String networks with junctions in competition models

String

networks

with

junctions

in

competition

models

P.P. Avelino

a

,

b

,

D. Bazeia

c

,

L. Losano

c

,

J. Menezes

a

,

d

,

e

,

∗

,

B.F. de Oliveira

f a_{CentrodeAstrofísicadaUniversidadedoPorto,RuadasEstrelas,4150-762Porto,Portugal}

b_{DepartamentodeFísicaeAstronomia,FaculdadedeCiências,UniversidadedoPorto,RuadoCampoAlegre687,4169-007Porto,Portugal} c_{DepartamentodeFísica,UniversidadeFederaldaParaíba,58051-970JoãoPessoa,PB,Brazil}

d_{InstituteforBiodiversityandEcosystemDynamics,UniversityofAmsterdam,SciencePark904,1098XHAmsterdam,TheNetherlands} e_{EscoladeCiênciaseTecnologia,UniversidadeFederaldoRioGrandedoNorte,CaixaPostal1524,59072-970Natal,RN,Brazil} f_{DepartamentodeFísica,UniversidadeEstadualdeMaringá,Av.Colombo,5790,87020-900Maringá,PR,Brazil}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received13November2016

Receivedinrevisedform18January2017 Accepted26January2017

Availableonline30January2017 CommunicatedbyC.R.Doering Keywords:

Populationdynamics Stringnetworks

Inthisworkwegivespecificexamplesofcompetitionmodels,withsixandeightspecies,whose three-dimensionaldynamicsnaturallyleadstotheformationofstringnetworkswithjunctions,associatedwith regions that have ahigh concentration ofenemy species. We study the two- andthree-dimensional evolution of suchnetworks, bothusing stochastic network and mean field theorysimulations. If the predation, reproduction and mobility probabilities do not vary in spaceand time, we findthat the networksattainscalingregimeswithacharacteristiclengthroughlyproportionaltot1/2_{,wheret isthe}

physicaltime,thusshowingthatthepresenceofjunctions,onitsown,doesnothaveasignificantimpact ontheirscalingproperties.

©2017ElsevierB.V.Allrightsreserved.

1. Introduction

Competitionmodelsarewidelyregardedasacrucialtoolto un-derstandthemechanismsleadingtobiodiversity

[1–4]

(seealso

[5,

6] fora review). Althoughthe simplestcompetition models usu-ally consider three species and allow for an equal number of basic microscopic actions (motion, reproduction and predation), many interesting generalizations, including further species and more complexinteraction rules, have beenconsidered in the lit-erature

[7–31]

.

In [14,15], a broad family of spatial stochastic May–Leonard modelswithanarbitrarynumberofspecieshasbeenintroduced. There,manyinterestingfeaturesofthisfamilyofmodels,including thedynamicsofcomplexnetworksofspirallingpatternsand inter-faces,havebeeninvestigatedindetail.Recently,in

[25]

,ithasbeen shownthatspecificsub-classesofthisfamilyofmodelsmaylead to theemergence ofstring networksinthree spatial dimensions. These strings are associated to predator-prey interactions which occurmainlyalonglinescorrespondingtoahighconcentration of enemy species. Thestrings studied in

[25]

do not havejunctions andtheirdynamicshasbeenshowntobecurvaturedriven.

In this paper we consider the emergence of string networks withjunctionsinthecontextofspecificspatialstochastic

competi-*

Correspondingauthor.

E-mailaddress:[email protected](J. Menezes).

tionmodelsbelongingtothegeneralfamilyintroducedin

[14,15]

. Weinvestigatethetwo- andthree-dimensionaldynamicsofthese modelsusingbothstochasticandmeanfieldnetworksimulations. The outline of this paperis asfollows. In Sec. 2 we introduce a sub-class of stochastic May–Leonardmodels allowing forthe for-mation ofstring networkswithjunctions inthreespatial dimen-sions.InSec.3weinvestigatethestochasticevolutionoftwo- and three-dimensionalnetworksinmodelsbelongingtotheabove sub-classwithN

=

6 orN

=

8 species.InSec.4westudythedynamics of suchsystems usingmeanfield theory simulations,considering alsotheparticularcaseofthecollapseofacircularloop.InSec.5

we constrainthe scaling parameter

λ

governingthe macroscopic dynamicsofthesemodels,consideringvariouschoicesforthe mo-bility,predationandreproductionparameters.Finally,weconclude inSec.6.

