Evolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population ﬂow

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Chaos, Solitons and Fractals

Nonlinear Science,andNonequilibrium andComplexPhenomena journalhomepage:www.elsevier.com/locate/chaos

Evolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population ﬂow

Junpyo Park

Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea

a rt i c l e i nf o

Article history:

Received 15 June 2020 Revised 21 September 2020 Accepted 2 November 2020 Available online 21 November 2020 Keywords:

Rock-paper-scissors game Population ﬂow Community paradigm Multistability Oscillatory behavior

a b s t r a c t

Classicframeworksofrock-paper-scissorsgamehavebeenassumedinaclosedcommunitythataden- sityofeachgroupisonlyaffectedbyinternalfactorssuchascompetitioninterplayamonggroupsand reproductionitself.Inrealsystemsinecologicalandsocialsciences,however,thesurvivalandachange ofadensityofagroupcanbealsoaffectedbyvariousexternalfactors.Oneofcommonfeaturesinreal populationsystemsinecologicalandsocialsciencesispopulationflowthatischaracterizedbypopula- tioninflowandoutflowinagrouporasociety,whichhasbeenusuallyoverlookedinpreviousworkson modelsofrock-paper-scissorsgame.Inthispaper,wesuggesttherock-paper-scissorssystembyimple- mentingpopulationflowandinvestigateitseffectonbiodiversity.Fortwoscenariosofeitherbalanced orimbalancedpopulationflow,wefound thatthe populationflowcan strongly affectgroupdiversity byexhibiting richphenomena.Inparticular, whilethe balanced flowcanonlyleadthe persistent co- existenceofallgroupswhichaccompaniesaphasetransition throughsupercriticalHopfbifurcation on differentcarrying simplices,the imbalancedflowstrongly facilitatesrichdynamicssuchas alternative stablesurvivalstatesbyexhibitingvariousgroupsurvivalstatesandmultistabilityofsolegroupsurvivals byshowingnotfullycoveredbutspirallyentangled basinsofinitialdensitiesduetolocalstabilitiesof associatedfixedpoints.Inaddition,wefoundthat,thesystemcanexhibitoscillatorydynamicsforcoex- istencebyrelativisticinterplayofpopulationflowswhichcancapturetherobustnessofthecoexistence state. Applyingpopulationflowin therock-paper-scissorssystemcan ultimatelychange acommunity paradigmfromclosedtoopenone,andourfoundationcan eventuallyrevealthatpopulationflowcan bealsoasignificantfactoronagroupdensitywhichisindependenttofundamentalinteractionsamong groups.

1. Introduction

Forthefastdecade,sinceapplied toelucidate complexbehav- iors inrealecosystems [1–5], therehave beengreatly challenged on population systems of non-hierarchical cyclic competition in bothmacroscopicandmicroscopiclevels.Ingeneral,fromtheper- spective of survival states, biodiversity in cycliccompetition systems have been interpreted based on a metaphor ofrock-paper- scissors (RPS) game by exploiting additional interactions includ- ing noise[6],mobility[7–13],habitatsuitability[14],intraspeciﬁc competition [15,16], mutation[17–22],quasi birthanddeath pro- cess [23], asymmetric niche and interactions [24–27], and inter- patchmigration[28,29]bypresentingvariousnonlineardynamical features[30–33].Inadditiontoimplementinginecosystems,rock- paper-scissorsgamehasbeenexploitedtounderstanddynamicsin socialsciences[34–37].

E-mail addresses: [email protected] , [email protected]

Typically,inclassicapproachesonRPSgames,fundamentalfac- tors toaffecta changeof adensityin eachpopulation group are interspecificcompetition(whichisusuallyregardedby predation) andreproduction to lead the decrease andincrease of a density, respectively, which imply that the change of population densi- tieswill depend on intracommunityinteractions. Thus the entire structure of interactions among three groups is of a closed hy- percycle.In real systems,however, inaddition to such intracommunityinteractions, a densityofeach group can bealso affected byinflowandoutflowofpopulation,i.e.,migrationofpopulation, which are commonly witnessed by various forms in ecological, biological, and social sciences [5,8,38–43]. For example, in mar- ket communities, thosewho want to start a business are willing to choose one of the promising businesses, and thus the start- up ofa particularbusiness mayincrease thesize ofthe business itself. In the caseof the establishment of the franchise industry, thisprocessmayincrease thenumberofbranchesoftheindustry.

