Analyzing the spatial dynamics of a prey predator lattice model with social behavior

Original Research Article

Analyzing the spatial dynamics of a prey–predator lattice model with social behavior

Mario Martı´nez Molina

^a,

*, Marco A. Moreno-Armenda´riz

^a

, Juan Carlos Seck Tuoh Mora

^b

aInstitutoPolite´cnicoNacional,CentrodeInvestigacio´nenComputacio´n,Av.JuandeDiosBatı´zs/nUnidadProfesionalAdolfoLo´pezMateos, Col.NuevaIndustrialVallejo,DistritoFederal,07738Me´xico,Mexico

bCentrodeInvestigacio´nAvanzadaenIngenierı´aIndustrial,UniversidadAuto´nomadelEstadodeHidalgo,Carr.Pachuca-TulancingoKm.4.5, PachucaHidalgo,42184Me´xico,Mexico

1. Introduction

The relationship between the long-term dynamics and the patternsobserved in population dynamics and in particular in prey–predator spatial models has been a matter of extensive research in theoretical ecology: reaction diffusion equations, cellular automata, patch models, coupled map lattices and individualbased modelsareonly a subset ofthe toolsused to analysephenomenasuchasphasetransitions(AntalandMichel, 2000;Bagnolietal.,2001),scalingandfinitesizeeffects(Suther- landand Jacobs, 1994; Pascualand Levin,1999; Pascualet al., 2002;Xuetal.,2005),oscillatorybehavior(Blasiusetal.,1999;

Lipowski,1999;Zhangetal.,2006),chaos(Jansen,2001;Lietal., 2005;Maionchietal.,2006;GibsonandWilson,2013)andnoise inducedeffects(Fiasconaroetal.,2004;LaCognataetal.,2010).

Giventhenatureofecologicalmodels,mostofthesephenomena arecloselyrelated.

Thedispersalofindividualsisoneofthecentralmechanisms behindpattern formationin spatiallyexplicitmodels(Hosseini, 2006; Filotas et al., 2008). A good approach to model such phenomenonis tousetransitionrulesthat describea diffusion process: in Comins et al. (1992), the authors study a host- parasitoidmodelonarectangulargridofpatches.Theeffectsofthe diffusion on the spatial dynamics of the model manifest as a wavefront of hosts traveling at constant speed; this event is followedbyafrontofparasitoidsthatconsumestheoriginalwave of hosts.Depending on thefractionof hosts thatdisperseeach generationseveralspatialpatternsmightbeobservedincluding spatialchaos,spiralsand‘‘crystal’’patterns.Theauthorsnotethat despitethefactthatthepresenceofanyofthesepatternsleadsto thecoexistenceofbothspecies,thereisathresholdforthesizeof thegridbelowwhichextinctionisalwaysobserved.

A comparative study of theeffects of diffusion processesin spatialmodelsappearsinSherratetal.(1997).Theauthorsanalyse the behavior of four different spatial prey–predator models (reaction–diffusion equations, coupled map lattices, cellular automata and integrodifference equations) where prey suffer theinvasionofpredators.Simulationsofone-dimensionalversions ofeachmodelshowtheexpectedwavefrontofpredatorsinvading theprey-onlystateand leavingbehinda coexistencestate.The ARTICLE INFO

Articlehistory:

Received22August2014

Receivedinrevisedform24March2015 Accepted29March2015

Availableonline

Keywords:

Latticemodel Prey–predator Scaling

Oscillatorybehavior Particleswarmoptimization

ABSTRACT

Alatticeprey–predatormodelisstudied.Transitionrulesappliedsequentiallydescribeprocessessuchas reproduction,predation,anddeathofpredators.Themovementofpredatorsisgovernedbyalocal particle swarm optimization algorithm,which causes theformation of swarms of predators that propagatethroughthelattice.Startingwithasinglepredatorinalatticefullycoveredbypreys,we observeawavefrontofpredatorsinvadingthezonesdominatedbypreys;subsequentfrontsariseduring thetransientphase,whereamonotonicapproachtoafixedpointispresent.Afterthetransientphasethe systementersanoscillatoryregime,wheretheamplitudeofoscillationsappearstobeboundedbutis difficultto predict. We observe qualitative similar behavior even forlarger lattices. An empirical approachisusedtodeterminetheeffectsofthemovementofpredatorsonthetemporaldynamicsofthe system.Ourresultsshowthatthealgorithmusedtomodelthemovementofpredatorsincreasesthe proficiencyofpredators.

ß2015ElsevierB.V.Allrightsreserved.

* Correspondingauthor.Tel.:+525557296000x56525.

E-mailaddresses:[email protected](M.M.Molina), [email protected](M.A.Moreno-Armenda´riz), [email protected](J.CarlosSeckTuohMora).

ContentslistsavailableatScienceDirect

Ecological Complexity

j ou rna l h om e pa ge : w w w. e l s e v i e r. co m/ l oc a t e / e coc om

http://dx.doi.org/10.1016/j.ecocom.2015.03.001 1476-945X/ß2015ElsevierB.V.Allrightsreserved.

authorsfocustheirattentiononthespatialdynamicsbehindthe initialwavefrontofpredatorswherethreedifferentphenomena areobserved:

Regular spatio-temporal oscillations. For this case, periodic travelling waves moving at a different speed than the originalfrontareobserved.Suchwavescorrespondtoafamily ofsolutionsforthemodelbasedonreactiondiffusionequations.

Irregularspatio-temporaloscillations.Forthereactiondiffusion equations, certainparametersmightforcethetravellingwave solutionintoirregularoscillations,theauthorsnotethat such pattern might be associated with spatial chaos. Irregularities expandfromthefocusoftheinvasionsuggestingagainthatsuch dynamicsarechaotic.

Irregular fluctuations.Here,there isa bandof periodicwaves immediatelybehindtheinvasivefront.Followingthisbandthere areirregularoscillationswithnoapparentpattern.Thisbehavior corresponds toa transientphase due toan unstable periodic wavesolution.

SimilarpatternswereobtainedinArashiroandTome´ (2007)for aprobabilisticcellularautomaton.Bycarryingnumericalsimula- tions,theauthorswereabletoobtainthecriticalexponentsforthe automaton,thusallowingtheclassificationofthemodelintothe directedpercolationuniversalityclass.

