australia.]oud o Biogeography10, l -10.nal 19, 369- 80.Ecology and Evolution18183-86.Plant Succession.Environmentalx nagement 3, 41- 50.

Multi le Organic Carbon Isotope Reversals across thehemo-Triassic Boundary of Terrestrial Gond-110,227-40.

1, 143-56.16,89-94.

Hen&, R and .M. McGlade.(1995) . The Role of Memory in Ecological Systems.Proceedings oftheHobbs, R. J . (1994) . Dynamcs of Vegetation Mosaics: Can WePredict Responses to Global Change?Royal Societyo ) hdn 259, 153-59.Ecosdence 1,346- 56.Holtmeier, F.-K., andG. Broll.(1992). The Influence of Tree Islands and Microtopogra hy on Pedoeco-lo cal Conditions in the Forest-Alpine Tundra Ecotone on Niwot Ridge, Coloraao Front Range,U%.A.Arctic 6Alvine Research 24. 216- 28.Hu ett, R. J . (1998). Soi l hronosequences,Soil Development, andSoil Evolution:A Critical Review.C%tena 32. 155-72.Ibanez, J. J ., S. De-Alba, F. F. Bermudez, andA. Garcia-Alvarez. (1995).Pedodiversity: Concepts andMeasures.Catena 24, 215-32.Illius, A., and T. G. OConnor. (1999).On the Relevance of Non-equilibrium Concepts to Arid and Sem-arid Grazing Systems.Ecological Appllcations 9,798-813,Jaffe,B. E., and D. M. Rubin.(1996). Using Nonlinear Forecasting to Learn the Magnitude and Phasingof Time-Varying Sediment Suspension in the Surf Zone. ournal of Geophysical Research l OC,Klbtzli,F. (1998) . Fluctuations, Chaos, and Succession in a Living Environment. nBiodiversity, editedbyW.Barthlott andM.Winiger,111-27. Berlin: Springer.

14283-96.


14/15

382 / Geographical AnalysisLapenis,A. G.,andM. V. Shabalova.(1994).Global Climate Changes and Moisture Conditions n the In-tercontinental Arid Zones.Cltmutic Change27,283-97.Lau,S. S. S., andS. N. Lane. (2002).Nutrient and Grazing Factors in Relation to Phyto lankton Level ina Eutrophic Shallow Lake:The Effect of LowMacrophyte Abundance.Water Reseam%36,3593-3601.La cock, W. (1991).Stable States and Thresholds of Range Condition on North American Rangelands:A

$iewpoint.**]mrnal f Range Management44,427-31.Lo ofet, D. 0. (1993).Matrfces and Graphs, Stability Pmblems in M athemuticd Ecology. Boca Raton,&a.: CRC Press.Mally, D., . P. Kimmns, andR. T. Busin .(2000).Disturbance and Succession in a ConiferousForest ofNorthwestern North America: Simulat!?ons with DRYADES, Spatial Gap Model.Emo& MohU ingMiles,J ., R. P. Cummns, D. D. French,S. Gardner, . L. Orr, andM. C. Shewrh.(2001).LandscapeSen-sitivity: AnEcologicalV ew Catena42,25-41.Mitchell, S. W., and F. Csillag.(2001).Assessin the Stability and Uncertain of Predicted Ve etationGrowth under Climatic Vanability: Northern di edGrass Prairie.EcologdM&Uing 139,181-21.Mossa,), andB. A.Schumacher. (1993).FossilTree Casts in South LouisianaSoh.]oud o Sedtmen-t a y etmlogy63,707-13.Nicolau, . M., A. Sol&-Benet, . Puigdefibregas, andL. GutiBrrez.(lQ96). Effects of Sol and Vegetation

on Runoff along a Catena in Sem-arid Span.Geomorphology14,297-309.Nyman, J .A., M. Carloss,R. D. DeLaune, and W.H .Patrick. (1994).Erosion Rather Than Plant Diebackas theMechanism of Marsh Loss in an Estuarine Marsh.Earth Sutjime ProcessesGL,andfom 19,Nyman, . A., R. D. DeLaune, H. H. Roberts, andW. H. Patrick.(1993).Relationshipbetween Vegetationand Soil Formation in a Rapidly Submerging Coastal Marsh.Marfne Ecology Progress Series 96,Nystriim, M., and C. Folke.(2001).Spatial Resilience of Coral Reefs.Ecosyst ans 4,406-17.Oono,Y. (1978).A Heuristic Ap roach to the Kolmogorov Entropyas a Disorder Parameter.Progress i nTheoretical Physics 60,1944-16.Orson, R. A., andB. L. Howes.(1992).Salt Marsh Development Studies at Waquoit Ba Massachusetts:Influence of Geomorphology on Long-Term Plant Community Structure.Estuari ne,&astd, and ShelfScience 35,453-71.Orson,R. A., R. L. Simpson, and R. E. Good. (1992).The Paleoecoloplical Develo ment of a LateHolocene, Tidal Freshwater Marsh of the Upper Delaware River EstuqEstuaries1% 130-46.Pahl-Wostl, C. (1995).The Dynamic Natureof Ecosystems: Chaos and Order Entwined. Chichester: ohnWdey.Parker, V.T., andS.T.A. Pickett.(1998).Historical Contingency and Multi leScales of Dynamcs withinPlant Communities. n Ecologtcal S&, edited by D. L. Peters andV $.Parker, 171-91. New York:Columbia University Press.Parsons, A.J., A. D. Abrahams, and Wainwright. (1996).Responses of Interdl Runoff and ErosionRates to egetation Change inSoukernArizona. Gemrphology 14,311-17.Perry, G. L . W. (2002).Landscapes,Space and Equilibrium: Shifting Viewpoints.Progress in PhysfcdGeography 26,339-59.Peterson,G.D. (2002).Conta ousDisturbance, Ecological Memory, and the Emergence of LandscapePattern.Ecosystems5,329-%3.Pethick, . (1974).The Distribution of Salt Pans on Tidal Salt Marshes.]ournal o Biogeography 1,Phillips, J. D. (1989).Erosion and Planform Irregularity of an Estuarine Shoreline.zettschrfftfur Geo-Phillips, J . D. (1992).Qualitative Chaos in Geomo hic S stems, with an Example from Wetland Re-Phillips, . D. (1993).ChaoticEvolution of SomeCoastal Plainsoh . Physical Geography 14,566-80.Phillips,J . D.(1999a).Earth Surface Systems: Complexity, Orderand Scale. Oxford: Blackwel.Phillips, J . D. (1999b).Divergence,Conve ence, and Self-organization n Landscapes.Ando &As-Phillips, J. D. (2000).Signaturesof Divergence and Self-organization inSoilsand Weathering Proflles.Philli s,J . D. (2001). TheRelative Importance of Intrinsic and Extrinsic Factors nPedodiversity.AnnalsPhillips,/. D.,,P. A. Gares, and M. C. Slattery. (1999).Agricultural Soil Redistribution and LandscapePhillips,J . D., and D. A. Marion(2004).Pedological Memory in Forest Soil Development.Forest Ewl-Pike, R. J . (2000).Geomorphomet3.-Diversity in Quantitative Surface Analysis.Progress In Physfcal