However, anincrease inthe numberofbranchesnaturallycauses

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Fig. 1. Schematic diagrams of different community structures on rock-paper- scissors game: (a) a closed structure in classic systems and (b) an open structure with population flow. Blue straight arrows indicate intergroup competition which occurs with a rate 1 in a cyclic manner. In each group, red dashed loop demon- strates intragroup competition with a rate p, and green straight and red dashed arrows describe the inflow and outflow of populations in each group, respectively.

(For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

intergroupcompetitionbetweenbranches,inwhichcasesalesmay be stoppedforbrancheswithpooroptions.Besides, dueto inter- nalcircumstances,itmayhappenthatthestore(orbranches)will decidetoshutdown itselfandthescaleofsuchbusinessmarkets maybe gettingsmaller. Thus,in socialsciences, such aflow pro- cessingroupsorpopulations isacommonfeatureandhasasig- nificant impact onthe survival ofeach group. Nevertheless,even if there were severalefforts toelucidate the effect ofpopulation flow on biodiversity ofcyclicgames within theframework of in- terpatch migration [28,29] which have been considered between finitepatches, tothebestofourknowledge,asanexternal factor toaffectadensityofagroupinmacroscopiclevels,thereisnoat- tempttoaddressthepopulationflowinrock-paper-scissorsgame.

Motivated from thispoint and inthe perspective ofchanging community paradigm, in this paper, we investigate the effect of population ﬂowon groupdiversity wherethegoverning relation- ship amonggroups is the metaphor ofrock-paper-scissors game.

We describe population flow consistingof inflow andoutflow in each group byimplementing themechanismofnaturalbirth and death processes inclassic Lotka-Volterra system[44,45], andde- velopan analytictheorybasedonthesystemofrateequationsto explain numerical evidencesand providethe effectofpopulation flow. Briefly,ourmain findingsare thefollowings.The firstresult isthat,incontrasttopreviousworksofrock-paper-scissorsgames under the closed communitystructure, balanced population flow can promote persistent coexistence by forming different carrying simpliceswhichwilldependonmagnitudesofinflowandoutflow.

The second resultis that imbalanced population can exhibit rich dynamicsofgroupsurvivalsbyexhibitingalternativestablestates consistingofvariousgroup survivalstates,emergenceofmultistability among singlegroup survivals, and oscillatory dynamics for coexistencedependingonrelativisticinterplayofﬂowmagnitudes.

InSection2,wemodelarock-paper-scissorsgamewithpopula- tion ﬂowinthemacroscopicframework.InSection3,we provide our resultswithin two perspectivesaccordingto consideration of population ﬂowsnumerically,andareanalyzedmathematicallyin Section4.ConclusionanddiscussionsareaddressedinSection5. 2. Model

Fundamental reactions of rock-paper-scissors games adopting intragroup competition (see Fig. 1(a)) can be deﬁned by the set offollowingrules[8,9,15,16,46,47]:

XY−→^σ X∅, YZ−→^σ Y∅, ZX−→^σ Z∅, (1) X∅−→^μ XX, Y∅−→^μ YY, Z∅−→^μ ZZ, (2)

XX−→^p X∅, YY−→^p Y∅, ZZ−→^p Z∅, (3) where∅presentsvacancieswhichcanallowreproduction(2).