2. Familyofmodels

Here, we consider a sub-class of the more general family of spatialstochasticMay–LeonardmodelsintroducedinRefs.[14,15]. We investigate models with an even number of species N

>

4, whereeachspeciescompeteswithN

−

4 other species.The com-petition diagrams for the simplestcases with N

=

6 and N

=

8 species are illustrated in the left and right panels of Fig. 1, re-spectively.Thedoublearrowsindicatethatthepredationbetween competingspecies isbi-directional. Exceptforthelabeling of the differentspecies,thediagramsdepictedin

Fig. 1

areinvariant

un-http://dx.doi.org/10.1016/j.physleta.2017.01.038 0375-9601/©2017ElsevierB.V.Allrightsreserved.

Fig. 1. Diagramsdescribingthepossiblepredator-preyinteractionsinthemodels withN=6 (leftpanel)andN=8 (rightpanel)differentspecies.(Forinterpretation ofthecolorsinthisfigure,thereaderisreferredtothewebversionofthisarticle.) derrotationsby

θ

n

=

2

π

(

n

/

N

)

,withn

=

0

,

±

1

,

±

2

,

. . . ,

±(

N

−

1

)

, indicatingthepresenceofa ZN symmetry.

InthesemodelsindividualsofN speciesareinitiallydistributed onsquareorcubic latticeswith

N

sites.Thedifferentspeciesare labeled by i

=

1

,

. . . ,

N, and the cyclic identification i

=

i

+

kN,

wherek isaninteger,ismade.Thesumofthenumberof individ-uals ofthe speciesi (Ii) with thenumber ofempty sites (IE) is equaltothetotalnumberofsites(

N

),thatis

N



i=1

Ii

+

IE

=

N

.

(1)

At each time step a random individual (active) is chosen to interactwithone ofits nearest neighbors (passive), alsoselected randomly.Thenumberofneighbors isequaltofour,inthecaseof thetwo-dimensional squarelattice, andtosix, inthe caseofthe three-dimensionalcubiclattice.Theunitoftime



t

=

1 isdefined asthetimenecessaryfor

N

interactionstooccur(onegeneration). Thepossibleinteractions areclassifiedasMotion i



→ 

i

,

Re-productioni

⊗

→

ii

,

orPredationi

(

i

±

α

)

→

i

⊗

,

where



may beanyspecies(i)oran emptysite(

⊗

)and

α

=

1

,

. . . ,

(

N

−

4

)/

2. A constant predation probability p between competing species, and constant reproduction and mobility probabilities (r and m,

respectively),commonto all species, isconsidered. Althoughthis classofmodelsisdefinedfora genericeven N

>

4,inthispaper weshallinvestigateexplicitlythemodelsillustratedin

Fig. 1

,with

N

=

6 and N

=

8 differentspecies.

Throughout this letter, we consider three different parameter sets,characterized byadominationofmotion[set M:(m, p,r)

=

(0.8,0.1,0.1)],predation[setP :(m,p,r)

=

(0.1,0.8,0.1)]or repro-duction[set R:(m,p,r)

=

(0.1,0.1,0.8)]interactions.

3. Stochasticnetworksimulations

Weperformedaseriesofstochasticnetworksimulationsofthe models with six and eight species in two and three spatial di-mensions(insquareandcubiclattices,respectively).Thenumerical resultsshow that,assoon asthesimulations start,individualsof thesame species, originally distributed randomly throughoutthe lattice,sharecommonspatial regions.The individualstendnot to be close to competitors, butin the vicinity ofindividuals of the samespeciesorofotherneutralspecies.

Intwo spatial dimensionssuchspatial configurations promote the coexistenceof species whichmay be organized either clock-wise orcounterclockwise,around roughly circular regions witha significantly higher density of empty sites. Attacks and counter-attacksbetweencompeting specieson opposite sidesof the bat-tle cores ensure the stability of such spatial patterns. Similar two-dimensionalarrangementsofthe specieshavebeenfoundin

Fig. 2. Illustrationofadefect/anti-defectpairwithfourspeciesinamodelwithan arbitrarynumberofspeciesN,wherei=1,...,N

2.

