Chaos, Cycles and Spatiotemporal Dynamics in Plant Ecology
Author(s): Lewi Stone and Smadar Ezrati
Reviewed work(s):
Source: Journal of Ecology, Vol. 84, No. 2 (Apr., 1996), pp. 279-291
Published by: British Ecological Society
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Journalof
Ecology1996,
84, 279-291
ESSAY REVIEW
Chaos,cyclesandspatiotemporal
dynamicsin plant
ecology
LEWI STONE and SMADAR
EZRATI*
Departments
ofZoologyand *Botany,Tel AvivUniversity,
RamatAviv69978, Tel Aviv,Israel
Summary
1 We reviewtheproblemsconcernedwithdescribing
ecologicalvariability
and explain
why nonlinearsystemstheoryand chaos may be of relevancefor understanding
patternand changein ecosystems.
2 The basic conceptsofchaos are consideredbriefly,
wherepossiblein thecontextof
plantecology.We outlinetheperioddoublingrouteto chaos and associatedconcepts
ofbifurcation,
universality,
sensitivity
to initialconditions,dimensionality,
attractors
and Lyapunovexponents.
3 Examples are selectedfromthe literatureto illustratehow nonlinearmodels and
timeseriesanalysishave alreadycontributed
to researchin plantecology.
4 Key methodsusedto detectthepresenceofchaos inecologicaltimeseries,including
determination
ofthe'returnmap,' theLyapunovexponent,thetechniqueofnonlinear
predictionand the methodof surrogatedata, are surveyed.We discusshow small
sample-size,as typicallyacquiredin field-studies,
can act as a formidablelimitation
in suchapproaches.
5 Models of plantcommunities,
whichemphasizetheimportanceof spatiotemporal
dynamics,are outlined.The conceptof deterministic
chaos is shownto be usefulin
thissettingalso.
Keywords:chaos, nonlinearmodel,plantdynamics,spatialmodel,timeseries
JournalofEcology(1996) 84, 279-291
attemptto reviewwhythe studyof not onlychaos,
but all formsof variability,
has alwaysbeen and will
Thestudyofnonlinear
dynamic
systems
hasbecome alwaysbe, importantin ecology.
a rapidly
expanding
research
discipline
thathasblosIn particular,we will discusssome of the specific
somedoverthelastdecade.Whileitsapplications
in
ways in whichchaos theorycan be applied to plant
thehardsciencesare firmly
established,
it is only
ecology,an area thatin our viewhas been somewhat
thatthemathematical
recently
theoryhas become neglectedto date. Until recently,conventionalwisusefulforecologists
on morethanjust an abstract dom dictatedthat chaotic and cyclicalpopulation
level.Infact,whether
ornottheesoteric
mathematics dynamicsare unlikelyto be found in plant comhas relevance
to thestudyofbiologicalpopulations munities(Crawley 1990). We will argue that this
hasbeena subjectofdebateforalmost20years(Hasevaluationmightwellbe outdated.For too longnow
selletal. 1976).Resistance
to thesetechniques
stems the almost mythicalnotion of stable equilibrium
largelyfromthedemanding
theoretical
conceptsof
(Connell& Sousa 1983),in theformof a competitive
chaosanditsassociated
technical
intimidating
succession leading towards an equilibriumclimax
jargon
(e.g. strangeattractors,
crises,repellors,
transients state,has beenthedominantmodeofthoughtinplant
andnumerous
otherexoticmathematical
all
objects),
ecology,at the expenseof othermechanismswhich
of whichare quiteforeign
to mostfieldecologists emphasizetheimportanceofdisturbance,
changeand
and generally
A numberof
outsidetheirtraining.
This is a viewpointthatis onlynow beginvariability.
ecologists
reasonably
question
whether
thesemethods ningto emergeand be takenseriously,as forinstance
willreallyhelpthemin theirquestto understand by Dodd et al. (1995) in theirlong-termstudyof the
communities
orwhether
ecological
theyarejustanoplantcommunitiesin the Park Grass Experimentat
thercase of 'physics
envy.'In thepresent
paperwe
Rothamsted,England.Althoughtheseplotsare con-
Introduction
? 1996 British
Ecological Society
280
Chaos,cyclesand
spatiotemporal
dynamicsinplant
ecology
sideredas possiblythe best candidatesanywherein
termsof beingstableequilibriumplantcommunities,
Dodd et al. (1995) suggest'thattheexistenceof outbreaksin a significant
numberof speciescalls fora
re-evaluationof the concept of stable plant communities.'
In termsof understandingvariabilityand nonequilibrium conditions, the theory of nonlinear
dynamicshas much to contributeto plant ecology,
and we suspectthattherathercautiousearlierassessmentsmayhave underestimated
thetrueexploratory
value of these techniques.In fact,one of the most
excitingand challengingareas in mathematicalphysics, the spatiotemporal dynamics of nonlinear
is currently
systems,
provingto be valuableforunderstandingspatialplantdynamicswhereprocessessuch
as the naturalmosaic cycle(Watt 1947) and habitat
fragmentation
(Tilman et al. 1994; Stone 1995) all
scales in
operatein a complexmanneroverdifferent
bothspace and time(Rand 1994;Hendry& McGlade
1995).
was particularlyinterestedin modellingthe apparentlyperiodicsunspotactivity.The latterphenomenonhad forlongbeenconsideredresponsibleforthe
cyclingoffamines,agricultural
production,tradeand
economicconditionsaroundtheglobe.More recently,
ecology's classic prey-predatorhare-lynxcycle as
well as thegermination
patternsin theYukon white
spruce, have been intimatelylinked with sunspot
activity(Sinclairet al. 1993). The sunspotcyclewas
deterministic
consideredessentially
and periodicwith
a 10-yearcycle(Fig. 1a) untilYule and his colleagues
that
challengedthe idea (Fig. Ib). By demonstrating
the apparentlyregularperiodicityof sunspotcycles
mightin realitybe nothingmore than an averaged
ratherlike
sumof randomeventswithcharacteristics
ofa shock
'Brownianmotion,'Yule caused something
in the scientific
community.For some,thissmashed
thehope thatNature'sregularcyclescould be understood or predicted mechanistically.The illusory
in timeserappearanceofcyclesand otherregularity
ies mightonlybe a formof 'orderin chance.'
Justas theprobabilisticoutlookwas gainingwide,
if not total,acceptanceby the scientific
community,
and werebeginningto be consideredfundamental
for
understanding
nearlyall aspectsof theuniverseright
down to the smallestelementary
particle,the tables
turned suddenly with the developmentof chaos
theory.In theearly1970sRobertMay demonstrated
thatthehighlyirregularfluctuations
we see so often
in thenaturalworld,and whichhaveall thehallmarks
of randomness,can paradoxicallybe generatedby
models.Processesthatto the
verysimpledeterministic
naked eye seemrandom,mightin realitybe strongly
deterministic
and, upon close inspection,exhibita
formor 'orderin chaos.' Not onlycan cyclicaloccurrences be explained by uncomplicatedmechanistic
models,but so too can seeminglyrandomphenomena.
Giventhesemajorparadigmshiftsin thehard scialike
ences,in whichphysicistsand mathematicians
itis little
to describechangeand variability,
struggled
wonderthatecologistshave been facingsimilartechnical problemswiththeirown work.In thecourseof
thisarticlewe hopeto exploreinmoredetailhowideas
connectedwithchaos theoryand nonlineardynamics
thevariability
mayhave relevanceforcharacterizing
foundin plantcommunities.
Historicaldiversion
Thereare countlessexamplesof how biologicalstudies have enrichedthemathematicaland physicalsciences(or viceversa).For instance,we needonlyrecall
thediscoveryof Brownianmotionlastcenturybythe
distinguishedbotanist Robert Brown, and its subsequentdevelopmentby Einsteinin his famous1905
paper. There are manysimilarexampleswhichhave
withscienas a commonthread,thegoal ofdescribing
tificrigourthe changeand variabilitythatoccursat
different
scales in the naturalworld.We emphasize
thatideas ofchance
thatitis onlyoverthelastcentury
and probabilitytheoryhave been fullyexploitedto
explainvariability.Priorto this,Sciencewas by and
world view,
large mesmerisedby a deterministic
handeddownin theformofNewton'slaws. Once the
step was made, however,and the fullweightof the
role of chance was recognized,major disciplinesin
Science were sometimescompletelyreappraised(as
forexample,thephysicsof quantummechanics).