For well-mixed population, reactions (1)–(3) can be generally describedbyrateequationsinthemean-ﬁeldframeworkofMay- Leonard limit. Let x(^t), y(^t), and z(^t) ^be ^the ^densities ^of ^three groups X, Y, and Z attime t, respectively. Then the determinis- ticsystemofthreegroupsincorporating(1)–(3)canbewrittenby thefollowingsetofrateequations[8,9,14,16,47]:

⎧ ⎪

⎨

⎪ ⎩

dx(^t) dt =x

(

^t

)

μ{

¹⁻

ρ (

^t

) }

⁻

σ

^z

(

^t

)

−^p₂x

(

^t

)

,

dy(^t) dt =y

(

^t

)

μ{

¹⁻

ρ (

^t

) }

⁻

σ

^x

(

^t

)

⁻^p2y

(

^t

)

,

dz(t) dt =z

(

^t

)

μ{

¹⁻

ρ (

^t

) }

⁻

σ

^y

(

^t

)

⁻^p2z

(

^t

)

,

(4)

where

ρ

(^t)=x(^t)+y(^t)+z(^t)^is^the^total^density^at^time^t.

In classic rock-paper-scissors models, changes of population densitiescanonly occureitherdecreasingbyintergroupcompeti- tion(1)orincreasingbyreproduction(2),whichmeansthedensity of each group ultimatelydepends on internal mechanisms. Thus, theentirestructure ofinterplaycan be regardedas“closedcom- munity” (see Fig. 1(a)). In real systems, however, the population densitycanbealsoaffectedbypopulationinflowandoutflow.For example,incasesofpoliticalparties,somenonaffiliatedindividuals maywanttojoinoneofpoliticalparties[48],ordefectedindividu- alsmaywanttojoinanotherparties[33].Inthiscase,adensityof certaingroup mayincrease byjoiningnewmembers.Inaddition, membersinagroup maywanttostop theirpoliticalactivities by intragroupcompetition orchanging their politicalideologies,and henceagroupcanbecomeshrinkbyoutflowofitsmembers.

Inthisregard, consideringpopulation flow ineachgroup may changetheinterplaystructureas“opencommunity” whichisillus- tratedinFig.1(b),andwethuswonder howsurvivorshipofcom- peting groupscan change by population flow. To address the ef- fectofpopulationflowongroupdiversityinthecycliccompetition system, we simply implementa similar waytonatural birthand death processes in the Lotka-Volterra system [44,45] to describe inflow andoutflowofpopulations in each group,respectively. To makeanunbiasedcomparisonwithpreviousworkswhichhasno population flow [8,9,16], we assume

σ

=

μ

=1. Then Eq.(4) can berewrittenby:

⎧ ⎪

⎨

⎪ ⎩

dx dt =x

(

¹⁻

ρ )

⁻^z⁻^p2x+

β

1−

δ

1

,

dy dt =y

(

¹−

ρ )

−x−₂^py+

β

2−

δ

2

,

dz dt =z

(

¹−

ρ )

−y−₂^pz+

β

³⁻

δ

³

,

(5)

where

β

iand

δ

i (ⁱ=1,2,3)îndicate^theînflowândôutflow^rates ofgroupsX,Y,andZ.

Suchadescriptiontoinflowandoutflowofpopulationsbyus- ing several parameters maybe eventually matched to the asymmetric rock-paper-scissors model [46], and may be regarded as naturalbirth anddeath in each group. Eveniftwo terms forin- flowandoutflowineachgroupcanbewrittenbyasingleparam- eter, we will distinguishtwo parameters to realizeand elucidate theeffectofpopulationflowindetail.

3. Results

Inthissection, we carry outnumerical simulationsto provide global aspects on the effect of population flow on biodiversity amongthree groups fromtwo perspectives, which can be classi- fiedbasedonthesymmetryofflows:

(a) balancedpopulation flowthat all groupshavesameinflowand outflowrates,i.e.,

β

i=

β

^and

δ

i=

δ

(ⁱ=1,2,3),wherethetwo rates

β

^and

δ

^may^benonuniformlygiven,and

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Fig. 2. (a–c) Phase transitions of attractors with different pand (d) maximum and minimum values of attractors of Eq. (5) depending on p. Parameters (β, δ)are given by (0 . 4 , 0 . 2) for tops and (0.3,0.5) for bottoms. The initial condition is given by (0.625,0.128,0.247). (a–d) As pincreases, the system exhibits phases transitions from asymptotically stable heteroclinic cycles to periodic orbits, and asymptotically stable sinks which occurs on different carrying simplices. (d) The phase transition recasts in a wide spectrum of pthrough bifurcation diagram by measuring maximum and minimum values of attractors versus p. Regardless of (β, δ), Eq. (5) always exhibits the phase transition between heteroclinic cycles and sinks via p = 1 .