Ref. [25].These can be described as defect/anti-defect configura-tions associated to clockwise/counterclockwise vortex states, re-spectively. The average area of the defect and anti-defect cores dependsontheinteractionprobabilities(m,p,r),butitisroughly constantintime.

In contrast with the model described in Ref. [25], where ev-ery species would compete with N

−

3 distinct ones, here each specieshaslesscompetitors (eachspecieshas N

−

4 competitors andthreeneutralspecies).Asa consequence,thenumberof con-figurationswhichpromotecoexistenceamongspeciesisenlarged. Infact,whileinRef.[25]allN specieshavetobegatheredaround the defectcores inorderto guarantee their stability,here, a sta-ble defect configuration requires only four species. Defects/anti-defectsarisewhenthedifferentspeciesdisposethemselvesinthe clockwise/counterclockwise vortex configurations shownin Fig. 2

{

i

,

i

+

N₂

,

i

+

1

,

i

−

N₂−2

}

,wherei

=

1

,

...,

N₂ (notethateachspecies

i does not interact with the species i

±

N₂−2 and i

+

N₂). These configurationsaccount for N₂ differentkindsofdefect/anti-defects withfourspecies.

Consider the competing species i and i

+

1 belonging to the defect/anti-defectconfigurationshownin

Fig. 2

.Theaverage num-ber ofattacks perunit timefromindividualsoftheouter species

i

+

1 supersedesthosefromindividualsoftheinnerspeciesi.This implies that individuals of the outer species tend to invade the territoryof theinner onescausingan approximation and annihi-lation of the defect/anti-defect pair. We shall show that the de-fect/anti-defectcoresattracteachother,havingavelocitywhichis, on average,inverselyproportional to thedistance betweenthem. Incontrast,apairofclockwise(orcounterclockwise)defects (anti-defects)cannotannihilateandrepeleachother.

Moreover, defect/anti-defectconfigurationswitha larger num-ber ofspeciesmayarise. Infact, defect/anti-defectconfigurations withn

=

4

,

...,

N₂+2 species, inwhicheach speciescompeteswith

n

−

3 otherspecies, mayappearinmodels withaneven number

N

>

6 ofspecies.

3.1. 6species

Let usfocus on the simplest casewith N

=

6. The left panel of

Fig. 3

showsonesnapshottakenfromatwo-dimensional10242 stochastic networksimulationof themodelwithperiodic bound-aryconditionsfortheparametersetP .

Each grid point is at mostoccupied by one individual which belongstothespeciesindicatedby thesamecolorasin

Fig. 1

.In addition,emptyspacesarerepresentedbywhitedots.

The spatial patterns show that all defects involvefour differ-ent species. Mostof the predator-prey interactions take place at the defect core, which is surrounded by two pairs of compet-ing domains, each domain being dominated by a single species. Morespecifically,one mayidentifythreetypesof defects-anti/de-fectswheredomainsdominatedbythespecies

{

i

,

i

−

1

,

i

+

1

,

i

−

2

}

(wherei

=

1

,

2

,

3)arrangethemselvesinthisorder(orthereverse) around thedefect(anti-defect) cores.Dueto theperiodic bound-aryconditions,thenumberofdefectsin eachsimulationis equal

worksimulationswithperiodicboundaryconditionsofthe N

=

6 model.We notice that the extension to threespatial dimensions of the dynamics presentedin the left panel of Fig. 3 gives rises toa string networkwithY-type junctions.Such strings represent regionswithasignificant largernumberdensityofempty spaces, whichappearasaconsequenceofthefrequentpredation interac-tionbetweencompetingspeciestakingplaceattheircore.

