OIKOS 113:497—505,2006
Canpreybehaviourinducespatiallysynchronicaggregationof solitarypredators?
VincenzoPenteriani, Miguel A.Fortuna, Carlos J.Melia´n,Ferm´ın Otalora andMiguel Ferrer
Penteriani, V.,Fortuna,M.A.,Melia´n,C.J.,Otalora, F.andFerrer, M.2006.Canprey behaviour inducespatiallysynchronic aggregation ofsolitarypredators? — Oikos113:
497—505.
Spatio-temporalpatterns ofpredator aggregations withintheirsettlementareas(i.e. temporarysettling zonesusedduring dispersal or, more generally, foraging patches) were studied.ByintegratingthemainbehaviouralrulesofjuvenilesofSpanishimperial eaglesAquilaadalberti during dispersal withthe temporal availability oftheeagles’ main prey, we have developed several individual-based models under different simulation scenarios. The results suggest that the spatially synchrony of the aggregation patterns that we observed were derived from a combination of behavioural responses of the prey and the distances between available settlement areas.Theseresultsarediscussedinthecontextofpredator—preyrelationships,optimal foraging andself-organizingprocesses.
V. Penteriani, Dept ofConservation Biology,Estacio´n Biolo´gica deDon˜ana, CSIC, Pabello´ndelPeru´, Avda.Mar´ıa Luisa s/n,Seville,Spain(). — M.A.Fortuna,IntegrativeEcologyGroup,Estacio´nBiolo´gicadeDon˜ana,CSIC,Seville, Spain. — C.J.Melia´n,NationalCenterforEcologicalAnalysisandSynthesis,735State St., Suite 300,Santa Barbara, CA93101-5504,USA. — F. Otalora, Laboratorio de EstudiosCristalogra´ficos,IACT(CSIC/Univ. ofGranada),FacultaddeCiencias,Spain.
— M.Ferrer, DeptofBiodiversityConservation,Estacio´nBiolo´gicadeDon˜ana,CSIC, Pabello´ndelPeru´,Avda.Mar´ıa Luisas/n,Seville,Spain.
Thequestionssurroundingpattern formation arecon- sideredtobesomeofthemostchallenginginmanyfields ofscience,withthecausesofheterogeneity inthespatial distributionofsystemcomponents still beingobscure (Petrovskii andMalchow 1999).Animalsdisplayawide rangeofcoordinated behaviours, asforexamplewhen foraging.Relationshipsbetweenpredators (forbetter hunting) and prey(forbetter predatordeterrence) represent typical examples (Rosenzweig et al. 1997, Bahr and Bekoff 1999, Brown et al. 2001). In this context,theresultssuggestthatpredatorsmayinfluence thebehaviour andspatialpatterns ofpreys(Lagosetal.
1995,BrownandKotler 2004,Yunger2004).Whenthis occurs,whataretheconsequencestothe behaviourand spatial arrangement of predators? Most studies of
predator—preyinteractions havefocusedonpreybeha- viour (Abramsky etal.1996,Lima 2002,Nelson etal.
2004),leadingtoanincompleteview ofpossible behavioural and ecological interactions. In addition, there isalackofknowledge about thewayinwhicha prey’sresponsetopredationpressuremayinturn affect thebehaviour ofthepredator.
Thestudyoftherelationships betweenpredators and
prey,aswellaspatchandpreymodels,isoneofthemost active fields of behavioural ecology. Over the past decade, the study ofanimal foraging has been guided by the awareness that simple models are often inade- quatetoprovideuseful predictions.Stochasticdynamic modelling and individual-based models wereidentified asusefultoolstoapproachcomplex,real-lifesituations
Accepted28October 2005
SubjectEditor: VeijoKaitala
Copyright #OIKOS 2006
ISSN0030-1299
(PerryandPianka 1997;butseeBart1995).Inthepast, attemptstoreconcileempiricists’ andtheoreticians’ approaches toanimalforaginghavebeenlargelyun- successful(Kareiva1989). Toobtainabetterunder- standingofbehavioural, ecologicalandevolutionary components of animal foraging it is necessary to combine both observational and theoretical work. In the past, predator—prey simulations have generally neglectedtheincorporation ofpreybehaviour under predation pressure (Giraldeau and Caraco 2000)and onlyrecently somechanges haveoccurred inthetreat- mentofthissubject(Lima2002).Therefore, itiscrucial tounderstandhowpredatoraggregations areinfluenced byvariations inthespatialscaleunderwhichthesystem evolves. Especially when aggregations are not deter- minedbysocialinteractions andforaging ingroups per se does not increase individual foraging efficiency (Valone1989,Beauchamp 1998).Infact,groupforaging behaviourwas mainlyanalysedinthecontextofsocial influencesonforaging(Giraldeau andBeauchamp 1999, GalefandGiraldeau2001,Dubois andGiraldeau2003) andtheeffectofgroupsizeontheefficiencyofresource exploitation (Be´lisle1998,RitaandRanta 1999,Coolen
2002).