(b) imbalancedpopulationflowthat allratesofinflowandoutflow maybe given nonuniformlywhich mayindicate that both in- flowandoutflowratesmaydiffertoeachgroup.

We implement a Runge-Kutta method and symbolic calculations on ourallnumerical simulationsto presentphase portraits, basinstructures,andcertainprobabilities.

3.1. Effectofbalancedpopulationﬂow:persistentcoexistencewith formingdifferentcarryingsimplices

AccordingtotheanalysiswhichwillbeexplainedinSection 4, wecanpredictthatEq.(5)mayexhibitaphasetransitionbetween asymptotically stableheteroclinic cyclesandasymptotically stable attractors constituted by an interior ﬁxed point as the parameter pvaries,wheretwostatesofheterocliniccyclesandattractors are noted by P₁ and P₃, respectively. To be concrete, regardless of choices of

β

^and

δ

, such a phase transition only dependson pifparameters

β

^and

δ

^satisfy ^the^common^condition

δ

<

β

+1. Since thephase transitioncan occurwhether pexceeds1ornot, we considerpby0.5,1,and1.5.Inthisregard,forgivendifferent parameters settings,e.g., (

β

,

δ

)=(⁰.4,0.2)^and^(0.3,0.5)^satisfying the condition

δ

<

β

+1, phase transitions and a bifurcation dia- gramwithrespecttothechangeofpareillustratedinFig.2.

AsshowninFig.2(a–c),Eq.(5)exhibitsaphasetransitionfrom heteroclinic cycles to sinks by showing periodic orbits as p increases. The interesting feature in phase portraits is the change of carryingsimplicesandinvariant manifoldsforsolutions ofthe system. Previous worksof rock-paper-scissorsgame withinternal interplay havebeendepictedwithout changing thecarryingsim- plex, andtheformationofattractorsstartedfromsimplex S₃ and ischanged graduallyby intragroupcompetition.Byapplyingpop- ulation ﬂow, however, we found that thephases ofsolutions are formedondifferentcarryingsimplices,andtheinvariantmanifold for attractorscan be also affected.Such a phase transitionof attractors ina wide spectrum of pare depictedconcretely by uti- lizing abifurcationdiagramversus passhowninFig.2(d),where a bifurcation diagram is derived by exploitingthemaximum and

Fig. 3. Basins of asymptotically stable heteroclinic cycles(blue) and attractors(green) which are distinguished by p = 1 for all cases. The white region indicates that associated fixed points of P 1and P 3are not defined since the existence conditions of fixed points are not satisfied. As βincreases, the basins move from left to right, and sufficiently high δis required to validate stable existence of two states. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

minimum value of the attractor in each p. In particular, similar to previous works [16,47,49], the phase transition can occur as Eq.(5)fallsintotheclassofHopfbifurcationinwhichissupercrit- ical,andit isacommonfeature regardlessofrates ofpopulation ﬂow (

β

,

δ

) ^whenever^the ^rates^satisfy ^the ^condition

δ

<

β

+1 to validate existencesoffixed pointsofP1 andP3,wheretwostates are defined by (9)–(11) and (15) in Section 4.1, respectively. The stabilityof heteroclinic cycles featured inFig. 2(a) can be determined by a calculation of the saddle value V which is defined by[50]:

V= 3

i=1

V_i= 2−p p

³

>1, ^∀p<1 (6)

forall ﬁxedpoints(9)–(11)ofP₁,andweeasily ﬁndthat therate ofintragroupcompetitionpistheonlyparametertodeterminethe stabilityofheterocliniccycles(seeTheorem1inSection4.1).Thus, asillustratedinFig.2,controllingpasabifurcationpointcanyield acommonphasetransitionregardlessof(

β

,

δ

)^:^fromheteroclinic cyclestoattractorsinwhichareasymptotically stableaccompany- ingwithperiodicorbits.