The snapshot shown in the left panel of Fig. 4 represents a 643 _{regionoftheentire256}3 _{three-dimensional network(the} re-sultsshownwere generatedbyassumingtheparameter set P ).It presentsthecontourplotsassociatedtoafixedvalueofthedensity ofempty sites, whichhighlight thepresence of a stringnetwork

Fig. 3. Snapshotsobtainedfromtwo-dimensional10242_{stochasticnetwork}

simula-tionsoftheN=6 (leftpanel)andN=8 (rightpanel)models.Thecolorscheme usedhereisthesameasrepresentedinFig. 1fori=1.Theresultswereobtained byassumingtheparametersetP .(Forinterpretationofthecolorsinthisfigure,the readerisreferredtothewebversionofthisarticle.)

Fig. 4. Snapshotstakenfromathree-dimensional643 _regionofa2563 _stochastic

networksimulationoftheN=6 model(leftpanel)andN=8 (rightpanel)models fortheparametersetP .

ofastringloop iscurvaturedriven anditisassociatedtothe ex-istence of a defect/anti-defect pair on anyplane intersecting the loop.TheloopcollapsewillbefurtherconsideredinSec.4.3.

Sinceanyplanardefectoranti-defectinvolvesonlyfourspecies, eachspeciesjoinstwodifferentstringsinthreespatialdimensions. ThisisresponsiblefortheappearanceofY-typejunctions,defined asthemeetingpointofthreedifferentstrings.Thestabilityofthe junctionsisensuredbypredator-preyinteractions,giventhateach specieshascompetitorsintwodifferentstrings.

Theupperpanelof

Fig. 5

illustratesthedifferentpossible con-figurations of a string junction in the N

=

6 model, where i

=

1

,

2

,

3.Notethat,inthismodel,allthespeciesareinvolvedinthe formationofaY-typejunctions,eachoneofthembeingassociated totwodifferentstrings.

3.2. 8species

TheresultsobtainedforN

=

8 arequalitativelysimilartothose presentedfor N

=

6,butwithan increasedcomplexity associated with the larger number of species. In this case, apart from the fourdifferentdefects/anti-defectswithfourspeciesrepresentedby

{

i

,

i

+

4

,

i

+

1

,

i

−

3

}

(wherei

=

1

,

..

4),thereare eight

defect/anti-defect configurations composed by five species. They are

com-posedbytheclockwise/counterclockwisedispositionofthespecies

{

i

,

i

+

4

,

i

+

1

,

i

−

2

,

i

+

3

}

(wherei

=

1

,

..

8)aroundthedefectcores. Wecallthedefectsformedby fourandfivespecies,typeIandII, respectively.

The right panel of Fig. 3 depicts one snapshot taken from a 10242 _{stochastic network simulation ofthe N}

₌

_{8 model forthe} parameter set P . The colors indicate the species that individuals belongto(seeFig. 1).Whitedotsrepresentemptysites.

In addition, the results provided by three-dimensional simu-lations are presented in the right panel of Fig. 4. The snapshot representsa643_{regionofathree-dimensional256}3_stochastic net-work simulationswithperiodicboundaryconditionsofthe N

=

8 modelfortheset P .

Analogously to the N

=

6 model, the extension to three spa-tial dimensions gives rises to a string network with junctions. Nonetheless, in the N

=

8 model there are two different types of strings, in which either four (type I) or five different species (type II) areinvolved.One stringoftype Iandtwooftype II are required in order to produce thestable Y-type junctions seen in

Fig. 5. IllustrationoftheformationofY-typejunctionsinthemodelsN=6 (leftpanel)andN=8 (rightpanel).Thecross-sectionsofthedifferenttypesofstringsjoining thejunctionsarepresented,wherei=1,...,N2.

Fig. 6. Snapshotsobtainedfromtwo-dimensional10242_{meanfieldsimulationsof}

theN=6 (leftpanels)and N=8 (rightpanels)modelsfortheparameterset P . (Forinterpretationofthecolorsinthisfigure,thereader isreferredtotheweb versionofthisarticle.)

thesimulationsofthe N

=

8 model.Thelower panel ofFig. 5 il-lustratestheformationofaY-typejunctioninthismodel.

4. Meanfieldtheorysimulations

LetusdefineN

+

1 scalarfields(

φ

0,

φ

1,

φ

2,

. . .