Thepresentsimulationstudymadeuse ofseveral individual-basedmodelsbuiltonwell-studiedaspectsof thebehaviour ofradio-tagged juvenilesofSpanish imperial eaglesAquilaadalberti (Ferrer 1993a, 1993b,
2001). Our results leads us to propose that observed spatio-temporal patterns ofeagleaggregations within settlement areas (i.e. temporary settling zones used during dispersal or, more generally, foraging patches usedduringdispersal)areinducedby behavioural modifications oftheeagles’ mainprey,therabbit Oryctolagus cuniculus, as well as distances between available settlement areas. Under spatial constraints duetotheevolutionof thesystemin whichtheyact, originallysolitaryforagersexploit foragingpatches similarlytosocialspecies,butwithnoawareness ofthe emergentaggregation patterns towhichtheycontribute.
Methods
Fieldinspirationandbackground
Along-term study oftheSpanish imperial eagle population of Don˜ana (southwestern Spain) inspired the approachused here. The informationavailable on thisbirdofpreyrepresentsoneofthemostextensivesets ofdataona vertebrate speciesavailabletoday(data collection started in1890).Furthermore,from 1986to
2000,30 first-yearjuvenilesofthispopulationwere radiotaggedtostudydispersaldynamicsandspace use withinsettlement areas(Ferrer 1993a,1993b,2001).
Theeagle-rabbitbehaviouralinteractions
Withindispersalareasexists acomplexrelationship betweeneagles,rabbitsandthetimeeagles spendina settlementarea(formoredetailsseeeagle-rabbitgamein Penteriani etal.2005a,2005b).Themainfactorinfluen- cingspatio-temporaloccupancy ofaforaging patch is the time (approx 12days) that the rabbit population takestochangeitsbehaviour (i.e.activitytimetable and useofspace)under predationpressure. Thecontinuous andpredictable presenceofthislargepredatorinasmall area (approx 450 ha) forces rabbits to temporarily modifytheirbehaviour(theyswitch fromdiurnalto nocturnal activity),consequentlydecreasingtheavail- abilityofpreyinthepatch(Ferrer1993a,1993b).Oneof theconsequences ofthis‘‘nonlethal’’predatoreffecton rabbits (i.e.trait-mediatedindirect interaction; Abrams
1995)isthateaglesusedifferenttemporarysettlingareas
inrotation(Ferrer 1993a).Wheneaglesareunsuccessful inhunting(rabbitsarefewornotavailableatall),despite increased effort, individuals change food patches and move to another settlement area to avoid starvation. Thisusuallyoccursafterapprox. 12days.Thisdynamic processofsuccessiveareaexploitation,asafunction of rabbitavailability,causescontemporaneousaggregations ofseveralsolitary eagles(inoneor fewareas), aswell asthesimultaneous disappearanceofthem from other areas(Ferrer2001,Penterianietal.2005a).Thispredator— preyinteraction isbasedonthebehaviouralresponseof thepreytopredationpressure, rather than onpredator density. Evidence does exist to show that foraging predatorsmayaffectpreyavailabilityand,consequently, capturesuccess (NeillandCullen1974, Parrish1992, LoggerwellandHargreaves 1996,Ainleyetal.2003).