Diffusion-liketransitionrulesofferasimpleandmathemati- cally tractableway todescribe thedispersalof individuals. In Filotasetal.(2008)theauthorsstate thatbecauseofa lackof commonrulesbehindthedispersalofspecies,ecologistsoften have to make the simplest assumptions (e.g., a density independentrateofdispersion)whenmodelingsuchphenome- non;evenifformanyspeciesfactorssuchasthelocalpopulation size, resource availability, or habitat quality influence the mobility of individuals. However, there has been efforts to developstrategies thatbetter mimicthe phenomenafound in natural ecosystems (Li et al., 2005; Boccara et al., 1994;

Rozenfeld and Albano, 2001; Szwabin´ski, 2012; Wang et al., 2012).Intheseapproaches,cooperationisneglectedinfavourof intraspecific competition, i.e.,only the negative effects of the aggregation of individuals are considered. However, it is reasonabletoexpectthatunder certaincircumstances,agroup of individuals has better chances of survival than those that remainisolated,e.g.,tofleefromapredator,ortohuntforprey.

Allee effects are a good example where aggregation leads to positivedensitydependence,albeitonlyforsmallpopulations;

howeverrecentworkshaveshownthatsucheffectsarekeyfor thestabilityofa prey–predatorsystem(Wanget al.,2011),or evendeterminethesuccessofaninvadingspecies(Mistroetal., 2012).Insomeanimalcommunities,socialbehaviorisalsothe source of many extraordinary patterns, e.g., bird flocking or insectswarms.Byincorporatingsocialbehavioronthedispersal rules of predators, we attempt to studythe effects thatsuch processhaveontheglobaldynamicsofanecosystem.

Inthepresentpaperweanalyseaprey–predatorlatticemodel wherealocalParticleSwarmOptimization(PSO)algorithmisused tomodelsocialinteractionsamongtheindividualsofthepredators species.PSOis anevolutionarycomputation algorithmtypically usedtofindanoptimalsolutioninasearchspacethatdefinesthe setofpossiblesolutionstoaparticularproblem.Thefoundationsof thealgorithmcomefromtheobservationofthesocialbehaviorof animalcommunities previouslymentioned:insectswarms,bird flocksorfishschools(Trelea,2003).InaPSOalgorithmthereisa populationofparticlescalledthe‘‘swarm’’,thepositionofeach particle determines a candidate solutionto theproblem under study.Typically, social interactions amongthe membersof the swarmoccursthroughoneoftwoinformationsharingschemes:

Global.Aparticlemovesaccordingtoitsownknowledgeofthe searchspace,andtheinformationitreceivesfromtheparticleat the location that represents the best solution found by the swarm.

Local.Inthisscheme,aneighborhoodcomprisingaparticleand someofitsnearestneighborsiscreated.Tomove,aparticleuses its own knowledge of the search space and the information provided by the particle with the ‘‘best’’ position among its neighbors.

Inourmodel,theseinteractionshelpapredatortodetermine the bestdirection ofmovement in order tosecure food for its survivalandreproduction.Cooperationamongpredatorsmanifest itself as an interesting spatial pattern: predators group into clustersthatmaintaincohesionastheymovethroughthelattice huntingforpreys;theanalysisofsuchphenomenonisthemain focus of the present article. In a previous work (see Martı´nez Molina et al., 2013)we showedthat thepopulation dynamics corresponding to theformation and propagation of clusters of predators is characterized by oscillations with a very regular period.Similarbehaviorhasbeenassociatedwithvariationsinthe mobility of the individuals of a species (Boccara et al., 1994;

Shigefumi et al., 2014), large migration rates in patch models (Blasius et al., 1999; Li et al., 2005), or the aggregation of populations atsmall or intermediatescales (Pascualand Levin, 1999;Pascualetal.,2001;Durrett,1994;Mobiliaetal.,2007).In lightoftheseresults,weinvestigatetherelationshipbetweenthe social behavior of predators and the observed population dynamics.Themainresultofthisworkisthatcooperationthrough alocalPSOincreasestheproficiencyofpredators,whichbehavior is characterizedby atransientphasefollowed byanoscillatory regime.Suchbehaviorwastakenintoaccounttobuildameanfield model that accurately predicts the mean densities of the populations.

TheproposedmodelisdefinedinSection2;here,wedescribe each stage ofthe model,and explain themain consequenceof theuseofalocalPSOalgorithmforthemovementofpredators,i.e., thegroupingofpredatorsintoswarms.InSection3weanalysethe invasion of prey dominated zones by predators using initial conditionsclosetotheabsorbingstatewherethelatticeisfullof preys. In Section 4 we show that the movement of predators reducesthedeathprobabilityofpredators,whichinturnincreases the death rate of preys. Finally, in Section 5 we explore some properties of the model for different sizes of the lattice. Our conclusionsappearinSection7.

2. Proposedmodel

Ourmodeldescribestheinteractionsbetweenasessileprey anditspredator,suchinteractionsarelocalinnatureandoccuron atwo-dimensionallatticeLwhereperiodicboundarieshavebeen implemented.Eachsiteofthelatticemaybeoccupiedbyaprey,a predator,bothorbeempty.Timeproceedsindiscretetimesteps.

Theevolutionofthemodeliscontrolledbyalifecycle,knownas

‘‘season’’,thatdeterminesthetransitionfunctionthatisappliedat eachtimestep.Dependingonthefunctionbeingapplied,preys andpredatorsmay interactwithina neighborhoodwhosesize (the number of sites within the neighborhood) is defined as follows:

jMrj¼ð2rþ1Þ² (1)

where r is the radius of the neighborhood. Thus an M1 neighborhood comprises the eight nearest neighbors of a particularsite,andthesiteitself;anM2neighborhoodthenearest

24neighbors,andthesiteitself,andsoon.Thetransitionfunctions areasfollows:

1.Intraspecificcompetition.Duringthisstagetheprobabilitythata preydiesduetocompetitivepressureisgivenby:

a

y

jMcj1 (2)

The quantity

a

2[0, 1] is the intraspecific competition coefficient,whichdeterminestheintensityofthecompetition exercisedbyotherpreysintheneighborhoodMcofasite,yisthe numberofpreyswithinsuchneighborhood.

2.Migration.Whenmigrating,predatorsmovefromzoneswhere thedensityofpreysislow,tozoneshighlypopulatedbypreys.