Inaddition,fromFig.2(d),we ﬁndthat thestablecoexistence canbesensitivelyaffectedbybothpopulationﬂowandintragroup competition.Forexample,for (

β

,

δ

)=(⁰.4,0.2),the solutioncan be validatedfor p>0.4 whilethat with (

β

,

δ

)=(⁰.3,0.5)îs ^defined for p≥0. In other words, Eq. (5) may not have any fixed

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pointsdependingonthechoiceof(

β

,

δ

)^with^respect^to ^p,which may implythat theremay be aphase transitionof basinsofpa- rametersforstableﬁxedpointsofP₁andP₃.Tobeconcrete,such phasetransitionsofbasinsofparametersconsistingof(

δ

,p)^by^ﬁx- ing

β

^forînstance âre ^presented ⁱⁿ ^Fig. ³ ^where ^borders âmong

distinct basins indicate the thresholds of existence and stability conditionsofcorrespondingﬁxedpointsof P₁andP₃.

From Fig.3, wefound that, fortheconsidered rangeof

δ

, the higher inflowraterequiresa relativelyhighoutflowrateforvali- datingP₁andP₃whichareindicatedbyblueandgreencolors.For whiteregionsineachpanelinFig.3,wehavenoparticularindica- tionsofassociateddynamicalbehaviors.Inthewhiteregions,fixed pointsofP1andP3arenotdefined,i.e.,theexistenceconditionsof fixed pointsoftwo statesarenot satisfied.Inthiscase, whenwe considerparameters inthewhiteregions, thesystemmayexhibit two features either(i) the trajectories ofsolutions of the system willconvergetotheoriginwheretheinteriorfixedpointofP₃has negativecomponents;or(ii) thetotaldensityofthreegroupsex- ceeds1evenifthedensityofeachgroupislessthan1.

Under balanced population flow, we found that the interplay between inflow and outflow of the population which is affected by intraspecificcompetitioncan lead tostablecoexistence.While theasymptoticallystableheterocliniccyclewhichisconstitutedby threeabsorbing fixedpointsofP₁ canbeappearedby theextinc- tion state on spatially extendedsystems [8,14,16,32],it mayindi- cateaweakcoexistencebecausesolutionsofEq.(5)areeventually traveling near the cycle. Nevertheless, it is obviousthat the bal- ancedflowcanfacilitatepersistentcoexistenceofallgroups.

3.2. Richdynamicsbyimbalancedpopulationﬂows

3.2.1. Basinstructuresofparametersofimbalancedﬂowfordiverse survivalstates

We now explore dynamical features withimbalanced popula- tionflow.Inasimilarapproachtothebalancedflow,wecanobtain that the Eq. (5) with imbalanced (nonuniform) population flows

canalsopossessthreetypesofsurvivalstates,Q₁,Q₂,andQ₃for describingsurvivalofonlyonegroup,existencesoftwointeracting groups, andcoexistenceof all groups, respectively,which are de- ﬁnedbythenumberofsurvivinggroups(seethedetailedformof associatedﬁxedpointsinAppendixA).