Non-social individuals oftenselectthesameforaging patches simultaneously butthechangeinthebehaviour oftherabbit seemstobeunaffected bythenumber of eagles contemporaneously in residence (Ferrer 2001). Thisisprobably duetothefactthat alltheindividuals sharing the same hunting territories (during dispersal they do not display territorialbehaviour, nor do they show any social behaviour during predation) hunt at approx thesametime,e.g.whendiurnal thermals occur. Because the thermal occurrence depends on several factors (e.g. temperature, weather condition, season), eaglespredationisnot predictable byrabbits and, asa result,therearenotfixeddiurnalhourswhererabbitsare bothprotected fromeagles andfreefromnocturnal predators.When diurnal predationpressure decreases (andnocturnal predationincrease,e.g. mortalityincrease because of owls and mammals predation at night), rabbits become diurnal again and, consequently, avail- abletotheeagles.Becauseinformationondynamics of nocturnalpredationwerenot available, weconsidered thatrabbitsbecameavailableforeagles afteratimeequal to the time that rabbits required to shift to nocturnal activity(i.e.12days).
Simulationmodels
Generalpremisesofasimplemodel
Oursimulationprovidedaparsimonious explanation, consistent with the known facts, of the apparently complexphenomenonof‘‘synchronized’’spatial aggre- gations ofsolitary predators.
Webuilt several individual-basedmodels simulating themovementsofeaglesamongseveralforagingpatches, distributed randomly on a regular 50x50 grid (with wraparoundboundaries),toexploreif: (a)theprey’s behaviouralresponsetopredatorpresencecouldexplain individual aggregations, and whether the aggregations differfrompatterns exhibitedby individualsmoving randomly (modelA);and(b)inter-area distancescould influenceindividualaggregation, thatisthelikelihoodof patchoccupancydecreasedexponentiallywith distance (modelB).Todothis,we addedtothemodelAa probabilityofarea occupancy negativelyrelated tothe area distances. Randommovement outputs weregener- atedby randommodels,oneforeachofthemodels describedabove.Intheserandom models:(a)individuals canstayinthesamepatchormoverandomly toanother foraging patch withthesameprobability; and(b)preys donotshowbehaviouralresponsetopredation.
Boththechangein therabbitbehaviourandthe consequent random movements among patchesindicate that our model differed from ideal free distribution models,inwhichindividualsarefreetosettleeverywhere and have a complete knowledge about the quality of eachhabitat (Cressman etal.2004).That is,eaglesare notforagershavingmemoryof therabbitprofitability within available patches. Moreover, no social interac- tions(e.g.conspecificattraction)areincludedtoexplain individualgrouping(e.g.groupforagerssearchforothers that havefound food and jointhem;Beauchamp etal.
1997,GiraldeauandBeauchamp 1999,Coolen 2002). Finally, to understandifaggregations patterns are a
property of small systems, large systems (e.g. sponta- neousaggregationsonlyemerge whenhighnumbersof individualsareinvolved),orboth,werantwoversionsof themodelAwithdifferentnumbers ofboth individuals andpatches,i.e.5 areas:10individualsand50areas:100 individuals. Changing sucharatio ofareas:individuals didnotqualitatively changetheresultspresented here.
ModelA(rabbitbehaviouralchanges)
Eachforaging patch wascharacterised byavalue indicating the diurnal activity of rabbits and ranging from1(totallydiurnalrabbits, i.e.theresourceisentirely available)to0(totallynocturnalrabbits, i.e.theresource isnotavailableatall).Atthebeginningofthesimulation severaleagleswererandomly distributedamongpatches. Whenasettlementareawas occupiedbyatleastone individual, therabbit profitabilityofeachsitestarted to linearlydecreaseinawayproportionaltothenumberof
time-steps an individual exploited that patch (as esti- matedfromempiricaldata,M. Ferrer,unpubl.).The decreaseinthediurnal activityofrabbit populationwas independent ofthenumberofindividualssimultaneously exploiting that patch and only the first individual to reach an empty patch started the decrease of rabbit activity.Whentherabbitdiurnalactivityina patch decreasedbelowagiventhreshold (50%ofrabbitdiurnal behaviour recovery, assuggested byfieldinformation), alltheeaglesintheareamovedrandomly tooneofthe otheravailableareas.Asindividualsleft asite,the abandonedpatch started torecoverafraction ofrabbit diurnal activityineachsuccessivesimulation time-step. Arearecoverywasofaquantity equaltotheoneitlost whentheeaglewentbeyondit.Changing thetime-steps needed to reduce the rabbit diurnal activity below a given threshold(and,consequently,tomakeanarea available for eaglesagain) did not qualitatively change theresultspresented here.