ThemovementoccursaccordingtoalocalPSOalgorithm,which givepredatorsthecapacitytointeractwithothermembersof theirspecies.

3.Reproductionofpredators.Duringthereproductionstage,each predatorchoosesrandomlyasiteinitsreproductionneighbor- hood.Ifthesiteisnotalreadypopulatedbyapredator,thena new predator is created at the chosen site, otherwise, no reproductionoccurs.Thisprocessisrepeatedasmanytimesas thereproductivecapacity

e

Zofthespeciesallows.

4.Deathofpredators.Ifapredatorisnotlocatedatasitecontaining aprey,itdieswithprobability1.

5.Predation.Ifapreyislocatedatasitecontainingapredator,it dieswithprobability1.

6.Reproduction of preys. Like predators, preys spawn new individualsatrandomwithinaMyneighborhood.Reproduction isonlysuccessfulifthechosensitesdonotcontainalready a predator.

Withexceptionofthemigrationrule,whichisappliedforfive consecutivetimesteps,allotherrulesareappliedduringasingle timestep.Therefore,aseasonisequivalentto10timesteps.The followingpseudocodeillustrateshowthetransitionfunctionsare scheduledintheproposedmodel.Such‘‘architecture’’isusefulto observethemodelafteratransitionfunctionhasbeenapplied,and notonlywhenaseasonends.

functionMAIN

INITIALIZE

forj 1,totalTimeStepsdo NEXTGEN

endfor endfunction

functionINITIALIZE

nextStage INTRASPECIFICCOMPETITION "Storeaddressoffunction migrationCounter=0

endfunction

functionNEXTGEN

nextStage "CallthefunctionpointedbynextStage endfunction

functionINTRASPECIFICCOMPETITION

fori 1,totalPreysdo

Determineifthepreyidiesduetocompetition endfor

nextStage MIGRATION "Storeaddressoffunction

endfunction

functionMIGRATION

fori 1,totalPredatorsdo

Update the position of predator i according to the PSO algorithm

endfor

migrationCounter migrationCounter+1 ifmigrationCounter=5then

migrationCounter=0

nextStage REPRODUCTIONOFPREDATORS

endif endfunction

functionREPRODUCTIONOFPREDATORS

fori 1,totalPredatorsdo forbirthCount 1,

e

Zdo

Ifpossiblespawnanewpredatoratarandomlychosensite endfor

endfor

nextStage DEATHOFPREDATORS

endfunction

functionDEATHOFPREDATORS

fori 1,totalPredatorsdo

Killallpredatorsnotlocatedinacellwithoutaprey endfor

nextStage PREDATION

endfunction

functionP^REDATION fori 1,totalPreysdo

Killallpreyslocatedinacellwithapredator endfor

nextStage REPRODUCTIONOFPREYS

endfunction

functionREPRODUCTIONO^FP^REYS fori 1,totalPreysdo

forbirthCount 1,

e

Ydo

Ifpossiblespawnanewpreyatarandomlychosensite endfor

endfor

nextStage INTRASPECIFICCOMPETITION

endfunction

Themigrationstagedeservesadditionalexplanation;however,a deepreviewofPSOalgorithmsisbeyondthescopeofthisarticle, detailed information can be found in Kennedy and Eberhart (2001).InordertousePSOasamigrationalgorithm,itisnecessary to enhance the capabilities of the predators. In particular each predatorrecordsthepositionofthesitewiththehighestdensityof preysvisitedsofar.Suchameasureisassignedtoeverysiteofthe lattice basedon thenumber ofpreysin a neighborhoodof size M_c.Eachpredatoralsopossessesavelocityvectorwhichindicates thecurrentmagnitudeanddirectionofitsmovement.Asstatedin Section1,PSOiscommonlyusedtosolveoptimizationproblems,

e.g.,findingthemaximumortheminimumofagivenfunction.We adaptsuchbehaviortomodelthemovementofpredators:when moving,apredatoraimstomaximizeitschancesofsurvival,itdoes sobymovingtozoneswithahighdensityofpreys.Incontrasttoa typicallocalPSOwheretheneighborsofagivenparticleremain fixed,inouralgorithmtheneighborhoodofapredatorchangesasit movesthroughthelattice.Duringthemigrationstage,theposition of every predator in the lattice is updated using the following equations:

V_i^tþ1¼

v

V_i^tþk1q₁ðP^ti X_i^tÞþk2q₂ðP^tl_iX^t_iÞ (3)

X_i^tþ1¼X_i^tþV_i^tþ1 (4)

whereX_i^tisthepositionofpredatoriattimestep

t

andV_i^tisthe velocityvectorofpredatoriattimestep

t

.EachtermofEq.(3) playsadifferentroleinthePSOalgorithm:

Theterm

v

V_i^tisresponsibleforkeepingthecurrentdirectionof movement of the particle, this term is known as the inertia component. Theparameter

v

2[0, 1]iscalledinertiaweight;

valuesof

v

closetooneproduceslongrangemovements,while valuesclosetozeroproducesmallmovements.Itmustbenoted thatthevalueof

v

islinearlydecreasedoverthedurationofthe algorithm.If

v

weretoremainconstant,thenapredatorwould movearoundazonewithahighdensityofpreys;howeveritis unlikelythatthefinalpositionofthepredatorwouldbewithin such zone.Thus, during thefirst iterationspredators execute long range movements that favours the exploration of their neighborhood. As migration nears its end, the movement of predatorsismoreconstrained,whichencouragesthesearchin zoneswithagooddensityofpreys.

Thetermk1q1ðP^t_i X_i^tÞisknownasthecognitivecomponentand servestoguidethemovementofapredatortowardszoneswhere ithaspreviouslyfoundagooddensityofpreys.P_i^tdenotesthe positionwiththehighestpreydensityfoundbypredatoriupto time

t

.Theparameterk₁isknownasthecognitivefactor,while q12[0,1]isauniformlychosenrandomnumber.Bothofthese parametersdeterminethemagnitudeofthecontributionofthe termtothenewvelocityvectorofthepredator.