Sincethere areseveralunknown parameters tobe considered, unfortunately,itisquitedifficulttogainoveralldynamicalfeatures simultaneously.Tohandlethisissue,wemayimaginethefollowing situation: Forexamplein social sciences, ina particular business situation,whencompetitionamonggroupsinsimilar fieldsisac- tiveandeachgroupisgrowingwell,theinfluxofnewparticipants to join (orinvest) each group can sometimesoverheat while the outflowmaydifferfromgrouptogroupbycharacteristics[51–54]. Inthis case,it is possibleto restrict excessiveinflow intogroups overall.Based onsuch a plausiblesituation,we mayassume that the inflow ratesin all groups maybe restrictedwithin a certain level,whichmeansthetotalamountofinflowratesmaybegiven byaconstant:3

i=1

β

i=,wheretheinitialoutflowrateineach group is givennonuniformly. In thispursuit, basedon the linear stabilityanalysisoffixedpointsofthreedistinctsurvivalstatesQ₁, Q₂,andQ₃,theoveralldynamicsofgroupdiversitybypopulation inflowcanbepresentedonatriangularphasespaceofinflowrates whichareillustratedinFig.4.

Fig. 4 shows the basin structure for each stable state in Eq. (5) for the given nonuniform outﬂow rates (

δ

1,

δ

2,

δ

3)= (⁰.1,0.3,0.4)^. ^According ^to ^the combination of

β

i (i=1,2,3), Eq.(5)exhibitsvariousdynamicalfeaturesonsurvivalstateswhich aredefinedbythenumberofsurvivinggroupswhilethebalanced populationflowcanonlyleadtopersistentcoexistence.Inparticu- lar,theimbalancedpopulationflowcanyieldsurvivalstatesofany twointeractinggroupsoncyclicdominantrelationshipswhichare notobservedinthebalancedpopulationflowinSection3.1.Inthis case,wefoundthat(a)thebasinstructureonthephasespacedoes notpresentallstatesofQ₂,and(b)anyoneofsurvivalstatescan onlyappearatmoderateratesofinflowsandintragroupcompeti- tion,which mayimplythat thesurvival stateswill be sensitively

Fig. 4. Basin structures for nonuniform inflow rates with fixed nonuniform outflow rates (δ1, δ2, δ3)= (0 . 1 , 0 . 3 , 0 . 4). In each panel, colors indicate basins of parameters for different survival states. According to the symmetry-breaking of inflow rates, Eq. (5) can possess various survival states which can be well-defined by both pand , and exhibit multistable states as shown in (a). On the other hand, if the total population inflow exceeds a certain level, the system does not show any survival state.

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determined bythecombinationofinteraction rates.Ineach given p,thealmostofinflowratesofasmallamountof^may^validate theexistenceofoneofthesurvivalstates.Asîncreases,^however, the portionofbasinsforstablesurvivalstatesare decreasingand certainvaluesof^relatively^larger^than^p^do^not^yieldâny^typeôf basinstructuresasshowninFig.4(e)and(j).Thereasonwhysta- ble survival statesare decreasingwiththe increase of ^may^be predictedbylinearstabilityanalysisofcorresponding fixedpoints andsensitiveinterplaybetweenintergroupandintragroupcompe- tition. Thus, specific types offixed points can be definedforthe givenparametersettingonoutflows.

It has been generallyaccepted that moderate orstrong intragroup competition can promote coexistence of cyclic competing three groups [16,32,47,49] which associates to Q₃ in our model.

However, eveniffor p=1,Fig.4showsthatnosurvivalstate in- cluding Q₃ isdefined.Tobe concrete,forthelarge amountofin- flow ratessuch as=2.5,thesystemdoesnotpresentanysur- vivalstateonthephasespaceofparametersforp=1.Onthecon- trary,forp=1.5whichisshowninFig.4(o),thelevel=2.5still exhibitbasinsofstatesQ₁,Q₂,andQ₃,evenifbasinsofQ₁andQ₂ areverysmall.Basedonthebasinstructure,wemayfindthatthe strengthofintragroupcompetitioncanincreaseasthetotallevelof inflows increases,andaddress thatthe viabilityofgroupscan be sensitively affected by bothpopulation flow andintragroupcom- petition.