ModelB(rabbitbehaviouralchanges+inter-patch distances)
Thismodelisidenticaltothe5:10modelA,exceptfor theaddition oftheinfluenceofthedistance factor. In fact,inthemodelB,afterpredatorsleaveanareadueto lowrabbit availability, theirprobabilitytooccupyanew patch depends on the distances between patches, with thenearest areahavingthehighestprobabilityofbeing occupied. Such a probability decreased exponentially withdistances.
Aggregationindexandstatistical analyses
Individualgroupingwithin foragingpatches(recorded during10000iterations afterthesimulation began)was calculated usingthefollowingaggregation index:
1 X2
N i=1
where A=total number offoraging patches, N=total number ofeaglesandni=numberofeaglesinthepatch i.Theindexrangesfrom0.2(noaggregation, 2eaglesin eacharea) and 1(allthe individuals assembled inone area)inthe5:10case,andbetween0.02and1inthe50:
100case.
Thenature andamount ofdepartureofthefrequency distributionsofaggregation from normality wasrepre- sented by asymmetry (Sokal and Rohlf 1981). High positive values of skewness indicate that the displace- ment of the curve tail is tilted to the right of the distribution,thatistowards thehighestlevelofaggrega- tion. Comparisons between aggregation distributions weremadebythenonparametricKolmogorov-Smirnov two-sampletest,particularlysensible todifferencesin samplesdistribution(e.g.skewness).
Results
Spontaneous emergenceofaggregations occurredat significantlyhigherlevel whenthemovementsofpre- dators wereaffected byboth preybehaviour and inter- areadistancesthanwhentheymovedrandomly (Table1, Fig.1a—c,2).Infact,both modelAandBshowedthe highestfrequenciesofindividual grouping (i.e.aggrega- tion index=1, that isallthe individuals assembled in oneareaonly).Allaggregation distributions varied significantlyfromarandomdistribution (Table1), suggesting thatthe aggregationsdidnotoccurhapha- zardly. Moreover, it isunlikely that rabbit availability alonecausedtheemergenceofspontaneousaggregations ofindividualsbecause,whenwe introducedthepatch occupation probability as a function of the distance among areasintothesimulation (modelB),thehighest levelsofaggregation becamesignificantlymorefrequent (Fig.1d,Table1).Infact,starting fromthevalueofthe aggregation index$0.32,moreeaglesoccupiedapatch simultaneously inmodelBthan inmodelA(Fig.1d).
Figure2showsanexampleofthedynamicpatterns of
individual aggregation formodelA(ratio 5:10),aswell ashowmodelA’spattern differsfromthecorresponding random model.Weobserved theappearance ofperiods ofalllengthsduring whicheaglesaggregated inonlya fewareas (thehighest valuesoftheaggregation index) whichwere followedbyreturnstolow activity(the intermittent bursts exhibited by the curve), which
correspondedto a random and scattered distribution oftheindividuals.
Discussion
Thesimulationoutputs,basedonthemostparsimonious assumption(i.e.rabbit availability), wereabletorepro- duce the patterns of eagle aggregations that were observedinthefield (Ferrer1993a,1993b).Initially, solitary wanderingindividualsaggregatedspontaneously withinsettlementareasandthey thenmovedsynchro- nously among them. Groups of predatorswere most likelytoforminpatchesincloseproximitytothoseareas thattheyhadjustleftduetolowpreyavailability.Sucha resultisparticularly interestingwhenwe considerthe absenceofanykind ofsocialinteractions (e.g.conspe- cific attractionandpublicinformation;Sergioand Penteriani 2005)inoursimulations toexplainindividual groupingwithinfeedingpatches.Thatis,wewereableto reproduce thepatterns ofaggregations observed inthe field withoutmakeuseofconspecificattractionasa mechanism that allowsanimals (withimperfect knowl- edgeoftheenvironment inwhichtheymove)tolocate highqualityhabitats by thepresenceofconspecifics (Stamps1988,ReedandDobson 1993,Beauchampetal.