Yule (1927) was one of several importantstatin promotingthisnewworldview
isticiansinfluential
earlierthiscentury.He attemptedto model the rises
recordsof naturalevents,and
and fallsin long-term
I
1750
60
II ' Ii
*70
80
90
IIi I '
ieo0
10
20
IIII
30
40
II
50
60
'
70
80
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90
1900
I
10
20
!I
numbers
iWbUers
(C 1996 British
Ecological Society,
JournalofEcology,
84, 279-291
i
<X/X/W\<>-<
</\\<
<//X
X J X,X
g\
~~~~~~~~~~~~Graduated'num
observedsunspotactivityovertheyears1750-1920.(b) The simulationof sunspotactivity
Fig. 1 (a) Wolfer'sdata portraying
model.Bothgraphsappear in Yule (1927).
by Yule's 'random'autoregressive
281
L. Stone&
S. Ezrati
Whatis chaos?
N
N,+I = F(N,)= rN,(1- N,)
r=3.77
,<
For theunfamiliar
reader,we beginbyverybriefly
describing
a particular
caseofchaosemerging
froma
wellknownbiological
equation.
Thoseseeking
a fuller
exposition
ofbasicconcepts
areadvisedtoreadmore
comprehensive
reviews
suchas Hastings
etal. (1993)
and May (1976,1986).Considerthenthenonlinear
anddensity-dependent
logistic
equation:
e
r 3,77
(1)
whereN,might
represent
thebiomassofa population
inthetthgeneration,
andritsnatural
rateofincrease.
The equationstatesthatthepopulationnextgeneration(N,+l) is somenonlinear
F (whichin
function
thiscaseis quadratic)
ofthecurrent
population
(N,).
Theabovedifference
ofthe
equationisrepresentative
morecommonly
utilizedcontinuoustimelogistic
of applications
in
equationwhichhas had a variety
plantecology(e.g. Lomnicki& Symonides
(1990)
d
r=3.5
modelledthepokeweedPhytolaccaamericana,Room
C
& Julien
(1994)theclonalweedSalviniamolesta,
and
Tilman(1994)describes
a forest
patch-model).
The timeevolution
ofeqn1 is foundbyiterating
theequationfroman initialpopulationN, > 0, to
yieldthesequence:
N, -N2 = F(N,)
=
F(N3)
r3.=
N3 = F(N2) -N4
2.1
* - --
It is naturalto askwhether
themodelpopulation,
if examinedovera long enoughintervalof time,
reachessomeequilibrium
level.To check
eventually
suchan occurrence,
one notesthatthepopulation
can onlyreachequilibrium
at somegeneration
t, if
populationis the
N,, = N,, i.e. whenthecurrent
sameas in thenextgeneration,
and forthatmatter
It is a simplematter
to
everygeneration
following.
calculatethis equilibrium
N*, by settingN*=
N,+, = N, intoeqn 1:
b
t~~~~~~~~~~~~~~~~~~
~
,r=2.I
a
0
10
20
30
40 t
of the modelpopulation
Fig.2 The first40 generations
(eqn 1) are plottedfor (a) r = 2.8; (b) r = 3.2; (c) r = 3.5;
400generations.
(d,e)r = 3.77.(e) Plotofthefirst
N* = rN*(1- N*).
Solvingtheaboveequationoneobtainsthenontrivial
N* = 1- 1/r.
equilibrium
Furthermathematical
analysis(but outsidethe
N*
scopeofthisarticle)revealsthattheequilibrium
willonlybe stablewhenever
rater liesin
thegrowth
1 < r < 3. Figure2a displays
theinterval
thegrowth
of thepopulationwhenr = 2.8 and theway it is
'attracted'
tothestableequilibrium
point.Iffollowed
in time,thepopulation
to the
eventually
converges
equilibrium.
It becomesinteresting
to ask whatmighthappen
rater be raisedbeyondtheequishouldthegrowth
r = 3? Fig.2 displays
librium's
ofstability
boundary
? 1996 British
Ecological Society,
JournalofEcology,
84, 279-291
some of the different
dynamicbehaviourpossibleas
the population'sgrowthratesincrease.For r = 3.2,
the equation oscillatesbetweentwo points(Fig. 2b),
indicatingthatthesingleequilibriumN* is no longer
stable,but has been replacedby a stable two-cycle.
beyondr = 3.5,
As thegrowthrateincreasesfurther
the two-cyclebecomes unstable and replaced by a
(Fig. 2c). Similar'period-doublings'
stablefour-cycle
follow with increasingrapidityas r increasesgeometricallygivingcyclesof period 1, 2, 4, 8, 16 ....
Whenr approachesa criticalvalue r, = 3.57,theperChaos sets in
iod of the cycle approaches infinity.
exactlyat thecriticalvalue r = r(,and irregularnoncyclicalbehaviourthatmightseemrandomto theeye
ensues when r is increased beyond this value
(Fig. 2d,e, r = 3.77). A summary picture of the
dynamicsof eqn 1 is givenin the associated bifurcationdiagram(Fig. 3 and legend)whichdisplaysthe
periodicand chaoequilibrium,
equation'sattracting
tic solutions,as theparameterr is varied.Even more
unusual 'period-halving'dynamicsare possible for
variantsofthissimplemodel,iftheequationis modi-
282
Chaos,cyclesand
spatiotemporal
dynamicsinplant
ecology
?) 1996British
Ecological Society,
JournalofEcology,
84, 279-291
or independent(Antonovics& Levin 1980; Hubbell
et al. 1990) is linkedto the controversy
of whether
0.8
populationdynamicsare chaotic.
2 A chaoticprocessmustexhibitsensitivity
to initial
0.7
conditions
with
nearby
trajectories
diverging
expo0.6
nentially.Suppose at the beginningof a study,two
N 0.5
populationsN and N' are almostidenticalin biomass
0.4
(i.e. similarinitialconditions),and differ
onlyby an
0.3
amountN- N' = 6 << 1. If bothpopulationsare gov0.2
ernedby eqn 1, thensensitivity
to initialconditions
means
that
the
measured
differencein biomass
0.1
betweenthetwo populationswillgrowexponentially
3
3.2
3.4
3.6
2.8
3.8
4
withtime.Using mathematicalnotation,the differr
ence at time t betweenthe two populationswill be
Fig.3 The bifurcation
diagramassociatedwitheqn1. For
approximatedby bt= boe2t,where A is a constant
valuesofr < 3, oneseesfromthebifurcation
diagram
that
representing
theLyapunovexponent.(A moreformal
the equationpossessesan attracting
equilibrium
point.
r to a valueslightly
Increasing
greater
thanr = 3,theequidefinition
of theLyapunovexponentcan be foundin
librium
bifurcates
and a two-cycle
is born;themodelthen
Wolf et al. (1985). We sketchhere only the general
alternatesbetweentwo populationlevelsN, -+ N2-+ N, -+
principle.)Sensitivity
to initialconditionsimpliesthat
thetwo-cyclebifurN2-+ N, -+ .... By raisingr yetfurther,
the
Lyapunov
mustbe positive(A > 0). This
exponent
cates and formsa four-cycleN, -+ N2 -+ N3 -+ N4-+ N, -+
ensuresthatthetrajectories
ofpopulationswithsimiN2-+ .... Increasingr in thissamemanner,yieldsan infinite
succession
ofbifurcations
so creating
a hierarchy
ofstable
lar initialconditionsrapidlyget further
and further
cyclesofperiods2, 4, 8, 16,..., 2n (n -+ oo).
apartfromeach other.
An importantcorollaryto the above is that the
behaviourofchaoticsystemsshouldbe predictablein
the shorttermbut any attemptto make long-term
fiedto introducesmall amountsof immigrationor
predictionswill only prove futile.When forecasting
dispersal(Stone 1993).
the behaviourof a chaotic system,the positiveLyaThe behaviourof eqn 1 has certainfeaturesconpunov exponentensuresthat predictionerrorsare
sidered to be universalin that they are generally
magnifiedexponentially.Long-rangepredictability
inheritedby all modelsthatexhibittheperiod-doubrapidly deteriorateswhen errors build up multilingrouteto chaos. The bifurcation
diagramin Fig. 3
as forecastsextendfurther
plicatively
and further
into
is one suchgenericfeature.Thus Inghe(1990) modithefuture.
fiedeqn 1 to model the irregularfloweringpatterns
3 Chaotic systemsare stablein the sense thatpopuwithinrametsof the perennialherbs Sanicula eurlation trajectories will not exhibit irreversible
opaea and Hepaticanobilis,onlyto finda bifurcation
exponentialrunawaygrowth.Instead, populations
diagramverysimilarto Fig. 3. By allowingthecosts
convergewithtimeonto whatis knownin technical
of reproductionto vary realisticallyin the model,
parlanceas an 'attractor',and theyfluctuateirreguInghe showed that chaotic dynamicscould 'occur
larlybetweentheboundssetbythesize ofthe'attracunder a wide range of environmentalconditions.'
tor'. When plottedgraphically,
it is sometimespossan analysisof empiricalfield-dataof S.