The termk2q₂ðP^t_l

iX^t_iÞis knownas thesocialcomponent,its objectiveistodirectthemovementofthepredatortowardsthe positionwiththehighestdensityofpreysamongtheneighbor- ingpredators(socialneighborhood).Theparameterk2isknown as the social factor,q22[0, 1]is a uniformly chosen random number; both parameters determine the contribution of the term to the velocity vector of the predator. P^t_l

i denotes the positionwiththehighestpreydensityfoundbytheneighborsof predator i at time

t

(prey density is measured using the competitionneighborhood).

The parameters q1 and q2 are randomly chosen in order to produce an uneven movement around a point which is the weightedaverageofP^t_i andP^t_l

i,thegoalistoforceapredatorto

‘‘explore’’theneighborhoodofapotentiallygoodzone.Oncethe contributionof each termhasbeen obtained, thenewvelocity vectorisaddedtothecurrentpositionX_i^tofapredatortoobtainits positionatthetimestep

t

+1.Fig.1showsthepositionupdating of a predator for a Ml neighborhood of radius l=2, and a Mc

neighborhoodofradiusc=1.

In a previous work (see Martı´nez Molina et al., 2013) we showed that the spatial dynamics of the proposed model is characterizedby theaggregationofpredatorsinto welldefined clustersthatpropagatethroughthelatticeasshowninFig.2.Here andthereafter,werefertosuchclustersas‘‘swarms’’inorderto

avoidconfusionwiththeusageoftheterm‘‘cluster’’inprevious works(cellsuand

v

belongtothesameclusterifbothhavethe samestateandtheyare‘‘adjacent’’toeachother,i.e.,

v

islocatedat oneofthenearestneighborsofu,SutherlandandJacobs,1994;Fu etal.,2010).Suchphenomenaisdrivenbythefollowingprocesses:

Formation.Thisprocessoccurswhenapredatorlocatedatazone withahighdensityofpreysattractspredatorswithinitssocial neighborhoodMltoitslocation.

Fragmentation.DuetothecognitivecomponentofthelocalPSO algorithm, it is possible for one or more predators to move beyondthesocialneighborhoodsoftheotherindividualsthat compriseaswarm.

Merging. When two or more swarms collide, the social neighborhoods oftheir predatorsoverlap, what follows is an aggregationofindividualsintoanevenbiggerswarm.However, suchaggregationsareshortlived:duetoanincreaseinthelocal density of predators, resources are rapidly consumed which produceslocalextinctionevents.

Fig.1.Localpositionupdating.

Fig.2.Swarmsofpredatorsmovingthroughthelattice.Preysaredepictedingreen, predatorsinred,andpreysinvadedbyapredatoraredepictedinyellow.(For interpretationofthereferencestocolorinthisfigurelegend,thereaderisreferred tothewebversionofthisarticle.)

3. Swarmdynamics

Computersimulationsoftheproposedmodelwereperformed using the parameters presented in Table 1. 10,000 seasons (100,000 time steps) were simulated using a lattice of

512512siteswithperiodicboundaries.Ourexperimentsstart withthesystemclosetotheprey-onlyabsorbingstate,i.e.,the latticestartswithaninitialdensityofpreys

C

0=1,andaninitial densityofpredators

F

0correspondingtoasinglepredator.Where appropriate,thefollowingcolorcodeisusedtoidentifythestateof asite:

Black:anemptysite.

Green:asitepopulatedbyaprey.

Red:asiteinhabitedbyapredator.

Yellow:asitecontainingapreyandapredatoratthesametime.

Fig.3showstheinvasionofthepreydomainbypredators.In Fig.3aseveralswarmsofpredatorsgroupedatthesourceofthe invasioncanbeobserved.Duetothehighdensity ofpreys,the formationofthesepatternsoccursquicklyafterthestartofthe simulation:ifthestartingpredatoranditsprogenysurvivethefirst

‘‘deathofpredators’’stage,thenthefirstswarmappearsafterthe reproduction of predators. Since the recently born predators inherittheirbestknowpositionfromtheirparent,andsincethe sizeofthereproductionneighborhoodissmallerthanthesizeof thesocialneighborhood,boththestartingpredatoranditsprogeny remain‘‘bounded’’.Wemaydefinethen(somewhatarbitrarily)the timeofcreationofthefirstswarmasthetimewhenthestarting predator and at leastone of its progeny survive the‘‘death of predators’’stage.Aspredationtakesplace,swarmsmoveoutwards Table1

Parametersusedincomputersimulations.ThevaluespecifiedforF0isthedensity thatcorrespondstoasinglepredator.Thevaluesforvspecifytherangeoverwhich theparameterislinearlydecreased.

Preys Value Description

C0 1 Initialdensity

eY 1 Reproductivecapacity

y 3 Radiusofthereproductionneighborhood

a 0.05 Intraspecificcompetitioncoefficient

c 3 Radiusofthecompetitionneighborhood

Predators Value Description

F0 3.814710^6 Initialdensity

eZ 3 Reproductivecapacity

z 1 Radiusofthereproductionneighborhood

Migration Value Description

k1 1 Cognitivefactor

k2 2 Socialfactor

l 7 Radiusofthesocialneighborhood

v 0.9–0.2 Inertiaweight

Fig.3.Invasionofpreysbypredators.

following the direction of highest prey density. However, the movementofswarmslocatednear thesourceoftheinvasionis limitedbyswarmslocatedfarfromthesourceoftheinvasion:ifan

‘‘inner’’swarmtriestoexpand,itispossibleforitsdestinationtobe alreadyoccupiedbyan‘‘outer’’swarm,oritmayalsobethecase thatitsdestinationisarecentlypredatedzone,sincethesezones havealowdensityofpreys,movingtoanyoftheselocationswould increasethedeathprobabilityofeveryindividualintheswarm.