1997).Asanendresult,weshowedthatthebehaviourof preyscandeterminetheaggregation ofpredatorsevenin
Table1. Spontaneouspatterns ofpredator aggregation withinforagingpatcheswhenindividualmovements(n=10000)were either random (random model),dependonthebehaviouralchangesofpreys(modelA)orontheeffectsofpreyavailabilitycombinedtoa negativecorrelationbetweenareadistance andoccupancy rate(modelB).Model Awastestedforboth theratios of5areas:10 predatorsand50areas:100predators,whereasmodelBwasonlybuiltfortheratio 5areas:10predators(seetextforadditional informationon the simulations). Kolmogorov-Smirnov two-sample test was used to compare the aggregation outputs of the differentmodels.
Model A (5: 10)
Randommovements / Rabbit behaviouralchangesx¯9SD / 0.2890.05 / 0.3390.09
Range / 0.20—0.68 / 0.20—1.00
Asymmetry / 1.40 / 1.62
Z=18.06,P=0.0001
Model A (50: 100)
Randommovements / Rabbit behaviouralchangesx¯9SD / 0.03090.002 / 0.03490.004
Range / 0.024—0.040 / 0.025— 0.070
Asymmetry / 0.52 / 1.31
Z=38.04,P=0.0001
Model B (5: 10)
Randommovements / Rabbit behaviouralchanges+distancex¯9SD / 0.2890.06 / 0.3890.11
Range / 0.20—0.82 / 0.20—1.00
Asymmetry / 1.49 / 1.23
Z=33.32,P=0.0001
Model A vs model B (5: 10) Z=17.26,P=0.0001
Fig. 1. The levelof individual aggregation increases (for both 5:10and 50:100ratios of areas: individuals) when random movements arecompared toasituation inwhichindividual movements areconstrainedbyrabbit availability (a)=5:10;(b)=50:
100or by the double effect of rabbit availability and inter-area distances (c).Comparedto the effect of rabbit behavioural modifications only,theintroductioninthemodelofanadditionalmovement-constrainingeffect(i.e.inter-area distances)increases theindividual grouping starting fromthevalueoftheaggregation index$0.32(d).In(a—c):whitebars=randommodeloutput;
blackbars=AandBmodels.In(d):greybars=modelA;blackbars=modelB.
absence ofsocialinteractions or facilitating factors on theforaging success.
In biological systems, aggregations due to local
instability (represented herebythemodification ofthe rabbit behaviour) areaknown mechanism (Kelsoetal.
1988).Theaggregation ofeaglesinforagingpatchescan beviewedastheemergenceofaspontaneous spatial pattern that permits individuals to best exploit the patches available for settling through aself-reinforcing
izedbydistance-dependent movements,themoretime eaglesspend hunting, the lower the rabbit availability andtherefore thegreater theemergenceofeaglegroup- ing.Thisservestoavoidthesimultaneous overfeedingof allareas and death duetostarvation.Asexpressedby Perry(1995),thisis‘‘...adynamicthatliterallyfeedson itself’’.
Severaltypesofanimalaggregations canbeexplained
by the evolutionary assumption that joining a group
1
0.8
0.6
0.4
0.2
0
Steps
(highervaluesofthe aggregationindex),priortothe
return ofactivityrepresenting random distributionsofindividuals. Thiscurvebehaviour, characterizedbyperiods ofstasis
interruptedbyintermittentbursts, isrepresentative ofapunctuatedequilibrium, atypicalbehaviour ofdecentralised systems
(Discussion).