Interestingly,
ible to see visually the population trajectory
thisgeneralpredictionso that
europaeadid notfalsify
'stretchingand folding'as it wanders throughthe
themodelreasonablycomplemented
thefield-study.
attractor.Aftera periodof 'stretching'
(exponential
growth)the population trajectory'folds' in phase
space reducing population levels, thereby comSome technicalcharacteristics
ofchaos
pensatingfor any burstsof runawaygrowth.The
processof stretching
and foldingensuresthe popuThe majorityof articlesin the ecological literature
lation trajectoryremainsbounded and preventsthe
concernedwith eitherthe theoryor applicationof
occurrenceof unstableand unmitigated
exponential
chaos only requirea knowledgeof surprisingly
few
growth.
additional concepts. We list some importantones
4 Stochastic(random)systemsare consideredto have
below.
infinitedimension,and a representative
population
1 In populationdynamics,thepresenceof a density- trajectorymightwanderunpredictably
in space and
dependent
timeas ifhavinginfinite
growthrateis generallyconsideredto be a
degreesoffreedom.A deternecessary(althoughnot sufficient)
preconditionfor
ministicsystem,in contrast,is of finitedimension.In
chaos. Density-dependence
simplymeansthattheper
factthe dimensionof the systemis indicativeof the
capita growthrateof a populationdependscrucially
numberofequationsrequiredto reconstruct
an accuon currentpopulationabundance.Thus the controrate model of its dynamics.Thus low-dimensionality
versyoverwhether
populationsaredensity-dependent is consideredgood evidencefordeterminism.
0.9
283
L. Stone &
S. Ezrati
in timeseries
The fluctuations
in time
fluctuations
thenatureofirregular
Identifying
seriesis a commonproblemfacedbyecologistswhich
as yet,has no simplesolution.Considerforexample,
in Fig. 2(e). It seems impossibleby
the fluctuations
the naked eye, and even by manywell knownstatwhetherthetimeseriesis:
isticaltests,to determine
of randomorigin;
(a) entirely
chaoticpro(b) theproductofsomelow-dimensional
cess;
(c) a combinationof both(a) and (b).
In actual factthe data was producedby iterating
the simple logisticequation (eqn 1) in the chaotic
regime.
On the otherhand, it may be wrongto diagnose
whichappear 'moreor less' cycfluctuations
instantly
periodicforat leasttwo reasons:
lical,as strictly
(a) As Yule demonstrated,cycles that are 'nearly'
periodiccan originatefroma randomprocess.Figure
l(a) graphs 175 years of sunspotactivity(Wolfer's
numbers)wheretheapproximate10-yearcycleis easilyobservable.However,Fig. 1(b) displaysa timeseries producedbyYule (1927) thatis nothingmorethan
randomprocess.It is difficult
a linearautoregressive
betweenthe real timeseriesand the
to differentiate
randomprocess.
(b) A chaotic process can oftencreate the illusion
of periodicity
too. Measles epidemicspeak annually,
usuallyat thebeginningof theschool yearwhensusceptiblechildrenbecomeexposedand easilyinfected.
Despite the naturalperiodicityof the epidemic,its
thoughtto be chaoticby
dynamicswerenevertheless
Sugihara& May (1990), eventhoughsimultaneously
lockedto a seasonal cycle.
All oftheabove combinationsshouldleaveanyone
intendingto analysea timeserieswiththe feelingof
in ecologicaltimeseries
confusion.The recentinterest
analysisarose exactlyforthisreason,and as will be
some advancedmethodsforidentexplainedshortly,
beingrefined.
ifyingecologicalsignalsare currently
blooms:a case study
Phytoplankton
? 1996 British
Ecological Society,
JournalofEcology,
84, 279-291
The dynamics of phytoplanktonin aquatic ecocase studysince these
systemsmakes an interesting
exhibiterraticand explosive'boom
algae frequently
and bust' growth(large r in eqn 1) and share many
chaos (Ascioti
featuresassociatedwithdeterministic
et al. 1993). Figure4 displaysa 20-yeartimeseries
(1970-90) detailingthe yearlyalgae bloom of Pyrrophytaspp. observedin Lake Kinneret(Israel). The
bloom is directly
developmentof thephytoplankton
correlatedwiththetimingand durationofthemixing
period in thiswarm monomicticlake (Pollingher&
Zemel 1981). The time seriesexhibitsfeaturesthat
chaos
could be associatedwith:(a) low-dimensional
byburstsofrapidand intensenonlinear
characterized
60C
500
400-
300:
0)200-
100-
70
75
80
85
90
biomass(mgC m-2)
phytoplankton
Fig.4 Lake Kinneret
overtheyears1970-90.
phytoplanktongrowth; (b) a noisy deterministic
(say the out(limit)cycle,i.e. stochasticfluctuations
erroror environmental
come of eithermeasurement
noise) superimposedupon a periodicsignal;(c) some
possiblecombinationof both(a) and (b).
blooms controla sigGiven that phytoplankton
nificantportionof theworld'sprimaryproductivity,
to generalwaterqualand are also oftendetrimental
ity,ecologistsare particularlyinterestedin underby whichtheseeventsare
standingthecircumstances
(Beltrami& Carrol 1994).Shoulda bloom's
triggered
dynamicsbe substantiallygovernedby low-dimensionaldeterministic
chaos,thenin theorythecomplex
can be modelledby simplemathematical
fluctuations
equationspossiblyof a formsimilarto eqn 1. As we
willexplain,in practicethesituationis not so simple
and at thepresenttimewe knowof no studythathas
carriedout such a programwithsuccess,eitherfor
bloomsor anyothernaturalphenomphytoplankton
enon.
Whyis chaosimportant?
Whyare scientistsso eagerto learnwhetherthefluctuatingdynamicsof the systemthey are studying
in originratherthanthe outmightbe deterministic
come of a randomprocess?For plantecologists,the
question is connected to the vexing problem of
or whetherthey
are structured
whethercommunities
completelylack organizationand are nothingmore
The issuecan be tracedat
thanchancearrangements.
least as far back as Gleason (1926), who regarded
communitiesas 'accidents' - loose assemblagesof
species that arise and are influencedmore by the
In
thanby speciesinteractions.
physicalenvironment
'is notan organhis own words,a (plant)community
ism, scarcelyeven a vegetationalunit,but merelya
coincidence.'The plant associationis 'merelya fortuitousjuxtapositionof plantindividuals'(quoted in
Watt 1947). This became knownas Gleason's individualistichypothesis.
284
Chaos,cyclesand
spatiotemporal
dynamicsinplant
ecology
? 1996British
EcologicalSociety,
JournalofEcology,
84, 279-291
In contrast,Clements's(1936) notionofsuccession
suggeststhatintheabsenceofdisturbance,
plantcommunitieswill eventuallyreach a predictable'climax'
equilibriumstatethatis stableand willbe maintained
overtime.The organizedstructure
of a climaxcommunityis thustheantithesisof Gleason's visionof a
loose assemblagesof species. Separatingout nonrandompatternsin analysesof spatial distributions
has thusbeen of concernforplant ecologists,and a
greatvarietyof statisticaltestshave been employed
for just this purpose. However, the assumptions
underlyingmost of thesetestsare oftenquite rigid
and most cannot adequatelydeal withthe irregular
chaotic signalsarisingfromnonlineardynamicprocesses.
There are many advantagesof knowingwhether
the fluctuations
of a systemare deterministic
rather
thanrandom.Most importantly,
it tellsus thatthere
is someunderlying
mechanistic
processwhichis worth
our whileelucidating.For example,Tilman& Wedin
(1991) show how the inhibitory
effectof plant litter
on theperennialgrassAgrostisscabra,is a mechanism
that mightexplain the plant's observedoscillations
and the sporadicbut pronouncedcrashesin growth.
A nonlinear(chaotic)model of theplant-litter
interactionservedto cast lighton thedynamicseven further.
If a data set is found to be governedlargelyby
deterministic
chaos, key dynamicparametersof the
systemunderstudyshouldbe extractable,
and at least
in principle,reasonablygood short-term
futurepredictionsare possible(Sugihara& May 1990).Because
oftheintrinsic
thequalityofpredictions
determinism,
should be betterthan those providedby traditional
timeseriestechniques.Thus theabilityto discriminate
betweendeterministic
signaland noise is not merely
of academicinterest,
forit should open the doors to
numerouspracticalapplications.