Suchprocessresultsinanexpandingwavefrontof‘‘outer’’swarms movingatthesamespeed,while‘‘inner’’swarmsremainbehind (see Fig. 3b). It must be noted that the wavefront is not homogeneous;i.e.,swarmsdonotcoverthewholeperimeterof thewave(seeFig.3c),whichinturnallowssomepreystosurvive the invasion. Zones with a low density of preys behind the wavefrontare repopulated by thesurvivors, thus creatingnew clustersof preys (see Fig.3d), which becomethetarget of the

‘‘inner’’swarmsofpredators.Thereisthenadivisionbetweena prey-onlystateaheadoftheinvasionfront,andacoexistencestate behind,whereswarmscontinuouslymigratefromzoneswitha lowdensityofpreystozoneshighlypopulatedbythem.Similar divisionshavebeenanalysedpreviously: foratwo-dimensional coupled map lattice and two-dimensional reaction diffusion equations,Sherratetal.(1997)observedperiodictravellingwaves thatpersistordecayintoirregularspatio-temporaloscillationsdue tobeingintrinsicallyunstable.ArashiroandTome´ (2007)obtained asimilardivisionforaprobabilisticcellularautomaton;behindthe initialwavefront theyobserveda coexistencecharacterizedby localtimecoupledoscillationswheresmallclustersofpredators invadelargeclustersofpreys.Asecondcoexistencestatewhere both populationsare grouped into small clustersisolated from eachotherwasobservedforalowdeathprobabilityofpredators whichiscompensatedbyhigherbirthprobabilities.

Thepopulation dynamicsfor theproposed modelunder the parametersspecifiedbyTable1isshowninFig.4.Theexpansionof theinitialwavefrontproduces amonotonicapproach toanon- trivialfixedpointduringthetransientperiodofthesystem;once suchapproach ends,it is possibletoobserve weak oscillations around the fixed point, which corresponds to subsequent travellingwavesthatareproducedwhensurvivingpreysrepopu- latethespacebehindthewavefront,onlytobeconsumedbythe

‘‘inner’’ swarms. Such process is short lived, and waves will eventuallydisappear;afterthetransientphase,bothpopulations oscillatewithaveryregularperiodandthetypical‘‘outofphase’’

behaviorofprey–predatorsystems(seetherightinsetofFig.4).

4. Meanfieldanalysis

It is possible to formulate the following set of mean field equations(detailsaboutsuchprocessaregiveninAppendixA)by consideringthechangesindensitythatoccurthrougheachseason ofthemodel:

C

I¼

C

t

aC

²t (5)

F

R¼

F

tþð1

F

tÞð1ð1PZÞ^eZ¯ztÞ (6)

F

tþ1¼

F

Rð1

C

IÞ

F

R (7)

C

D¼

C

I

C

I

F

tþ1 (8)

C

tþ1¼

C

Dþð1

C

DÞð1ð1PYÞ^eY¯ytÞ (9) where:

tismeasuredinseasons.



F

tisthedensityofpredatorsattimet.



C

tisthedensityofpreysattimet.



C

I is the density of the prey species after the intraspecific competitionstage.



F

R is the density of predators after the corresponding reproductionstage.



C

Disthedensityofpreysafterthepredationstage.

 ¯y_tisthemeannumberofpreysinaneighborhoodofsizejMyjofa site.

 ¯ztisthemeannumberofpredatorsinaneighborhoodofsizejMzj ofasite.

PY=1/jMyjistheprobabilitythatasitethatdoesnotcontaina preyisselectedforreproductionbyoneofitsneighboringpreys.

PZ=1/jMzjistheprobabilitythatasitedevoidofapredator is selectedforreproductionbyaneighboringpredator.

A pairofmean fieldequationsthat describethechanges for bothpopulationscanbeobtainedfromEqs.(5)–(9);however,since wewerenotabletoobtainananalyticalsolutiontosuchsystem, wekeeptheequationsintheiroriginalform.Notethatsinceno changeindensityoccursduringthemigrationstage,thereisno meanfieldtermassociatedtothisrule.Theseequationswillbe used toprovide a comparison of thebehavior of the proposed modelagainstamodelthatconsidersa‘‘wellmixed’’environment.

Wewillalsoshowthatbyconsideringtheeffectsofthemigration stageinthemeanfieldequations,anaccuratepredictionofthe meandensitiesofbothpopulationscanbemade.

In agreement with computer simulations of the proposed model, the mean field equations have two fixed points that correspondwiththeabsorbentstatesoftheproposedmodel:the first corresponds to an ecosystem where both populations go extinct; thesecond is obtained when alltheindividualsof the predatorspeciesdie,leavingbehindanecosystemonlypopulated by preys. Numerical simulations of the mean field model, accordingtotheparametersofTable1,alsoshowan evolution towardafixedpointcharacterizedbythecoexistenceofpreysand predators(seeFig.5);itisworthnotinghowever,thatthelong- termdensityofthepopulationsisnotaccuratelypredicted,andno oscillatorybehaviorcanbeobserved.Bothoftheseoutcomesareto beexpected,itiswellknownthatmeanfieldmodels,suchasthe onegivenbyEqs.(5)–(9),provideonlyaroughdescriptionofthe dynamics of a latticemodel, since theydo nottake space into accountandassumea‘‘wellmixed’’environmentwhereindividu- alsarerandomlydistributedonthelattice.

Fig.4.TemporaldynamicsoftheproposedmodelundertheparametersofTable1.

Theleftinsetshowsamonotonicapproachtoafixedpointduringthetransient phaseofthesimulations,whichcorrespondstotravelingwavefrontsthatappear duringtheinvasionprocess.Therightinsetshowsthatoscillationshaveavery regularperiodandareoutofphase.

4.1. Theroleofthemigrationstage

Ourmeanfieldmodelhasanadditionalshortcoming:itdoes nottake intoaccount theeffects ofthemigration stage on the populationsofpreys andpredators. Animmediateeffectof the movementofpredatorsfromzoneslowonresourcestozoneswith ahigherdensityofpreys,isthatthedeathprobabilityofpredators mustbedifferentfromthat when no migrationoccurs. Ideally, migrationshouldprovidebetterconditionsforthesurvivalofa predator,orthesurvivalofitsprogeny;however,theeffectsofsuch change on a global scale are not easily predicted. To test our hypothesis,wehaveselectedtheparametersetsshowninTable2:

setAdisablesthemovementofpredators,soparameterslikek1

andk2aresettozero;parametersinsetBareidenticaltothose showninTable1andarerepeatedtoprovideaneasyreadingofthe article.