,

φ

N) represent-ingthefractionofspacearoundagivenpointoccupiedbyempty sites (

φ

0) and by individualsof the species i (

φ

i), satisfying the constraint

φ

0

+ φ

1

+ . . . + φ

N

=

1. For an even N

>

4 the mean fieldequationsofmotion

˙φ

0

=

D

∇

2

φ

0

−

rφ0 N



i=1

φ

i

+

p N



i=1

⎛

⎜

⎝

N−4 2



α=1

φ

i

φ

i+α

+

N

_

−1 α=N+4 2

φ

i

φ

i+α

⎞

⎟

⎠ ,

(2)

˙φ

i

=

D

∇

2

φ

i

+

rφ0

φ

i

−

p

⎛

⎜

⎝

N−4 2



α=1

φ

i

φ

i+α

+

N

_

−1 α=N+4 2

φ

i

φ

i+α

⎞

⎟

⎠ ,

(3)

describethe average dynamicsofthe modelsstudied inthe pre-vious section (see [7],for moredetails).Here, a dot representsa derivativewithrespecttothephysicaltimeandD

=

2m isthe dif-fusionrate.

4.1.Twoandthree-dimensionalnumericalresults

Fig. 6depictstwosnapshotstakenfrom10242 meanfield net-worksimulationswithperiodicboundaryconditionsoftheN

=

6 (left panel) and N

=

8 (right panel) models for the parameter set P .Initialconditionswith

φ

i

=

1 ifi

=

s and

φ

i

=

0 ifi

=

s were setateachgridpoint(

φ

0 wassettozeroateverygridpoint).Here

s isaspeciesdrawnrandomlyateachgridpoint.

Theresultsprovided bythemeanfieldsimulations(Fig. 6)are similar to those obtained from the stochastic network evolution (Fig. 3),exceptforthenoise(thecolorschemeisthesameinboth figures).Thiscorrespondencealsooccursinthethree-dimensional simulations. In particular, the macroscopic dynamics depicted in the snapshot shown in Fig. 7 obtained from three-dimensional 2563 meanfieldtheorysimulationsoftheN

=

6 and N

=

8 mod-els,issimilar(again,exceptforthenoise)tothatshownin

Fig. 4

.

4.2.Defectprofile

Letusnowconsiderthedefectprofiles,i.e.,thestationary num-ber density of empty spaces in and around the defect cores. In other words,let us studythe spatial distribution of

φ

0 around a defectcenter,consideringthatthedefecthasradialsymmetryand iscenteredatr

=

0.

Fig. 7. Snapshotstakenfromthree-dimensionalmeanfieldsimulationsoftheN=6 (leftpanel)and N=8 (rightpanel)models.Thefiguresrepresent643_regionsof

2563_{simulationsfortheparametersetP .}

Fig. 8showsthevalueof

φ

0,asafunctionofthedistancer to thedefectcore,obtainedfordefectsassociatedwithfourandfive species from mean field simulations ofthe N

=

6 (upper panel) and N

=

8 (middleandlower panels)models. The disposition of the species around the defect cores is shown in the inset plots. Thenumericalresultsshowastrongdependenceofthedefect pro-fileonthemodelparameters.Specifically,thedefectprofileheight is larger in the case of set P, dominated by predation interac-tions.Onthecontrary,alargerreproductionrateleadstoasmaller numberdensityofemptyspaces(set R).Finally,alargermobility parameterleads tothebroadeningofthedefectprofile,sincethe individualsaremorelikelytomoveoutsidethecoreofthedefect intoenemyterritory(set M).

Fig. 8alsoshowsthat,inthemodelwithN

=

8 speciesandfor fixed(m, p,r),thedefectisbroaderwhenthenumberofspecies composingthedefectisincreasedfrom4 to5.

4.3. Stringloop

We now investigate the collapseof a circular string loop us-ing mean field theory simulations in a cubic lattice. To identify thestringwedefineanewvariable

ϕ

(

r

,

t

)

≡

max

(φ

0

(

r

,

t

))

− φ

c₀

,

0

)

. Here,

φ

₀c represents a threshold which guarantees that only grid points with a high number density of empty sites, close to the core of the string, are identified as belonging to the string. The averagenumberdensityofempty sitesassociatedtothe stringis thendefinedby

ρ

(t

)

=

1

N

2



r

ϕ

(



r,t

) .