3.2.2. Emergenceofmultistabilityamongsolesurvivalstates Even if basins are quite narrow, the feature to focus on the basin structure in Fig. 4 is the emergence of multistability. To be concrete, Fig. 4(a) exhibits two basins of parameters to lead multistable states between two fixed points in Q₁: Q₁(^X^&^Y) ând Q₁(^X^&^Z)^. ^From ^the ^phase ^transition ôf ^basin ^structures âs în- creases,wemayexpectthatthelower^mayâlso^yield^multista- bilitiesamongfixedpointsofQ1.Sinceitisstill ambiguousofthe multistability of differenttypesof fixed points,to investigatethe emergence of multistability indetail by considering casesconsti- tuted by fixed pointsof differentclassesfor thegivenparameter foroutflow, we measurethe probability P_M that how multistability state can arise frequently dependingon the flow mechanism, whereP_Misdefinedby[33,47,49,55]:

PM= n

{ ( β

¹^,

β

²^,

β

³

) | ∃

multistableamongQ_i

(

ⁱ⁼¹^,²^,³

) }

n

( β

¹^,

β

²^,

β

³

)

³

i=1

β

ⁱ⁼

, ^∀

β

ⁱ^≥⁰

, (7)

and the P_M exhibits that the multistability can only emerge between any twofixed points ofQ₁ for specificranges of ând ^p aspresentedin Fig.5(a)–(c).In ourmodel,thereisno possibility toobservethemultistablestatesofallfixedpointsofQ₁asshown inFig.5(d).

Forthegivenoutflowrates,wefindthat,specificcasesofmul- tistablestatesbetweentwostatesinQ₁,inparticularQ₁(^X^&^Y)ând Q₁(^X^&^Z), can be revealed more frequently. In particular, for two statesQ₁(^X^&^Y)ând^Q1(^X^&^Z),phaseportraitsofsolutions andcor- respondingbasinstructuresofinitialconditionsunderspecificpa- rametersettingsarepresentedinFig.6.

The emergenceofmultistability amongsolegroup survivalsin systems of evolutionary dynamics has been already reported in Refs. [31,33], where the interplays in the proposed systems are slightlychangedeitherbidirectionalcompetitionortransferringin- dividualsamonggroups.Theinterestingfeatureonmultistabilityin ourmodelistheformationofbasinsofinitialconditionsformulti- stablestates.While thephase spaceofinitialconditionshasbeen fully coveredby anyone ofbasinsforsole stateswhenthe mul- tistabilityofextinctionoccursinpreviousworks,we observethat, accordingtothemultistablestate, basinstructurescanbe discon-

Fig. 5. Degree of multistability P Mamong fixed points of Q 1in combination with and p. In the given outflow rates, when the strength of intragroup competition is quite weak, Eq. (5) can exhibit multistable states of any two points in Q 1with respect to inflow rates at small ranges, i.e., an inflow rate βiof each group is quite small.

Fig. 6. Phase portraits and corresponding basins of initial conditions for the multistability in Q 1with (, p)= (0 . 5 , 0 . 5) associated to Fig. 4 (a). In each figure, a trian- gle and a bullet indicate the initial condition and one of fixed point of Q 1, respectively. Basins of initial conditions for corresponding fixed points are presented by different colors. (a) For parameters (β1, β2, β3)= (0 . 0055 , 0 . 476 , 0 . 0185) satisfying =0 . 5 , solutions of Eq. (5) starting from different initial conditions can converge to one of fixed points either Q 1(X ) or Q 1(Y ). (b) Similar to (a), the system exhibits multistable states of Q 1(X )and Q 1(Z ) depending on the initial condition where the parameter set of inflow rates is given by (β1, β2, β3)= (0 . 214 , 0 . 003 , 0 . 283).

tinuous dependingon inflow rates.To be concrete, forthe given settingofinflowrates,asdepictedinFig.6(b),twobasinsofinitial conditionsforQ₁(^X) ând^Q1(^Z) âre^spirallyêntangledând^do ^not coverthephasespaceS₃entirelywhilethoseforQ₁(^X)ând^Q1(^Y) fullycoverthephase spaceS₃ andaredistinguishedintotwo re- gionsasshowninFig.6(a).Thus,wecanconcludethatmultistabil- ityemergesamongthesolegroupsurvivalsandismoresensitive totheinitial densitiesofthethreegroupsunderpopulationflow, especiallyatlowlevelsofinflow.