If thesignalis trulychaoticand of low-dimension
thenthe theorystatesthat only a small numberof
equationsare requiredto modelthesystemprecisely.
With thismotivation,Nicolis & Nicolis (1984) and
Tsonis & Elsner(1988) attemptedto analyseseveral
importanttimeseriesof climaticdata. Theyclaimed
to identify
intrinsic
deterministic
withinthe
properties
data set that should allow the salientfeaturesof a
climate systemto be describedby a simple model
involvingonlyfourto eightcoupled ordinarydifferential equations. These are remarkable findings
which, if true, should have the potential to revolutionizethe whole scienceof meteorology.Unforboththemethodsand resultsofthesestudies
tunately,
are controversial
and accordingto some evaluations,
should at thisstage be consideredspeculative(Procaccia 1988).
The possibilitythat the climatesignal is chaotic
wouldcertainly
haveecologicalramifications.
The climate systemmightbe viewedas a forcingfunction
thateffectively
acts to drivemanyecosystems.Since
thereis nowgrowingevidencethatat leastsomemajor
climatesignalsare chaotic (e.g. the El-Nifiomechanism;Tzipermanetal. 1994),itwouldtherefore
seem
reasonablethatthesesignalsmanifestthemselves
in a
chaoticwayat theecosystemlevelas well.More than
that,one cannotignoretheknownecosystem-climate
feedbackloop, whichmakes the separationof cause
fromeffect
sometimesimpossible.In thelightof this,
it mightbe wrongto referto the climatesignalas a
ifecosystemresponseplaysan equforcingfunction,
ally importantrole in shapingtheoverallecosystemclimate dynamics.This symbioticfeedback mechanismis, afterall, theessenceof theGaia hypothesis
and indeed,Lovelock's model of a daisyworldprovides a novel case of a chaotic ecosystem-climate
interaction
(Zeng et al. 1990).
Cyclesand chaosin plantcommunities
Therehavealreadybeena numberofseriousattempts
to utilize and incorporateideas from nonlinear
dynamicsand chaos theoryinto the realm of plant
ecology.One of the mostvisual and impressivefield
studiesof cyclesand potentialchaos can be foundin
Symonideset al. (1986). They provide an example
of a naturalplantpopulation,Erophilaverna,which
exhibits two year cycles in the field, apparently
because of its over-compensating
nonlineardensity
dependence in fecundity.The cycles were of an
extremenature,jumping betweendensitiesof 1-2
individualsand 55-65 individualsper 0.01 m2 (very
similarto Fig. Sd), but neverthelesspredictedby a
simple, empiricallyderived,iterativemodel qualitativelysimilar to eqn 1. Germinationconditions
tendedto promotethemodel'spropensity
forcycling
in a mannerthatconformswiththeperiod-doubling
route to chaos (see Fig. 5 and discussion below).
levelswerefoundto be of
Indeed,local germination
sufficient
magnitudeto inducechaoticdynamics.
Silander& Pacala (1990) and Thrall et al. (1989)
have presenteda numberof valuable studieswhich
attemptto determinethe conditionsfor oscillatory
and chaoticdynamicsin annual plants.Such behaviour, theyargued,would be more likelyto arise in
annualsthathave low seed dormancy,highseed survivorship,minimumplant-sizethresholdsfor seed
production,largefruitedindividualswithmanylarge
seeds, or grow in locations withhigh soil fertility.
Theirmodelsdemonstrate
a rangeofdynamicbehaviour for annual plants from stable positive equilibriumwithdamped oscillations,to sustainedoscillations,to apparentchaos. One ofthemoreinteresting
findingsof theirdetailedmodellingexerciseis that
seed dormancy(and itsassociatedtimelags) acts as a
controlon chaos, tendingto cause a plantpopulation
whichwould otherwiseappear oscillatory,to have a
locally stable equilibrium.Nevertheless,they concludethat'contrary
to Watkinson(1980) and Crawley
& May (1987), oscillatorydynamicsmay be a com-
285
L. Stone&
S. Ezrati
(a1)
1000
(b)
0.5%
6000
25
/s20
15,
Ca 10 I
0
6
11
16
21
26
31 36
Seedling
41
46
51
1
56
3
5
4
Generation
(c)
(d)
6000
%
sooo0
so
S
Fig.5 Symonides et al. (1986)
graphicalmodel of E. vernaexam-
40
4000
o
'~
X
3000
Q 2000
w2000
o
0
fi
20 /
\
6
_____________.__,__,__,__,
1
11 16 21 26 31 36 41 46 51 56
Seedling densityj
expressedas N,whereis thefraction
ofseedsthatsuccessfully
germinate.
2
4
Generation
The returnmaps on the left(a,c,e)
are obtainedfromplottingN,+, vs.
N, (i.e. seeds/plot vs. seedling
density).The point where the 45
(f)
degreelinemeetsthereturn
map,
I
90
80
Sooo0
-20004
\
ation.The lattervariablecan be
(e)
CL 4000
inesseedsperplotinthet+ 1thgeneration(i.e. N,+ ) as a functionof
seedlingdensityin the tth gener-
/
\ /
10
6000-
XM3000-
\
30/
/
1000
/
~
f
o 90
\
/
2%
\
>
>
\
70
/
60
Xe 40
30 1\/
\
/
20
1000^
o0
c
0 6 11 16 21 26 31 36 41 46 51 56
Seedling
? 1996 British
Ecological Society,
JournalofEcology,
84, 279-291
2
density
densitg
10
0
1
2
3 4 5
Generation
mon featureof some annual plants' (Silander &
Pacala 1990).
The oscillatorynatureof populationsof the perwas examinedin a nonennialherb Violafimbriatula
linearmodel by Solbriget al. (1988), whichallowed
sizedseedbankand
forthepossibility
ofa realistically
made use of fielddata forcalibratingall parameters.
Because of the combinedeffectof nonlineardensity
dependentseedlingsurvivaland theincorporationof
a timelag forseedlingsas theygrowinto adults,the
model population exhibitedoscillatoryratherthan
equilibriumbehaviour.AlthoughSolbriget al. (1988)
do not discuss the possibilityof chaos, theirpredictions concerningthe widespread occurrenceof
these oscillationsare of obvious relevance:'If the
greatereffectof densityon seedlingand youngage
classes, which has been documented repeatedly,
resultsin the observedoscillationsand if we have
simulateddensity,thentheoscillationsthat
correctly
we record should be the rule in equilibriumplant
populationsand we should be able to observethem
innature.'That oscillationsmightbe theruleforsome
plant communitiesis a predictionthat breaks with
indicatestheequilibrium
=
N,+, N,
\
which
is only stable in model (a).
in(b,d,f)arederived
Thedynamics
byiterating
therespective
graphical
returnmap for five generations.
= 5%; (c,d)
(a,b) germination
germination=1%; (e,f) germin=, 2%.
*ation
5l
, , <Reprinted
with per1,
_,
,,o.
missionfromSymonideset al.,
Oecologia 71, 156-158.
historicaltradition.The nextstepin theprocessis to
of the prevalenceof these
gain some understanding
oscillationsin thefieldand whethertheyreallyare of
mechanisticorigins.
itis feasiblethatnonlinear
As indicatedpreviously,
dynamicsand chaos may arise fromthe couplingof
two different
processes.That is to say,two processes
chaotic
may not be intrinsically
whichby themselves
or oscillatory,could well be so if coupled together.
Tilman & Wedin's (1991) fiveyearstudyof the perennialgrassAgrostisscabra,forexample,showedthat
oscillationsand chaos could resultfromthe interactionbetweenplantgrowthand theirlitter.The timedelayed inhibitoryeffectof plant litter, which
inhibitedthe
withsoil fertility,
increasedsignificantly
growthof plants in futuregenerations,and thus
caused oscillatorycrashes(sometimes6000-fold)in
biomass levels.These oscillatorygrowthphases and
crashes gave the appearance of chaos. To support
thisspeculationTilman& Wedin(1991) modelledthe
dynamicsof the inhibitoryplant-litterinteraction,
and the model indeed predictedchaos under high
regimes.
productivity
286
Chaos,cyclesand
spatiotemporal
dynamicsinplant
ecology
?