(4)

We recallthat theaveragenumberofempty sitesper unit string length(

μ

)doesnotchangesignificantlywithtime.Therefore,the loop perimeteris proportional to

ρ

(

t

)

andthe area ofthe string loopa

(

t

)

evolvesproportionallyto

ρ

2

₍

_t

₎

_.

Thetimeevolutionoftheareaa

(

t

)

ofthecircleenclosedbythe loopisdeterminedusingmeanfieldtheorysimulations ina 2563 cubiclattice.Werunsimulationsfortwodifferentthresholdvalues inorder toensure theaccuracyandreliability oftheresults. The upperpanel of

Fig. 9

displaystheevolutionoftheareaofastring loopformed byfourdistinctspeciesa4

(

t

)

intheN

=

6 modelfor theparameterset P .Notethattheresultsarealmostidenticalfor both choicesofthresholdvalues,

φ

₀c

=

0

.

06 and

φ

₀c

=

0

.

15,which correspondapproximatelyto25% and60% ofthemaximumof

φ

0 (

φ

₀max)atthecoreofthestring.Similarresultswerefoundforthe evolutionofthearea ofthecircleenclosed bystringloops associ-atedtofourandfivedistinct speciesa4

(

t

)

anda5

(

t

)

inthe N

=

8 modelfor theparameter set P (see lower panel of

Fig. 9

).

Fig. 9

showsthattheloopareadecreaseslinearlyintimeaccordingto

a(t)

=

a0

1

−

t tc

,

(5)

Fig. 8. DefectprofilesobtainedfordefectswithfourandfivespeciesfrommeanfieldsimulationsforN=6 (leftpanel)andN=8 (rightandlowerpanels).Thered,green andbluelinesrepresenttheresultsgeneratedbyassumingtheparametersetsP ,M andR,respectively.Theinsetplotsrepresentthedispositionofthespeciesaroundthe defectcores.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 9. MeanfieldevolutionoftheareasofstringloopsintheN=6 (leftpanel)andN=8 (rightpanel)models,computedbytakingdifferentthresholdvalues.Notethat a4(t)anda5(t)representtheareasoftypeIandtypeIIstringloops,respectively.TheresultswereobtainedbytakingtheparametersetP .

wheretc isthe collapsetime or, equivalently,that radius of cur-vature decreasesproportionally to t1/2_{. Asimilar resulthas been} obtainedinRef.[15]foradifferentmodelallowing forstring net-workswithoutjunctions.

5. Scalingbehavior

Finally,weconsiderthescalingbehaviorofthemodels investi-gatedintheprevioussectionsusingmeanfieldtheorysimulations. Tothispurpose,letusdefinethecharacteristiclength L ofthe de-fectnetworkas L

=



μ

ρ

∝



1

ρ

,

(6)

where

μ

isthe averagenumberofempty spacesassociated with thedefects(perunitstringlength,inthreespatialdimensions)and

ρ

istheaverage numberdensityofempty sitesassociatedtothe defectnetworkdefinedbyEq.(4).

We constrain the evolution of the characteristiclength of the stringnetwork withtimeusingsets of10meanfieldsimulations of themodels M, R and P . Eachsimulation starts withdifferent random initial conditionsand foreach ofthem we compute the scalingexponent

λ

associatedwiththescalinglawL

∝

tλ.

The upperand lower panels of Fig. 10 show the dependence of

λ

onthethreshold

φ

₀c forthe sets M, P and R,forN

=

6 and

N

=

8,respectively(inunitsof

φ

c₀

/φ

max₀ ).ForN

=

8,

φ

₀max istaken asthemaximumof

φ

0atthecoreofthestringoftypeI.Theerror bars representthe standard deviationin an ensembleof 10 sim-ulations. Fig. 10showsthat ifthe

φ

c₀ is between20% and60% of

φ

max₀ , the scaling constant

λ

doesnot show a significant depen-denceonthethreshold.Outsidethisinterval,thisisnolongerthe case.Forlowervaluesof

φ

₀c,thishappensbecauselowdensity re-gions farawayfromthecorearebeingtakenasbelongingto the defect.Ontheotherhand,thenumberoflatticepoints associated withthedefectmaybecometoosmallforhighervaluesof

φ

₀c.