For both simulations, we have measured the following quantities:(a)thedensity of preys atthestart of thedeath of predatorsstage

C

I;(b)thedeathprobabilityofpredators;(c)the densityofpredatorsatthestartofthepredationstage

F

t+1;and(d) thepredationprobability.Fig.6showsthesemeasurements:itis easytoseethatwhensetAisused,ourdataconcentratesinan smallarea,which correspondstoanorbittowardsa fixedpoint obtained when no migration occurs. Due to the oscillatory behavior resulting from the movement of predators, the data obtained for set B follow a pattern akin to a limit cycle, thus spanning a wider area. It is worth noting that most of the

measurementsobtainedforthesimulationsusingsetBliebelow thedataobtainedforsetA(seeFig.6a).Whenthemigrationstage isenabled,thereisadecreaseinthelong-termdensityofthepreys species;however,evenunderthese‘‘harsh’’conditions,thedeath probabilityofpredatorsislowerwithrespecttothecaseinwhich nomigrationoccurs.AsimilarbehaviorcanbeobservedinFig.6b.

MeasurementsobtainedusingsetAliebelowtheresultsobtained fromsetB,duetothemovementofpredatorsthereisanincreasein thepredationprobability.

Toincludethechangesinthedeathprobabilityofpredatorsand predation probability in our mean field model, we follow a procedurefirstdevelopedbyPascualetal.(2001)todeterminethe long-termdynamicsatdifferentspatialscalesofastochasticlattice model;themethodpartsfromtheresultthatcertainquantities, e.g., the growth rates of preys and predators, preserve their functionalformatdifferentspatialscales.Byfirstmeasuringsuch quantities directly from the computer simulations, and then performingsuitablestatisticalregressionmethodsontheobtained data,itispossibletoadjusttheparametersthatcontrolthemean fieldtermsofthequantitiesofinterest.

Toapplythismethodologyinoursimulationsitisnecessaryto alsomeasurethegrowthrateofbothpopulations.Thefunctional formofthesequantitiesissimilar:ð1

C

DÞð1ð1PYÞ^eY¯ytÞfor thepreys and ð1

F

tÞð1ð1PZÞ^eZ¯ztÞfor thepredators, these termsarecontrolledbytheparameters

e

Yand

e

Zrespectively.To obtaina fittedcurveforthedataaleast-squaresregressionwas used. Theresults obtained showthat the growthrateof preys behaves as if in average the reproductive capacity of preys

e

Y=0.8744; meanwhile the reproductive capacity of predators behaves as if

e

Z=1.4693. It is interesting to note that both Table2

Parametersusedtoanalysethemigrationstage.

Parameter SetA SetB

C0 1 1

eY 1 1

y 3 3

a 0.05 0.05

c 3 3

F0 0.00000384147 0.00000384147

eZ 3 3

z 1 1

Migrationparameter SetA SetB

k1 0 1

k2 0 1

l 1 7

v 0 Lineardecrease

from0.9to0.2

Fig.6. Effectsofthemigrationstage onthedeath probabilitiesofpreys and predators.(a)Duetothemovementofpredatorsfollowingthedirectionofhighest preydensity,thereisadecreaseonthedeathprobabilityofpredatorswithrespect toasimulationwherenomigrationoccurs.In(b)itispossibletoobservethat,when predatorsdonotmigrate,thepredationprobabilityislowerwithrespecttothecase inwhichpredatorsmove.

Fig.5.Long-termmeandensitiespredictedbythemeanfieldmodelgivenbyEqs.

(5)–(9);dashedlinescorrespondtothemeanfieldmodelthatomitstheeffectofthe migrationstage,solidlinescorrespondtotheadjustedmeanfieldmodelthat considerssucheffects.

parametersarelowerthanthevaluesspecifiedforthesimulation, such results is to be expected given the local nature of the interactionsbetweenindividuals.

Themeanfieldtermforthedeathprobabilityofpredators1

C

I

assumesalinearrelationshipwithrespecttothenumberofsitesnot populatedbypreys.Initspresentform,thetermisonlyvalidfora

‘‘wellmixed’’environmentwhereindividualsarerandomlyplaced onthelattice.However,asshownbyFig.6athedataforthedeath probabilityofpredators,obtainedusingsetsAandB,clearlyshowa linearrelationshipwith

C

I;theslopeofthedatashowninFig.6b suggestsasimilarbehaviorthatdependson

F

t+1.However,thedata obtained from set B exhibit a greater deviation from the line representingsucharelationship;thisoutcomehastobeexpected giventhemagnitudeofthe oscillationsduetothemovementof predators.Sincewearedealingwithlinearfunctions,apolynomial regressionisenoughtoadjustourmeanfieldterms.Notethatthe onlyparametersthatdefinethebehaviorofthedeathprobabilityof predatorsandthepredationprobabilityarethecoefficientsofthe firstorderpolynomialforeachdataset.Letuscallsuchcoefficientsa, b,d,ande,thereforethemeanfieldtermforthedeathprobabilityof predatorscanbewrittenasfollows:

bþa

C

I (10)

with a=1 and b=1, meanwhile the term for the predation probabilitycanbeexpressedas:

eþd

F

tþ1 (11)

withe=0and d=1. Itis possibletorewriteEqs.(7)and (8)as follows:

F

tþ1¼

F

Rð1

C

IÞ

F

R¼

F

Rðbþa

C

IÞ

F

R (12)

C

D¼

C

I

C

I

F

tþ1¼

C

I

C

Iðeþd

F

tþ1Þ (13) TheresultsoftheregressionmethodforthedataofsetBgivethe coefficients a=0.9434, b=0.8626, d=1.5034, and e=0.1726 (additionaldetailsabouttheseresults canbefoundin Table3).

Theeffectsoftheadjustedparametersonthemeanfieldmodelare showninFig.5,anon-trivialfixedpointwherebothpopulations coexistcanbeobserved;itiseasytosee,thatthedensitiespredicted bytheadjustedmeanfieldmodelcorrespondtothepointsaround which the densities of preys and predators oscillate when the migrationstageis enabled(seeFig.4). We maysummarize the effectsofthemovementofpredatorsonthecarryingcapacityofthe populationsas follows:when movingtowards zones wherethe densityofpreysishigh,apredatorincreasestheirchanceofsurvival;

also,itincreasestheprobabilitythatitsprogenysurvivesthe‘‘death ofpredators’’stage.Withagreaternumberofpredatorssurviving, thereisanincreaseinthepredationprobability,whichincreasesthe deathrateofpreys.Itisinterestingtonotethatbyincreasingtheir odds of survival, the population of predators create harsher conditionsfortheirspecies.Despitethefactthatpredatorsareable tosurviveundersuchconditions,thecarryingcapacityofbothpreys andpredatorsislowerthanthedensitiesobservedundera‘‘well mixed’’environment.