1996British
EcologicalSociety,
JournalofEcology,
84, 279-291
The classicexampleofthesprucebudwormsystem
providesanotherinteresting
case studyof how the
interconnectedness
of ecosystemcomponentsmay
lead to oscillatoryand chaotic dynamics.In these
systems,coniferousforestsare periodicallysubjected
to severebutinfrequent
(- every40 years)defoliation
by outbreaksof an insectpest,thesprucebudworm.
Holling(1973) detailedthedynamicsoftwoeast Canadian forestsin whichbalsam firis usuallythedominanttreespecies.It is also themostsusceptibleto the
destructive
sprucebudworm.Spruceand whitebirch
are less abundant,withtheformerless susceptibleto
thebudwormthanthefir,and thelatterneverbeing
attacked.Duringa budwormoutbreak,thereis major
destructionof Balsam fir(youngindividualsare less
damaged), and the spruceand whitebirchcome to
dominate. In between outbreaks the budworm is
exceedinglyrare,probablybeing limitedby natural
enemies,therebyallowingthe forestto recoverproducingstandsof matureand overmaturetreesof all
threespecies. If budwormoutbreaksdid not take
place, the balsam firwould competitively
exclude
dominatetheentire
spruceand birch,and ultimately
forest.Thus thebudwormoutbreaksare essentialfeaturesin maintaining
thepersistence
ofthisecosystem.
Whilethesporadicirruptions
of thebudwormmight
leave theimpressionthattheforestcommunity
is an
unstablesystem,paradoxicallyit is because of these
periodsofdefoliationthattheforestsystemhas enormous resilience.May (1977) has used a nonlinear
modelto explorethedynamicsofthebudwormpopulation showinghow such a systemmight,underthe
rightconditions,jump betweentwo alternative
stable
states.Despite the success of the model, therehave
beena numberof recentstudieswhicharguethat'the
existence[ofsuchalternative
states]. . . remainpoorly
researched'(Grover& Lawton 1994) and whichsuggestthatthesesortsof two-statemodelsmightneed
further
biologicalrefinement
(Royama 1992). NeverthelessMay's modelhas otherpointsof interestthat
are oftenoverlooked.For example,whenexamined
as a difference
equation, chaotic population oscillations(as opposedto multiplestablestates)are easily
sustained(Stone,L. personalobservation).
A morerecentdemonstration
oftheeasilyinduced
chaotic dynamicsin the growthof insectpests was
providedby Cavalieri& Kocak (1994) in theirstudy
of the European corn borer. In theirmodel simulations,Cavalieriand Kocak foundthatthedynamics
of thesepestscan entera chaotic statewithrelative
ease aftersmallchangesin parametersthatare affected by weather.When chaotic, the simulatedEuropean corn borer population reaches higherlevels
of abundance than in an equilibriumor (periodic)
oscillatoryscenario.The erraticchaoticbehaviourof
theirmodelhas manycharacteristics
in commonwith
real field-datathathas been gatheredforthesepests.
The effects
ofchaos mayariseat a varietyofdifferentecologicaland biologicalscalesand notjust at the
populationor community
level.Luttge& Beck (1992)
demonstrate
how theendogenousrhythm
ofthephotosynthetic
processitselfmightjump to a chaoticstate
in certainenvironmental
scenarios.They examined
net CO2 exchangein the Crassulacean-acid-metabolism(CAM) plant,Kalanchoedaigremontiana
under
laboratory conditions. When irradiance or temperaturewereraised above a criticallevel,irregular
oscillationswerenoted.In an attemptto understand
the cause of the unusual dynamics,Luttge& Beck
modelledtheCAM networkusinga systemofcoupled
nonlineardifferential
equations.In keepingwiththe
experimental
results,thesimulationmodel showeda
similartransitionto oscillatorybehaviourand then
to chaos as irradiancewas increased.The modelalso
displayeda verysharptransition
fromorderto chaos
when ambienttemperature
was increasedby only a
fewtenthsof a degree.The CAM cycle,withits two
different
routes to chaos (via temperatureand via
irradiance),was explainedas an outcomeoftheinteractionoftwointrinsic
stableoscillationsin theinfluxefflux
dynamicsoftheplantvacuole in a mannerthat
operates ratherlike the physicist's'double-pendulum.'
Methodsto detectchaos
All of theseexamplesindicatethe possible presence
of chaoticdynamicsin real plantsystems.However,
provingunambiguouslythat this is the case, is a
difficultif not impossiblemethodologicaltask. A
numberof ingeniousmethodshave been put forward
as testsforchaos, but as yet,no techniquehas been
devisedthatprovidesincontestable
proof.We briefly
outlinefourpopular methods,all usefulforprobing
forchaos and nonlinearity
in ecosystemanalyses,and
- thesmallsample
thendiscusstheirmajorlimitation
size of fieldstudies.
DETERMINING
THE
'RETURN
MAP'
There is a verysimpleyetreasonabletool forecologistswho suspectthe presenceof a density-dependent relation in their data sets of the form:
N, I = F(N,) (suchas foundineqn 1). It requiresonly
plottingthe variablesN,+I vs. N,. Any stronglinear
or nonlinearrelationshiparisingfromthe potential
presenceof the functionF, should be immediately
apparentto the eye. Once the formof the so called
'returnmap' F is identified,
thenitis possibleto probe
further
formechanismsthatproduce oscillationsor
chaos.
In Fig. 5 we reproducethe work of Symonideset
al. (1986) who plottedthetotalviableseedproduction
per plot (y-axis) against initialseedlingdensity(xaxis). There are 11lpoints in total whichsuggesta
'humped'density-dependent
curveforthefunctionF.
Iteratingthismap leads to thedynamicsdisplayedin
(b) (d), and (f) whichrepresentgermination
ratesof
287
L. Stone&
S. Ezrati
Note how theperiod
0.5%, 1%, and 2%, respectively.
doublingroute to chaos is observedwithcycles of
period1,2 and possibly4. The actual plantdynamics
verycloselyresembledthatdisplayedin (d), with1%
germinationrate althoughrates of up to 3% were
thereturn
observedin some fieldplots.Interestingly,
map in Fig. 5 suggeststhat levelsof germinationof
2% (possibly even less), might produce chaotic
dynamics.
CALCULATING
THE
LYAPUNOV
EXPONENT
A numberof techniqueshave emergedwhich are
designed to estimatethe Lyapunov exponent(see
above) ofa timeseries(Wolfet al. 1985;Ellner1991).
If the Lyapunov exponentproves positive,then it
seems reasonableto take this as evidenceof chaos.
These techniquestake advantageof the factthatfor
a chaotic system,trajectorieswithverysimilar(but
not thesame) initialconditionsshoulddivergeexponentiallyat a rate determinedby the Lyapunov
exponent.Withtheaid of a computerit is possibleto
subdividea long timeseriesinto a large numberof
smallsegments.The computerthenisolatessegments
whichare nearbyand determineswhethertheirtrajectoriesdivergein time fromone anotherexponentially.If so, it is possible to calculate the rate of
separationand so obtaintheLyapunovexponent.
NONLINEAR
? 1996British
Ecological Society,
JournalofEcology,
84, 279-291
PREDICTION
Nonlinearpredictionmethodsattemptto probe for
in time series.The undershort-term
predictability
chaos shouldhave
lyingidea is that,low-dimensional
bettershort-term
than a randompropredictability
cess (Farmer & Sidorowich1987; Sugihara & May
1990).The techniqueis oftensuccessfulevenifa chaotic data set is contaminatedby noise. To sketchone
commonmethod,supposethetimeseriesunderanalysis consistsof N data points{y,}. For each pointy,in
the time series,a computeralgorithmattemptsto
predictthe followingpoint y+, . The predictionso
obtained,is thencomparedto theactualknownvalue
e,,can be deduced.
yi, and theerroroftheprediction,
The endproductis thusa setofforecastsforone timeas wellas errorterms{ei}.
step(T = 1) intothefuture,
r betweenthe forecasts
The correlationcoefficient
and theobservedvalues maythenbe computed.The
whole process is repeatedforpredictionstwo steps
(T = 2) intothefutureand thenforT = 3,4... k steps
into the future(wherek is some predefinedinteger)
so thatthescalingofthecorrelationcoefficient
r,and
theerrortermscan be examined.
to initialconIf thereis some formof sensitivity
ditions then predictionerrors should grow exponentiallywithT (thenumberofstepsforecastintothe
r should fall
future),whilethecorrelationcoefficient
dramatically.For a stochasticprocess,on the other
hand,thecorrelationcoefficient
and predictionerrors
to thispattern.It is these
will scale verydifferently
different
scalingpatternsthatallowthedifferentiation
of chaos fromnoise. This is essentiallytheargument
used by Sugihara& May (1990) whentheyattempted
to provideevidenceforchaoticdynamicsin real epibloomtime
demiologicaldata setsand phytoplankton
series (see also Tsonis & Elsner 1992). In practice,
themethodcan becomeproblematicalsincecomplex
noise sourcesalso appear to have severalof the sigto chaos,and arecapable offooling
naturesattributed
the best testsavailable (see e.g. Ellner 1991; Stone
ifthe
1992). The problemis exacerbatedevenfurther
timeseriesunderanalysisis poorlysampledor lacksa
sufficient
numberofdata points,as we discussshortly.