TheaverageevolutionofL withtimet wasobtainedforsetsof 10 distinct two- and three-dimensionalmean field network

sim-Fig. 10. Thedependenceofthescalingexponentλonthethresholdφc

0 inameanfieldsimulationwith N=6 (leftpanel)and N=8 (rightpanel)species.Theresults

wereobtainedbycarryingout10 simulationsof20482_{two-dimensionalnetworksforawiderangeofφ}c

0,fortheparametersetsM,P andR.Theerrorbarsrepresentthe

standarddeviationinanensembleof10simulations.

Fig. 11. ScalingbehaviorforthemodelswithN=6 andN=8 differentspeciesobtainedusing20482_{two-dimensional(leftpanel)and256}3_{three-dimensional(rightpanel)}

meanfieldnumericalsimulationsfortheparametersetP .

ulations(20482 _and2563_{) withrandom initial conditionsforthe} parameterset P .

Fig. 11

showsthat thecharacteristiclengths L6S (6 species) and L8S (8 species) evolve in reasonable agreement withthe scaling law L

∝

tλ_{, with}

_λ

₌

₁

_/

_{2, characteristic of} net-worksin whichthe dynamicsis curvature driven.Still, the devi-ationwithrespect to

λ

=

1

/

2, alreadypresentinthe caseofthe stringnetworks without junctions studied in[25], appears to be significantanddeservesfurtherinvestigation.Here,thepoints de-notetheaveragevalueofL computedfromthesimulationandthe errorbarsprovideinformationontheroot-mean-squaredeviation foreachsetof10 simulations.InallcasesthevalueofL was nor-malizedtounityatt

=

100,theparameters (m, p,r)were setto (0.10,0.80, 0.10) and thethreshold

φ

₀c was fixed at 40% of

φ

max₀

(forN

=

8 thethreshold

φ

c₀wasobtainedbyconsideringthevalue of

φ

max₀ obtainedfortypeIIstrings).

These results are consistent with those obtained in Ref. [25]

formodelsallowingforstringnetworkswithoutjunctionsinthree spatialdimensions.Asimilarbehaviormayalsobefoundinother physicalsystems,inparticularinthecaseofcurvaturedriven dy-namicsofstringnetworksincondensedmatter.

6. Commentsandconclusions

Inthisworkwehaveshownthattherearespecificsub-classes, withanevennumberofspecies,ofamoregeneralfamilyofMay– Leonard models which lead to the formation of string networks withjunctions,associatedtoregions withahighconcentrationof empty spaces. We have investigated the dynamics of these net-works using stochastic and mean field network simulations, as-sumingthatthepredation,reproductionandmobilityprobabilities areconstantinspaceandtime.Wehavefoundthatthepresenceof junctionsdoesnothaveasignificantimpactonthescaling behav-ior ofthecharacteristicmacroscopic scaleof thenetwork L with

the physical time t, showing that it grows roughly proportional to t1/2_.

Acknowledgements

We thankCAPES, CNPq,CNPq/Fapern, andFCT-Portugal for

fi-nancial support. The work of PPA was supported by Fundação

para a Ciênciae a Tecnologia(FCT) throughthe Investigador FCT contract ofreferenceIF/00863/2012 andPOPH/FSE(EC)by FEDER funding through the program ProgramaOperacionaldeFactoresde Competitividade,COMPETE.

References

[1]B. Kerr,M.A.Riley,M.W.Feldman,B.J.M. Bohannan, Mobilitypromotes and jeopardizesbiodiversityinrock–paper–scissorsgames,Nature418(2002)171. [2]J.J.Leisner,J.Haaber,Intraguildpredationprovidesaselectionmechanismfor bacterialantagonisticcompounds,Proc.R.Soc.B,Biol.Sci.279(2012)4513. [3]H.Cheng,N.Yao,Z.Huang,J.Park,Y.Do,Y.Lai,Mesoscopicinteractionsand

speciescoexistenceinevolutionarygamedynamicsofcycliccompetitions,Sci. Rep.4(2014)7486.