5. Oscillatorybehaviorandspatialscale

Besides changing the mean density of the populations, the movementofpredatorsalsocausesatransitiontotheoscillatory behavior. The oscillationshave a very regular period,however, their amplitude,whilebounded, isvery difficulttopredict(see Fig.4).Previousworkshaveshownthatsuchoscillationsinaprey–

predatorsystemshouldvanishinthethermodynamiclimit (see Antal and Michel, 2000; Lipowski, 1999; Boccara et al., 1994;

Durrett,1994;Mobiliaetal.,2007;Petrovskiietal.,2004),i.e.,their amplitude diminishesas thesize ofthe latticeis increased.To investigatesuchphenomenon,weperformedadditionalcomputer simulationsforlatticesofsize2⁶⁺ⁱfori=0,1,2,3,4,5,6,7,i.e.,from alatticeofsize6464sitestoalatticeofsize81928192sites;

10000seasonsweresimulatedusingtheparametersofTable1.To obtainameasureoftheamplitudeoftheoscillations,wecalculated thestandarddeviationofthetimeseriesofthedensityofpreys.

Sincethepredatorsspecieshasasimilarbehavior,wedonotreport thosedata.Fig.7showstheresultsofoursimulations,asexpected, theamplitudeoftheoscillationsdecaysasthesizeofthelattice increases: forsmall lattices,an evolutiontowards anoisylimit cycle can be observed; as the size increases, the limit cycle destabilizes and the pattern of Fig. 4 appears. Additional increments in the size of the lattice further diminish the amplitude;however,instarkcontrasttotheresultsreported in Boccaraetal.(1994),wherenoisyoscillationsoflowamplitudeare observed for large lattices, we have foundthat thequalitative behaviorofthedensityofpreysremainsunchanged:theamplitude oftheoscillationsisstillbounded,butdifficulttopredict,andtheir periodispreserved.Suchbehaviorraisestheneedtoemphasize the differences between our model and previous work where scalingeffectsarestudied.Firstitmustbenotedthatourpredators haveamobilityhigherthanthatcommonlyfoundinotherlattice models, where dispersal (whether by movement, reproductive processes,orboth)isrestrictedtofirstneighbors.Itiswellknown thatlargedispersalrangesbringthedistributionofindividualson thelatticeclosetothe‘‘wellmixed’’case(DurretandLevin,1994;

Bra¨nnstro¨mandSumpter,2005),sooscillationsmightariseasa consequence of long range movement. Second, the spatial dynamics of oscillating prey–predator systems are commonly characterizedbypatternsakintotheinitialinvasionprocess,i.e., therearefrontsofpredatorsthatinvadepreyrichareasleaving behind a few site populated by preys. These survivors quickly repopulatepredatedareasandtheprocessstartsagain.Rozenfeld

Table3

Additionaldetailsforthelinearregressions.

SetB R² p-Value

Regressionforaandb 0.9595083345163955 0.0

Regressionfordande 0.48146049419825371 0.0

Fig. 7. Scalingbehaviorof theamplitude ofthe oscillationsobserved in the populationofpreys.Smalllatticesproduceorbitstoanoisylimitcycle;asthesizeof thelatticeincreases,theamplitudeoftheoscillationsdiminishesandthelimitcycle destabilizes. Forlarge latticesa new decreaseofthe amplitude is observed;

however,theoscillationsstillhavearegularperiodandtheiramplituderemains bounded,yetdifficulttopredict.

andAlbanoreferstothesephenomenaas‘‘alternatingpercolation events’’(RozenfeldandAlbano,2001).Asnotedinsection3,inthe proposed model such events are only observable during the transient phase; afterwards, we observe predators grouped together as swarms moving through the whole lattice while searching for zones with a ‘‘good’’ density of preys (a curious enoughreadermightwishtocomparethespatialpatternsfoundin AntalandMichel(2000),Boccaraetal.(1994),Mobiliaetal.(2007) and Rozenfeld and Albano (2001) with Fig. 2). The oscillatory behaviorobservedwhenmigrationisenabled,suggeststhat the timelapsebetweenthepredationofaparticularzonebyaswarm andthemomentuntilsuchzoneisqualifiedagainas‘‘good’’bya swarmisregular.

6. Futurework

Themean-fieldequationsdevelopedinSection4areonlyable topartiallypredictthebehavioroftheproposedmodel.Wehave shown that by including additional information about the populations of preys and predators in these equations, such predictionbecomesmoreaccurate.Isitpossibletomakefurther modificationstothemean-fieldequations,insuchawaythethey predicttheoscillatorybehavioroftheproposedmodelatsmall andintermediatescales?Considerasanexamplethesizeofthe swarms,itisreasonabletoexpectthatindividualsbelongingto themaresubjectedtoAlleeeffects.Inparticularthereshouldbea thresholdforthenumberofpredatorsinaswarm,belowwhicha swarmcannotsurviveforanyextendedperiodoftime,i.e.,alocal extinctionoccurs.Abovesuchthresholdtheswarmcoexistsand move. In (Pascual et al., 2011) it is shown that when local densitiescanbedescribedasfunctionsofglobaldensities,the dynamicsofthesystem,notonlyinthelong-term,butalsoduring thetransientphase,canbeapproximatedbyameanfieldmodel whosefunctionalformsincludepower-lawsofthelocaldensi- ties. Itremainsto be investigatedif a similar process can be applied in our model. The size of the clusters of predators dependsonthevaluetakenbytheparametersk1,k2andl,soitis reasonabletoexpectthatthatquantitiessuchasbirthrate of predators and predation rate can be estimated using this information.

7. Conclusions

Inthepresentworkwehaveanalysedthebehaviorofaprey–

predatorlatticemodelwherepredatorshuntaccordingtoalocal PSOalgorithm.Wehaveshownthatforinitialconditionsnearthe absorbingstate,wherethelatticeisfullycoveredbypreys,thereis aninitialwavefrontofpredatorsmovingoutwardsatthesame speed.Suchbehaviorisreflectedasamonotonicapproachtoa non-trivial fixed point where preys and predators coexist.

However, there are weak oscillations around the fixed point due to subsequent travelling waves produced by preys that survivetheinitialinvasionofpredators.Oncethetransientphase endsthesystemsexhibitsoscillationswitharegularperiod,but withanamplitudewhichisdifficulttopredict.