SURROGATE
DATA
TESTS
FOR NONLINEARITY
The methodofsurrogatedata,as conceivedbyTheiler
ifnot
thedifficulty,
et al. (1992), startsbyrecognizing
chaotic
of convincingly
demonstrating
impossibility,
dynamicsby any of the methodssuggestedto date.
Instead Theiler et al. are less ambitious,and only
attemptto seek out strongevidence for the nonlinearityof a timeseries.Whilethisnecessarilyleads
abouttheunderlying
timeseries
to a weakerstatement
dynamics,it has the advantage of providingstatresultsthatcannotbe misleading.
isticallysignificant
A null hypothesisis firstposited statingthat the
observedtimeserieshas been generatedfromsome
typeoflinearGaussianprocess(LGP) and is therefore
random.One thenteststhe null hypothesisby comparingthe observedtimeseriesto a largenumberof
surrogatetimeseriesthatare knownto be 'random'.
Each of the surrogatetimeseriesis generatedon the
computerbya trueLGP randomprocess,and is thus
guaranteedto be randomby construction.As well,
each surrogatetimeseriesis generatedso thatit simultaneously inherits important qualities of the
specobservedtimeseries(e.g. an identicalfrequency
trum).(As a visual example,a typicalsurrogatefor
the actual sunspotdata in Fig. 1a mightbe Yule's
random autoregressivemodel in Fig. lb. Not only
does the surrogateappear almost identicalto the
observeddata set,butithas thesame frequency
spectrum.)The nullhypothesisis thentestedby checking
whetherthe observedtimeserieshas identicalstatisticalcharacteristics
to the set of proxytimeseries
whichare trulynull. If nothingexceptionalis found,
thenull hypothesisthatthe observedtimeseriesis a
randomprocesscannotbe rejected.
factor
Samplesize - thelimiting
The inescapable problem with all of the above
methodsis thatthenumberofdata pointsin thetime
seriesunderanalysismustbe verylarge forreliable
assessments.Clearlya 10-yeartimeseriescomprising
of plant abundance would
10 annual measurements
tell us very littleabout plant dynamics.Indeed it
288
Chaos,cyclesand
spatiotemporal
dynamicsinplant
ecology
would be extremelydifficult
to deduce whetherthe
data displayeda lineartrendor was insteadcyclical.
It would be harderstillto decidewhethertheannual
fluctuations
in thedata setwerechaotic.
Eckmann& Ruelle (1992), both highlyrespected
theoreticalphysicists,have argued that there will
always be fundamentallimitationsfor estimating
dimensionsand Lyapunov exponents.Accordingto
theircalculations,the accurateestimationof a Lyapunov exponentin a time series requiresapproximatelyN = 1000000 data points in a systemwith
dimension(d = 5), and more than N = 1000 data
pointsford = 3. Such data sets are trulylarge,and
we knowof no ecologicaltimeseriesthatcomesclose
to theselengths.Most ecologistswould treata time
seriesof 100 data pointssimplyas a treasure.
Despite thedepressingassessmentsof Eckmann&
Ruelle (1992), testson knownchaotic and random
data sets,have shownthatin practiceall oftheabove
methodsare reasonablyfaithful
fortimeseriesof the
order of 500 data points. Since severalnoteworthy
ecologicaltimeseries(usuallyepidemiological)ofthis
size are available, theyhave made interesting
testcases. Althoughclaimshave been made formethods
withfarless stringent
data requirements,
we feelthat
manyoftheseclaimsshouldbe treatedagainsta backgroundof caution.
In all,-thisis particularly
bad newsforplantecologists whose data sets usually consist of annual
measurements
takentypicallyovera fewyears.Some
ofthebest'long-term'
data setsonlycompriseofsome
40-60 measurements(e.g. Watt 1981; Dodd et al.
1995), which immediatelypoints to where future
researchdirectionsmightlie both in fieldwork and
methodologicalareas. For the former,we presume
thatGIS (geographicalinformation
systems)and satelliteremotesensingwill provideverylarge,quality
data setsin thenot too distantfuture.
Spatiotemporal
dynamics
? 1996British
EcologicalSociety,
JournalofEcology,
84, 279-291
Until now we have focusedon a varietyof ways in
whichplantpopulationsmightfluctuatein time.But
thisin itselfrevealsonlya fractionof thewholeecologicalpicture,forin thestudyofplantcommunities,
interestalso centreson patternsthatoccur over the
entirespatiallandscape.Traditionhas itthatthespaare fairly
tiotemporaldynamicsofplantcommunities
simple,and largelyaccord withthe classical climax
theoryin whicha fixedand predictablecommunity
structure
determined
will
bycompetitive
equilibrium,
alwayseventuallybe attained(Clements1936). After
a disturbancethe plant communityis predictedto
pass througha seriesof successionalstates,each with
differing
speciescomposition,untilthe climaxequilibriumstateis reached.Watt (1947), however,challengedtheclimaxtheory,arguingthatmoreemphasis
needs to be placed on the highlydynamicnatureof
plant communitiesratherthan focusingon abstract
notionsof stableequilibria(see e.g. Connell& Sousa
1983).Accordingto Watt,thecommunity
is not spatiallyhomogeneousand is bestunderstoodby breakingitdownto a setof localizedpatches.The patches,
or phases as Watt sometimestermsthem,consistof
aggregatesof individualsand of species,and change
dynamicallyoftencyclingin a progressionof states
(e.g. pioneer-+ building-+ mature-+ degenerate).
They'forma mosaicand together
constitute
thecommunity... thepatchesare dynamically
relatedto each
other.Out of this arises that orderlychange which
accountsforthepersistence
ofthepatternin theplant
community.'
Watt describesseven communitieswhich never
appear to reach an equilibriumclimaxstate;instead
each showsunusualcyclicaldevelopmentamongstits
constituent
mosaic of patches.The patchescyclein a
regular pattern affectingadjacent patches in a
dynamicway thusforminga 'resultantregeneration
complexas a community
ofdiversephases[orpatches]
forminga space-timepattern.Although there is
changein timeat a givenplace,thewholecommunity
remainsessentiallythe same; the thingthatpersists
in the
unchangedis the process and manifestation
sequenceof phases.' In contrast,a singlepatchitself
is not stableoverlong periodsof time.Furthermore,
because of the cyclicalprocessof the patches,Watt
shows that it is sometimespossible to observethe
ofchangeat a givenpatch,recordedin
time-sequence
thespatialdistribution.
Sprugel (1976) and Sprugel & Bormann (1981)
foundthe mosaic cycleconceptveryusefulin their
studyof Abies balsamea (balsam fir)forestsin the
north-eastern
UnitedStates.The wind-induced
cycles
of death, regeneration,and maturationappear to
move constantlythroughtheseforestspredictablyat
intervalsof some 60 years. These repeatingspatial
waves(Fig. 6) are difficult
to explainbyusual notions
ofclimaxequilibrium
butreadilyconformwithWatt's
theoryofphases.Sprugelhypothesized
thatprevailing
regenerating intermediate
Mature
Wind
crosssectionthrough
a regeneration
wave
Fig.6 A typical
in thebalsamfirforests
described
bySprugel& Bormann
(1981).The wavepattern
dynamics
leavesbandsof dead
treesinterspersed
withbandsof regenerating
and mature
firs.Reprinted
withpermission
fromSprugel& Bormann
(1981),Science
211, 390-393.Copyright
1981American
Association
fortheAdvancement
ofScience.
289
L. Stone &
S. Ezrati
(C 1996British
Ecological Society,
JournalofEcology,
84, 279-291
wind disturbancesincreasedeath amongstexposed
trees located at the wave-edge.Because balsam fir
forestsvigorouslyregenerate,
stripsof dead treesare
rapidlyreplaced by seedlingsand young firssoon
appear. The wave-edgemarchesforwardsas thewind
disturbanceaffectsa newlyexposed layerof mature
firs.A spatialwave patternthusemerges,withstrips
ofdead treesfoundat regulardistances,and withthe
waveslowlymovingthroughtheforestinthedirection
ofthewind.Note thatthetime-sequence
in growthis
evidentin the spatial-distribution
(see Fig. 6), just as
predictedby Watt'stheory.