[4]A.J.Daly,J.M.Baetens,B.D.Baets,Theimpactofinitialevennessonbiodiversity maintenanceforafour-speciesinsilicobacterialcommunity,J.Theor.Biol.387 (2015)189–205.

[5]R.C.Sole,J.Bascompte,Self-OrganizationinComplexEcosystems,PrincetonUP, Princeton,2006.

[6]M.A.Nowak,EvolutionaryDynamics:ExploringtheEquationsofLife,Harvard UniversityPress,2006.

[7]R.May,W.Leonard,Nonlinearaspectsofcompetitionbetweenthreespecies, SIAMJ.Appl.Math.29(1975)243.

[8]T.Reichenbach,M.Mobilia,E.Frey,Mobilitypromotesandjeopardizes biodi-versityinrock–paper–scissorsgames,Nature448(2007)1046.

[9]T.Reichenbach,M.Mobilia,E.Frey,Noiseandcorrelationsinaspatial popula-tionmodelwithcycliccompetition,Phys.Rev.Lett.99(2007)238105. [10]M.Peltomaki,M.Alava,Three- andfour-staterock–paper–scissorsgameswith

diffusion,Phys.Rev.E78(2008)031906.

[11]W.-X.Wang,X.Ni,Y.-C.Lai,C.Grebogi,Patternformation,synchronization,and outbreakofbiodiversityincyclicallycompetinggames,Phys.Rev.E83(2011) 011917.

[12]A.Dobrinevski,E.Frey,ExtinctioninneutrallystablestochasticLotka–Volterra models,Phys.Rev.E85(2012)051903.

[13]A.Roman,D.Konrad,M.Pleimling,Cycliccompetitionoffourspecies:domains andinterfaces,J.Stat.Mech.TheoryExp.(2012)P07014.

[14]P.P.Avelino,D.Bazeia,L.Losano,J.Menezes,VonNeummann’s andrelated scal-inglawsinrock–paper–scissors-typegames,Phys.Rev.E86(2012)031119.

[18]A.Dobrinevski,M.Alava,T.Reichenbach,E.Frey,Mobility-dependentselection ofcompetingstrategyassociations,Phys.Rev.E89(2014)012721.

[19]C.Rulquin, J.J.Arenzon,Globallysynchronizedoscillationsincomplexcyclic games,Phys.Rev.E89(2014)032133.

[20]P.P.Avelino,D.Bazeia,L.Losano,J.Menezes,B.F.deOliveira,Interfaceswith internalstructuresingeneralizedrock–paper–scissorsmodels,Phys.Rev.E89 (2014)042710.

[21]B.Szczesny,M.Mobilia,A.M.Rucklidge,Characterizationofspiralingpatterns inspatialrock–paper–scissorsgames,Phys.Rev.E90(2014)032704. [22]L.Varga,J.Vukov,G.Szabó,Self-organizingpatternsinanevolutionaryrock–

paper–scissorsgameforstochasticsynchronizedstrategyupdates,Phys.Rev.E 90(2014)042920.

[27]M.F.Weber,G.Poxleitner,E.Hebisch,E.Frey,M.Opitz,Chemicalwarfareand survivalstrategies inbacterialrangeexpansions, J.R. Soc.Interface11 (96) (2014)20140172.

[28]D.Grošelj,F.Jenko,E.Frey,Howturbulenceregulatesbiodiversityinsystems withcycliccompetition,Phys.Rev.E91(2015)033009.

[29]B.Intoy, M.Pleimling,Synchronization and extinction incyclicgames with mixedstrategies,Phys.Rev.E91(2015)052135.

[30]G.Szabó,K.S.Bodó,B.Allen,M.A.Nowak,Fourclassesofinteractionsfor evo-lutionarygames,Phys.Rev.E92(2015)022820.

[31]A.Roman,D.Dasgupta,M.Pleimling,Atheoreticalapproachtounderstand spa-tialorganizationincomplexecologies,J.Theor.Biol.403(2016)10–16.