groupmembers(Parrish andHamner 1997,Parrish and Edelstein-Keshet 1999, Bonabeau etal.1999).For examples,in foraginggroups,foodsearcharemore successful,compared totheeffortsofaloneindividual, duetothegreater amount ofinformationthat agroup cangather andanalyse.Despitethisfact,andgiventhe knowledgethatinanimateobjectscan aggregatethem- selvesandcreateimpressiveemergentpatterns, itishard to argue that all animal aggregations must have a functionalpurposearising froman individualdecision. Starting from a situation of full prey availability, predatorsarefrequently aloneintheirforaging patches, and preybehaviour determines lowaggregations. How- ever,asthenumber ofavailable areas decreasesdueto thebehaviouralresponseoftheprey,theretendstobea fewpatches that areoccupiedbygroups ofindividuals. Intheend,itiscommon tofindthat allthepredators haveunconsciously aggregate inonly1—2patches. Co- evolution isanintegral part ofpredator—preycommu- nities,illustrating themutual evolution ofpredatorand preystrategies:predators respondstrategicallytoprey behaviour and viceversa. Such a reciprocal influence is crucial in attempting to understand behavioural predator—prey interactions (Lima 2002).For example, co-evolution has frequently been invoked for benefits and costs of aggregations for hunting predators(Lett etal.2003).Thesystemthat weexplored isinteresting becauseitispartfunctional aggregation (duetothefact that groups of eagles can better exploit the rabbit resourcewithina foragingpatch)andpartsimple, spontaneouspattern (theindividuals havenoawareness ofthepattern that theycreate).Inaddition, thepattern of patch use that emerged fits well with models of optimalpatchexploitation thatformthecoreofclassical foraging theory (Giraldeau and Caraco 2000). Non- socialaggregations ofsolitarypredators inalimited numberof thespatiallyavailableareasallowthe recuperationoftherabbitpopulation oftheremaining ones, forming a perfect example of optimal patch
exploitation.Anyway,becausewe presentedthesimplest scenario that could be responsible of the observed predator—preypatterns (e.g.wedidnot assume inour modelmigration/immigration rates,reproduction,prey population reduction by predation pressure), future works are needed to explore the possible influence of theintrinsicdynamicsofbothpredator andprey populationsontheemergenceofsuchaggregations.
AsemphasizedbyLima(2002), whenapproaching predator—prey interactions from the behaviour ofpre- dators, newemergentbehaviour mightchangethewayin whichwe thinkabout suchinteractions. Forexample, predator—preysystemshavealready shownthepossibi- lityofexhibitingself-organizationcapable ofproducing stabilizingheterogeneitiesin preyspatialdistributions (Hasseletal.1991, Jansen1995,deRoosetal.1998, Gurney etal.1998,vandeKoppel 2005)andhierarch- icalstructures (Sakaguchi2003), duetothedynamic instabilityofthenon-linear mutual interactions between predatorsandprey.Aboveall,suchspatialpatterns may facilitatepersistenceof unstableprey-predatorinterac- tionsandincreasestabilityoverlargespatial scales(van deKoppel 2005).
The study of self-organization, despite the large amount literature onthetopic,isarelativelynewfield, especially whenconsideringtheemergenceofdecen- tralizedpatterns inecologicalsystems(reviewed by Camazine et al. 2001). We consider it appropriateto defineself-organization as theemergenceof complex patterns at the global levelofasystemdue to simple, local interactionsbetweenindividuals(orbetweenin- dividualsandtheirimmediateenvironment — bioticor abiotic)that havenoawarenessoftheoverallpictureto whichtheycontribute.
Inthepast,self-organizationwas assumedtobea phenomenonmainlylinkedtolargenumbers, i.e.thou- sands,hundreds ofthousands ormillionofagents, elements or events involved (Nicolis and Prigogine
1977, Deneubourg et al. 1986). Furthermore, social
specieshavebeenthemostinvestigatedgroupofanimals (Deneubourg andGoss1989, Camazineetal.2001), especially inthecontextofaggregation (Parrishand Hamner 1997,ParrishandEdelstein-Keshet 1999) and foraging(Seeley 1987,Portha etal.2002).However, evidencehasshownthatself-organizedpatterns canalso arisefromsmallnumbers ofindividualinteractions with an absence of sociality and cooperative behaviour (Rivault etal.1999,Ferrer andPenteriani 2003).