WehavealsoshownthatourlocalPSOalgorithmincreasesthe proficiencyofpredators,whichproducesadecreaseintheirdeath probability with respect to a simulation where no migration occurs.Ahigherdensityofpredatorsresultsinanincreaseinthe predationprobabilitywithacorrespondingdecreaseinthedensity ofpreys. By takinginto account thesechangesin a mean field model, we wereable toaccurately predictthe mean densities aroundwhichthedensitiesofbothpopulationsoscillatewhenthe migrationstageisenabled.

Finallywehaveexploredthebehavioroftheproposedmodel fordifferentsizesofthelattice;wefoundthateventhoughthe

amplitudeoftheoscillationsdiminishesasthesizeofthelatticeis increased,thesamequalitativebehaviorisstillobservedforlarge lattices,andtheperiodofoscillationsispreserved.

PSOalgorithmshavetheadvantageofbeingtheresultofthe studyofcooperativebehaviorfoundinanimalcommunities(bird flocksbeingamajorinfluence).Moreover,weareonlyproposinga verysimpleusecaseforthealgorithm;itiscertainlypossibleto adaptourPSOalgorithmstomorecomplexscenarios,e.g.,tomodel socialhierarchiesinacommunity;here,anindividualtakesonly intoaccounttheinformationthatmembersof‘‘highrank’’provide.

So farwe havefoundthatthroughcooperationit ispossibleto observeaninterestingspatialpatternthatisquitedifferentthan thepatternsreportedinpreviousworks.Yet,suchphenomenon also leads tooscillatory behavior, which hasbeen a matter of extensiveresearchinecologicaltheory.

AppendixA.Meanfieldequations

1.Intraspecificcompetition.Inordertoderiveameanfieldequation thatdescribes theintraspecificcompetitionstage, letusask thefollowing: what is the expectednumber of deaths in a seasonduetotheintraspecificcompetition?Asitwasshown previously, the death of a prey depends on the number of individualsofthepreyspeciescontainedintheneighborhood Mc,itis reasonablethentoexpectthatthemean numberof deaths each season depends on the mean number of individualsintheneighborhoodofeachcell,let ¯yt represent thisnumber.Tocalculate ¯ytweproceedasfollows:itisclear that if

C

t=0, then the mean number of preys in each neighborhood Mc is also zero, similarly if

C

t=1, then the mean number of preys within each neighborhood Mc

(excludingthepreyatthecenter)isjMcj1,aninterpolation betweenthepoints(0,0and(1,jMcj1)(seeFig.8)givesthe equation:

¯yt¼ðjMcj1Þð

C

tÞ (14)

Then,theprobabilitythatapreydiesasaconsequenceofthe intraspecificcompetitionis:

¯yt

jMcj1¼ðjMcj1Þð

C

tÞ

jMcj1 ¼

C

t (15)

Thedeathoftheallpreysinthelatticeisasequenceofevents that can be described through a random variable with binomialdistributionX.IfYtdenotesthenumberofpreysin

Fig.8.TheinterpolationassumesproportionalitybetweenCtandthenumberof preysintheneighborhoodMc.

thelatticeattimet,thentheexpectednumberofpreydeathsis givenby:

EX¼

C

tYt (16)

whereEistheexpectedvalueoperator,andYtisgivenby:

Yt¼

C

tjLj (17)

thus:

EX¼

C

²tjLj (18)

Toincorporatethisquantityinthemeanfieldequation,itmust beconvertedintoadensity,thisisdonebydividingEXbyjLj,and scalingtheresultbythecoefficient

a

,obtainingthatthedensity ofpreysthatiseliminatedeachseasonis

aC

²_t.Finallythemean fieldequationthatdescribestheintraspecificcompetitionis:

C

tþ1¼

C

t

aC

²t ¼

C

tð1

aC

tÞ (19) 2.Reproductionofpreys.Consideracellwithno prey,letuscall suchcellu.During thereproductionstage,allneighborsofu containing a prey might change its state, the number of neighborsbeingequaltojMyj1.Sinceeachneighborchooses thelocationfornewpreysatrandom,theprobabilitythatacell choosescelluduringreproductionis:

Pu¼ 1

jMyj1 (20)

Themeannumberofpreysintheneighborhoodofuisgivenby

¯y_t,thusthetotalnumberofreproduction‘‘attempts’’is:

e

Y¯yt (21)

where:



e

Yisthereproductivecapacityofpreys.

 ¯yt¼ðjMyj1Þð

C

tÞ is the mean number of preys in the neighborhoodofcellu.

LetXdenotetherandomvariablethatcountsthenumberoftimes celluischosen forreproduction,itisclearthatXisabinomial randomvariablewithparametersðPu;

e

Y¯ytÞ.Nowð1PuÞ^eyy¯tisthe probabilitythatnoneofthepreysintheneighborhoodofuchooses itforreproduction,then1ð1PuÞ^ey¯ytistheprobabilitythatcell uischosenatleastonceduringthereproductionstage.Themean densityofcellswithnopreys thatarechosen forreproduction duringthecorrespondingstageis:

ð1

C

tÞð1ð1PuÞ^eY¯ytÞ (22) 3.Deathofpredators.Eachseason,ifapredatorisnotlocatedata cellcontainingaprey,itdieswithprobability1.Thenumberof predatorsthatdieeachseasonisproportionaltothenumberof cellsnotcontainingaprey,theprobabilityoffindingacellwith suchapropertyisgivenby:

jLjYt

jLj ¼1Yt

jLj¼1

C

t (23)

Thentheexpectednumberofdeathsthatoccureachseasonis givenby:

ð1

C

tÞZt

where Zt is the number of predators at time t. Finally the expecteddecreaseindensityis:

ð1

C

tÞ

F

t (24)

4.Reproduction of predators. This process behaves just as the reproductionofpreys.

5.Predation.Eachseasonifapreyislocatedatacellcontaininga predator,itdieswithprobability1.Theprobabilityoffindinga cellwithapredatoris:

Zt

jLj¼

F

t (25)

Theexpectednumberofpreysdyingeachseasonis:

F

tYt (26)

Andtheexpecteddecreaseindensityisgivenby:

F

t

C

t (27)

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