127,183-205.

69-84.264-79.

57-62.morphologtesuppl.73,59-71.sponse to Sea LevelRise.]ourndo Geology lOo,%s-7~

sociation of American Geographers89,4%-88.]OUWWL of Geobgy 108.91-102.oCompexltyL,andscape Ecology 14,197-211.ogy and Management 188,363-80.Geography 24, 1-20.

Assodation of American Geographers91,609-21.


15/15

J onathan D. Phillips / 383Polakow, D. A., and T. T,Dunne. (2001).Numerical Reci es for Disaster: Changing Hazard and theStand-Origin Map.Forest Ecology and Management 147,783-96.Puigdehbregas, .. andG .Sanchez.(1996).tkomorphological mplicationsof Vegetation Patchiness onSemi-aridSlopes. In Advances in Htllsbpe Processes vol. 2, edited by M. G. Anderson and S. M.Brooks,1027-60. Chichester: Wiley.Robertson,A. (1994).Directionality, Fractals, and Chaos in Wind-Shaped Forests.Agricultural6 ForestMeteorology72,133-66.Robertson,K . M., and C. K . Augspurger.(1999).GeomorphicProcesses andS atial Patternsof PrimaryForest Succession on the Bogue Chitto River, USA.]ouml o Ecology 87,1852-63.Sangoyom, T. B., U. Lall, and H. D. I. Abarbanel. (1996).Nonlinear Dynamcs of the Great SaltLake:Dimension Estimation.Water Resources Research 32, 149-59.Schlesinger, W. H., A. Raikes,R. E. Hartley, andA. F. Cross. (1996).OntheSpatial Pattern of Soil Nu-SchlesinRW.H ,?F.Reynolds,G .L. Cunningham,L. F, Huenneke, W. M. Jarrell,R. A. VirW.G. tford. 1990).Biological Feedbacks in Global Desertification.Science247,1043-giaandSinton,D. S., J . A. Jones, . L. Ohmann and F. J . Swanson.(2000).Windthrow Disturbance, Forest Com-position, and Structure n the Bullrun Basin, Oregon.Ecology 81,2539-56.Sivakumar,B. (2004).Chaos Theory in Geophysics: Past, Present, and Future.Chaos, Solitons, and F rac-

tals 19,441-62.Stone,L ., andS. Ezrati.(1996).Chaos, Cycles, and SpatiotemporalDynamcs in Plant Ecology.]mmlo Ecology 84,279-91.Sullivan,S., andR. Rohde. (2002).On Non-equilibriuminArid and Sem-arid Grazing Systems.Journalof Biogeography29,1595-1618.Tausch,R. J., P. E. Wigand, and J. W. Burkhardt.(1993).Viewpoint Plant Communi Thresholds, Mul-ti leSteady States, and Multiple Successional Pathways: Legacy of the Quatemaryv ouml o Rangedznagement 46,439-47.Thomes, . B. (1985).The Ecology of Erosion.Geography 70,222-35.VanDeusen, P. (1990).Stand Dynamcs and Red Spruce Decline.Canadianjournal o Forest Research20,743-49.VanLear, D.H., P. R. Kapeluck, andW. D. Carroll.(2000).Productivi of Loblolly Pineas Affected byDecomposing Root Systems.Forest Ecology and Management 138,&-43.Walker,B. H. (1993).RangelandEcology: Understandingand Managing Change.Ambw 22,80-87.Westob ,M., B. H. Walker, and I. Noy-Meir. (1989). rtunistic Management of Rangelands Not atEqui~brium.*]mrnul Range Management 42,266ypWhittaker,R . J ., K. .Willis, andR. Field. (2001).Scale andS cies Richness: Towards a General, Hier-Wilson, J . B., andA. D.Q. A ew. (1992).Positive-Feedback Switches in Plant Communities.Adounces

trients in Desertkcosystems. Ecology77,364-74.

archicd Theoryo . Species Diversity. ournal ofBiogeogmp~28,453-70.in Ecological Research 23,%3-336.

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