Itispossiblethat spontaneousaggregations ofeagles
represent anexampleofaself-organizedpattern? With- outattemptingacomprehensiveexplanation,letusspell outsomeessentialfeaturesofourpattern formation that coincidewiththosetypicalofdecentralized phenomena. The pattern of aggregations we reproduced resulted from: (a) internal constraintsof the system (Kelso et al.1988, Camazineetal.1990),representedbyprey availabilityandinter-area distances(i.e.localcuesofour system);(b)simpleinteractions (Seeley1987)betweena predator and its main prey; (c) changes of prey behaviour, giving theforagingpatchesanunstableand fluctuating environment, capable ofdevelopingtempor- ary decentralized structures (Nicolis and Prigogine
1977);(d)behaviouralchanges ofpredatorsduetothe information (i.e.preyavailability)given byareacondi- tions(i.e.stigmergy,Kelsoetal.1988);and(e)feedback loops (Seeley1987,Kelso et al. 1988)represented by the fact that not only individual interactions shape the overall pattern ofthe system(i.e.prey behaviour), butthattheconditionswithinthesystem(i.e. prey availability) also determine the behavioural responses of predators.In addition, the self-grouping of preda- torsdetermined:(a)theemergenceofnew properties (Bonabeau etal.1995),suchasindividual aggregations andcoordinatedmovementsamongareas;(b)anoverall pattern ofaggregation andmovementsthatcouldnotbe predicted bythe behaviouralrules ofthe parts ofthe system(i.e.nonlinearity ofthe properties, Kelso etal.
1988,Bonabeau etal.1995);(c)criticality (Kelsoetal.
1988, Bak and Paczuski 1995), illustrated by a shift towardaself-organizedconfigurationwhenapproaching thecriticalthresholdofthesystem(i.e. preydiurnal availability),whichevolvesintoacriticalnonequilibrium state (Fig. 2);and (d)apunctuatedequilibrium beha- viour(BakandPaczuski1995),i.e.asaconsequenceof criticality, the system exhibits periods of stasis inter- rupted byintermittentbursts ofactivity(Fig.2).
When invoking self-organizationasapossibleexpla-
nation ofpredatoraggregations, we alsoconsiderit importanttostressthat: (a)Gurney and Veitch(2000) provided evidencethat self-organizationisacomponent ofsometypesofcyclicpredator—preyrelationships,in whichpreycanrecoverwhenitsdensityfalls belowa threshold as a consequence of the reduction in local predatordensity;and(b)acommon mechanism under-
published predator—preymodelsisastrong interaction betweenpredatorandprey(vandeKoppel 2005),asin oureagle-rabbit model.
Ifself-organizationcanexplaineagleaggregation, this couldbe thefirsttime,toourknowledge,thatself- organization has emerged as a regulating element of smallsystems(i.e.our model testing aggregation for5 areas and 10individuals). Thismaybeduetothefact thatwe wereusinganon-socialspecies whereasthe preferredmodelstostudytheemergenceofdecentralized patterns inanimalaggregations havebeensocialspecies, forwhichsmallnumbers ofindividuals donoallowthe emergenceofself-organization(Deneubourgetal.1986). Intheend,oneof themostfascinatingaspectsof decentralisedsystemsistheirabilitytocreatecomplexity fromsimplicity,merely reflectingsimpleindividual interactions with their surrounding environment and notindividualcomplexity.However,thequestionofhow spontaneouspatterns andevolutioninteractstillremains unanswered inthestudyofbiologicalsystems.
Acknowledgements — Thispaperhasgreatlybenefitedfrom conversationsandcommentsbyJ.Bascompte, M.Be´lisle,L.-A. Giraldeau,P.JordanoandR.Jovani.Duringthisresearch,V.P. has been supported by a Marie Curie Fellowship of the EuropeanCommunity programme‘‘Improving theHuman ResearchPotentialandthe Socio-EconomicKnowledgeBase’’ under contract number HPMF-CT-2000-01098.Theauthoris solelyresponsibleforinformationcommunicated andthe European Commission is not responsible for any view or resultsexpressed.M.A.F. and C.J.M. havebeensupported byPh.D.FellowshipsBES-2004-6682 andFPI-2000-6137, respectively.TheJunta deAndaluc´ıa alsocontributed tothe financialsupport ofthisproject.
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