AlthoughWatt's ideas largelywentforgotten,
in
recentyears therehas been a revivalof interestin
the patch-mosaicecosystemconcepthe put forward
(Remmert1991).Mathematicaland computermodels
have been foundto be particularly
usefulforexaminingpatchdynamicsin ecosystemsforwhichspecies
and resourcesinteractlocally accordingto a set of
verysimplerules (Rand & Wilson 1995; Hendry&
McGlade 1995). The resultingand changingstructuresoftenlook haphazardand random,and seemto
lack any organizationalqualitieswhatsoeverto the
naked eye. However,subjectto mathematicalanalysis, thesemodel ecosystemsshow surprising
internal
structureand organizationbecause the simplerules
they are derived from may sometimesyield spatiotemporalchaos, i.e. a formof chaos that carries
over into both space and time.The new techniques
available forexaminingspatiotemporal
chaos (Rand
1994; Rand & Wilson 1995) are extremelyexciting
fortheyprovidea way of probingfordeterministic
in spatiallysampledecosystems;
patternand structure
patternsthat would most likelygo undetectedwith
moreusual statisticalapproaches.Suchmethodshave
greatpotentialin ecosystemanalyses,especiallygiven
thenew shifttowardsremotesensingsurveysby satelliteand othermeans.
Spatial modellinghas also led to thedevelopment
of the conceptof metapopulationdynamics.A metapopulationconsistsof a set of local subpopulations
locatedovera largespatialterrainthatare linkedby
dispersal.In his book The Diversityof Life,Wilson
(1992) providesa veryeloquentdescriptionof metapopulationdynamics:
'Watchedacross longstretches
of time,the
speciesas metapopulationcan be thoughtof
as a sea of lightswinkingon and offacrossa
darkterrain.Each lightis a livingpopulation.
Its locationrepresents
a habitatcapable of
supportingthespecies.Whenthespeciesis
presentin thatlocationthelightis on, and
whenit is absentthelightis out. As we scan
theterrainovermanygenerations,
lightsgo
out as local extinctionoccurs,thencome on
again as colonistsfromlightedspotsreinvade
thesame localities.The lifeand deathof
speciescan thenbe viewedin a way that
invitesanalysisand measurement.
If a species
managesto turnon as manylightsas go out
fromgenerationto generation,it can persist
indefinitely.
Whenthelightswinkout faster
thantheyare turnedon, thespeciessinkto
oblivion.
The metapopulationconceptof species
existenceis cause forbothoptimismand
despair.Even whenspeciesare locally
extirpated,
theyoftencome back quickly,
providedthevacatedhabitatsare leftintact.
But iftheavailablehabitatsare reducedin
sufficient
number,the[speciescan disappear
forall time].A fewjealouslyguardedreserves
maynot be enough.Whenthenumberof
populationscapable of populatingemptysites
becomestoo small,theycannotachieve
colonizationelsewherebeforetheythemselves
go extinct.The systemspiralsdownwardout
of control,and theentiresea of lightsturns
dark.'
Currentlytheoreticalecologistsare attempting
to
understandthespeciesextinction
processusingmodellingapproachesto identifyprocesseswhichcause
systemsto spiralout of control(Tilman et al. 1994;
Stone 1995). Wilson's conceptualpictureof habitat
destructionis in some ways counter-intuitive
and
mightbe consideredto be at odds withnotionsof
linear systemstheory.In a 'linear world' it seems
reasonable to conjecturethat species populations
might drop proportionallyas habitat destruction
increases,butwithextinctions
neveroccurringunless
thehabitatis completelydestroyed.As theTilmanet
al. (1994) model of forestecosystemsmakesevident,
thedynamicsofspeciesextinctions
are farmorecomplicated.In a nonlinearworld,whenspeciesare dispersingacrossa landscapeand competingforpatches
on which to thrive,the spatiotemporalpictureis
extremely
complexand as recenttheoreticalworkis
sometimeschaoticin structure.
suggesting,
Discussion
The possibilitythat populationdynamicsmightbe
chaotichas radicallyshiftedtheemphasisin ecology
away fromitsfoundingcore principles.Ideas of ecosystemsthatsitat somepossiblymythicalstableequilibrium,or expressionsof faithin tenuouslinear(or
linearizations
of)ecosystemmodels,usuallylead only
to conclusionsthatare based in sciencefiction.Similarly,it would be unrealisticto believeunwaveringly
thatchaos is some sortof genericecosystembehaviour; forthe timebeingit is stillimpossibleto point
to one singlestudyin whichpopulationdynamicsare
unambiguously
chaotic.
Nevertheless,
theexampleswe havediscussedmake
clear the changingvision in ecology. Not only has
therebeenan increasedinterest
in exploringthemany
waysinwhichpopulationschangeovertime,butthere
290
Chaos, cyclesand
spatiotemporal
dynamicsinplant
ecology
has been a concertedeffortto also understandtheir
changesin spatialorganization.
Our reviewhas attemptedto synthesizethe case
way
fornonlineardynamicsin the mostenthusiastic
is an ubiquitous
possible. In our view,nonlinearity
featurein naturalsystemsmakingitnotunreasonable
to expectassociatedsignalsof oscillatingand chaotic
dynamics, especially in ecosystemsgoverned by
growthprocesses.However,it
strongdeterministic
possiwould be wrongto ignorethe straightforward
bilitythat much of nature'svariabilityarises from
random(possiblynonlinear)stochasticprocesses,or
in other words, is simplyan expressionof unpredictablechanceevents.To whatextentthelatterpossibilitypredominatesovertheformeris stillverymuch
an open question.
thata numberof the major
We findit interesting
themesdiscussedin thisreview,werepromoteddecades ago by earlypioneeringecologists,buthave not
sincethen.Thus Watt's
beendevelopedmuchfurther
(1947) researchon the 'dynamicconcept' and the
'mosaiccycle'inplantecology,largelywentforgotten.
Perhaps this only reflectsthe absence of adequate
modellingand statisticaltools requiredto examine
the ideas in the depth theydeserve.The theoryof
nonlineardynamicsand chaos should help fillthis
gap, sinceitprovidesa noveland excitingframework
withwhichto explorethe variabilityand change in
plant communitiesfrom both a theoreticaland
applied perspective.Success in many ways depends
ofecologiststo acceptthechallenge
on thewillingness
and welcomewithan open mindtheseusefulmathematicaltools.
Acknowledgements
We would like to thankDr Deborah Hart forcarefullyreviewingthe manuscriptand ProfessorMark
withthisproject.
Williamsonforhis encouragement
References
? 1996British
Ecological Society,
JournalofEcology,
84, 279-291
Antonovics,J. & Levin, D.A. (1980) The ecological and
regulationin
geneticconsequencesofdensity-dependent
plants.AnnualReviewof Ecologyand Systematics,11,
411-452.
Ascioti,F.A., Beltrami,E., Carrol,T.O. & Wirick,C. (1993)
JournalofPlankton
Is therechaos inplanktondynamics?
Research,15, 603-617.
Beltrami,E. & Carrol,T.O. (1994) Modelingtheroleofviral
blooms. Journalof
phytoplankton
disease in recurrent
MathematicalBiology,32, 857-863.
Cavalieri, L.F. & Kocak, H. (1994) Chaos in biological
control systems.Journalof TheoreticalBiology,169,
179-187.
Clements,F.E. (1936) Nature and structureof the climax.
JournalofEcology,24, 252-284.
Connell,J.H. & Sousa, W.P. (1983) On theevidenceneeded
to judge ecological stabilityor persistence.American
Naturalist,121,789-824.
Crawley,M.J. (1990) The populationdynamicsof plants.
oftheRoyal SocietyLondon,
PhilosophicalTransactions
B330, 125-140.
Crawley,M.J. & May, R.M. (1987) Population dynamics
and plant communitystructure:competitionbetween
annuals and perrenials.Journalof TheoreticalBiology,
125,475-489.
J.,McConway,K., Potts,J.& CrawDodd, M., Silvertown,
ley,M. (1995) Communitystability:a 60-yearrecordof
trendsand outbreaksin theoccurrenceof speciesin the
Park Grass Experiment.Journalof Ecology,83, 277285.
Eckman,J.-P.& Ruelle,D. (1992) Fundamentallimitations
forestimatingdimensionsand Lyapunovexponentsin
dynamicalsystems.PhysicaD, 56, 185-187.
chaos in popuEllner,S. (1991) Detectinglow-dimensional
lationdynamicsdata: a criticalreview.Chaos and Insect
Ecology(eds J.Logan & F. Hain), pp. 65-92. University
VA.
of VirginiaPress,Charlottesville,
Farmer,J.D. & Sidorowich,J.J.(1987) Predictingchaotic
timeseries.PhysicalReviewLetters,59, 845-848.
conceptoftheplant
Gleason,H.A. (1926) The individualistic
association.BulletinoftheTorreyBotanicalClub,53, 726.
studieson
Grover,J.P.& Lawton,J.H. (1994) Experimental
communityconvergenceand alternativestable states:
commentson a paperby Drake et al. JournalofAnimal
Ecology,63, 484-487.
Hassell, M.P., Lawton,J.H. & May, R.M. (1976) Patterns
of dynamicalbehaviour in single species population
models.JournalofAnimalEcology,60, 471-482.
Hastings,A., Hom, C.L., Ellner,S., Turchin,P. & Godfray,
H.C.J. (1993) Chaos in ecology: is mothernature a
strangeattractor?AnnualReviewof Ecologyand Sys24, 1-33.
tematics,
Hendry,R.J. & McGlade, J.M. (1995) The role of memory
in ecologicalsystems.Proceedingsof theRoyal Society
ofLondon,B259, 153-159.
Holling,C.S. (1973) Resilienceand stabilityof ecological
4,
systems.AnnualReviewof Ecologyand Systematics,
1-23.
Hubbell, S.P., Condit, R. & Foster,R.B. (1990) Presence
and absenceof densitydependencein aGneotropicaltree
community.PhilosophicalTransactionsof the Royal
SocietyofLondon,B330, 269-281.
rhythms
Inghe,0. (1990) Computersimulationsofflowering
in perrenials- is therea newarea to explorein thequest
forchaos?Journalof Theoretical
Biology,147,449-469.
Lomnicki,A. & Symonides,E. (1990) Limitedgrowthof
individualswithvariablegrowthrates.AmericanNaturalist,136,712-714.
and chaos
Luttge,U. & Beck,F. (1992) Endogenousrhythms
in crassulceanacid metabolism.Planta,188,28-38.
May, R.M. (1976) Simplemathematicalmodels withvery
complicateddynamics.Nature,261, 459-467.
inecosystems
May, R.M. (1977) Thresholdsand breakpoints
of stable states.Nature,269, 471witha multiplicity
477.
May, R.M. (1986) The Croonian Lecture,1985: Whentwo
and two do not make four: nonlinearphenomenain
ecology. Proceedingsof the Royal Societyof London,
B228, 241-266.
Nicolis,C. & Nicolis,G. (1984) Is therea climaticattractor?
Nature,311, 529-532.
Pollingher,U. & Zemel,E. (1981) In situand experimental
evidenceof the influenceof turbulenceon cell division
processes of Peridiniumcinctumfa. westii.(Lemm.).
Lefevre.Brittish
PhycologicalJournal,16, 281-287.
Procaccia, I. (1988) Complex or just complicated?Nature,
38, 121.
Rand, D.A. (1994) Measuring and characterizingspatial
patterns,dynamicsand chaos in spatially extended
291
L. Stone&
S. Ezrati
? 1996 British
Ecological Society,
JournalofEcology,
84, 279-291
dynamicalsystemsand ecologies.PhilosophicalTransactionsof theRoyal SocietyofLondon,A348,497-514.
Rand, D.A. & Wilson,H.B. (1995) Using spatio-temporal
chaos and intermediate-scale
determinism
to quantify
spatiallyextendedecosystems.Proceedingsof theRoyal
SocietyofLondon,B259, 111-117.
H. (ed.) (1991) TheMosaic-CycleConceptofEcoRemmert,
systems.New York. SpringerEcological Studies85.
Room, P.M. & Julien,M.H. (1994) Population-biomass
dynamicsand the presenceof -3/2 self-thinning
in the
clonal weed Salvinamolesta.AustralianJournalofEcology,19, 26-34.
Royama,T. (1992) AnalyticPopulationDynamics.Chapman
and Hall.
Silander,J.A.& Pacala, S.W. (1990) The applicationofplant
population dynamic models to understandingplant
competition.
inPlantCompetition
Perspectives
(eds J.B.
Grace & D. Tilman),pp. 67-91. AcademicPress. San
Diego.
Sinclair,A.R.E., Gosline,J.M.,Holdsworth,G., Krebs,C.J.,
Boutin, S., Smith,J.N.M., Boonstra,R. & Dale, M.
(1993) Can the solar cycleand climatesynchronise
the
snowshoe hare cycle in Canada? Evidence fromtree
ringsand ice cores.AmericanNaturalist,141, 173-198.
Solbrig,O.T., Sarandon,R. & Bossert,W. (1988) A densitydependentgrowthmodelof a perennialherb,Violafimbriatula.AmericanNaturalist,131,385-400.
Sprugel, D.G. (1976) Dynamic structureof wave-regenerated Abies balsamea forestsin the north-eastern
UnitedStates.JournalofEcology,64, 889-911.
Sprugel,D.G. & Bormann,F.H. (1981) Naturaldisturbance
and the steadystatein high-altitude
balsam firforests.
Science,211, 390-393.
Stone, L. (1992) Colored noise or low-dimensional
chaos?
Proceedingsof theRoyal Societyof London,B250, 7781.
Stone, L. (1993) Period doubling reversalsand chaos in
simpleecologicalmodels.Nature,365, 617-620.
Stone, L. (1995) Biodiversityand habitat destruction:A
comparativestudyof model forestand coral reefecosystems.Proceedingsof the Royal Societyof London,
B261, 381-388.
Sugihara,G. & May, R.M. (1990) Nonlinearforecasting
as
a way of distinguishing
chaos frommeasurement
error
in timeseries.Nature,344, 734-741.
Symonides,E., Silvertown,
J.& Andreasen,V. (1986) Popu-
lation cycles caused by overcompensatingdensitydependencein an annual plant.Oecologia,71, 156-158.
Theiler, J., Eubank, S., Longtin,A., Galdrikian, B. &
Farmer,J.D. (1992) Testingfor nonlinearityin time
seriesdata: The methodof surrogatedata. PhysicaD,
58, 77-94.
Thrall,P.H., Pacala, S.W. & Silander,J.A. Jr(1989) Oscillatorydynamicsin populationsof an annualweed species Abutilontheophrasti.
Journalof Ecology,77, 11351149.
Tilman,D. (1994) Competitionand biodiversity
in spatially
structured
habitats.Ecology,75, 2-16.
Tilman, D., May, R.M., Lehman, C.L. & Nowak, M.A.
(1994) Habitat destructionand the extinctiondebt.
Nature,371, 65-66.
Tilman,D. & Wedin,D. (1991) Oscillationsand chaos in
thedynamicsofa perennialgrass.Nature,353,653-655.
Tsonis,A. & Elsner,J.B. (1988) The weatherattractorover
veryshorttimescales.Nature,333, 545-547.
Tsonis, A. & Elsner,J.B. (1992) Nonlinearpredictionas a
way of distinguishingchaos from random fractal
sequences.Nature,358, 317-220.
Tziperman,E., Stone,L. & Cane, M. (1994) El-NinoChaos.
Science,264, 72-74.
Watkinson, A.R. (1980) Density-dependencein singlespecies populations of plants. Journalof Theoretical
Biology,83, 345-357.
Watt, A.S. (1947) Pattern and process in the plant
community.
JournalofEcology,35, 1-22.
Watt, A.S. (1981) Furtherobservationson the effectsof
excludingrabbits fromgrasslandA in East Anglian
breckland:The patternof changeand factorsaffecting
it (1936-73). JournalofEcology,69, 509-536.
Wilson,E.O. (1992) TheDiversityof Life. BelknapPressof
HarvardUniversity
Press.Cambridge,Massachusetts.
Wolf,A., Swift,J.B.,Swinney,H.L. & Vastano,J.A.(1985)
DeterminingLyapunov exponentsfroma time series.
PhysicaD, 16, 285-317.
Yule, G.U. (1927) On a methodofinvestigating
periodicities
in disturbedseries,with special referenceto Wolfer's
sunspot numbers. Philosophical Transactionsof the
Royal Society,A226, 267-298.
Zeng,X., Pielke,R.A. & Eykholt,R. (1990) Chaos in daisyworld.Tellus,42B, 309-318.
Received19 June1995
revisedversionaccepted20 November1995