3.2.3. Characterizingrobustnessofcoexistencebyinterplaybetween ﬂowandintragroupcompetition

EveniftheeffectofimbalancedflowisexploredthroughFig.4, understanding balance of flows according tointragroup competi- tionis stillambiguoussince Q₃ can bestablesensitively dependingon bothflows andintragroupcompetitionasshowninFig.4. Tobe concrete, when arate ofintragroupcompetition pis fixed whichissufficientlylargetoyieldstablecoexistence,thebasinof parameters (

β

1,

β

2,

β

3)^forcoexistence isshrinking andmaydis-

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Fig. 7. The probability S coex(8) in a combination with and for different p. (a-c) For lower passociated to weak intragroup competition, there is no parameter combina- tions of population flow to yield stable coexistence. (d) As pincreases, population flow can affect coexistence even if small amounts of sets are constituted. (e-h) For p ≥1 , the flow can strongly lead stable coexistence with weak levels of and . Each panel may be classified by four regions of parameters with respect to the sensitivity of coexistence on population flow. In each panel, the red bullet indicates the threshold of that Q 3is not defined which also increases as pincreases. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

appearastheamountofîncreases.În^this^regard,^to^predict^the stablecoexistenceandcharacterizationoftherobustnessofcoexis- tencewhenitoccursstably,itisnecessarytoexploretheinterplay betweenpopulationflowandintragroupcompetition.Characteriz- ingtheinterplayandrobustnessmaybeevaluatedbythenumber ofcomponentsofallratesforstablecoexistence,referredtoasthe sensitivity of coexistenceScoex, whichcan imply thedegree of coexistence based on the basin area. To implement the sensitivity, we assume that both inflowandoutflowcan varyat certainlev- els:3

i=1

β

i=^and³i=1

δ

i=,respectively.Inthisregard,sim- ilarlytoP_M(7),thesensitivityofcoexistencecanbecalculatedby [31,33,47,49,55]:

Scoex= n

{ ( β

1,

β

2,

β

3,

δ

1,

δ

2,

δ

3

) | ∃

^stablecoexistence

}

n

⎧ ⎨

⎩ ⁽ β

¹^,

β

²^,

β

³^,

δ

¹^,

δ

²^,

δ

³

)

3

i=1

β

ⁱ⁼

, 3

i=1

δ

ⁱ⁼

,

∀

β

i,

δ

i≥0.

⎫ ⎬

⎭

, (8)

where ^and ^range ^on⁰≤,≤3 forthe given p≥0, and thelandscapesofScoexwithdifferentpareillustratedinFig.7.

From thelandscapeofS_coex inFig. 7,wefound thatstableco- existenceissensitivelyaffectedbypopulationflowandintragroup competition. In particular, the classification of robustness of stable coexistence can be captured by oscillatory dynamics of solutions dependingontotalmagnitudesofinflowandoutflowwhich hasbeenalsofeaturedinotherframeworksofmathematicalmod- els [56,57]. For the weak intragroup competition, as shown in Fig.7(a)–(c),thereisnoattemptthatallgroupscancoexistforany frequency of population flow. The stable coexistence, i.e., the Q₃ becomesa stableattractor,can bepossible as pincreaseseven if small amountsofparameter setscan validate whichare detected by low probability as in Fig. 7(d). As illustrated in Fig. 7(e)–(h), suchstablebehaviorcansimilarlyappearaspincreases,andsmall levels of

λ

ând ⁱⁿêach ^p^with ^p≥1 can yieldthe robust coexistence which corresponds to Scoex=1.The interesting point is the transition ofa critical ^(which îs îndicated ^by ^the ^red ^bullet)thatthedegreeScoex becomes0.Tobeconcrete,forp≥1,the criticalpoint increasesatthesame timeas pincreases.Evenifa smallareaofparameters ofând^can^validate ^Q3,thecoexis-

tencecanappearsensitivelydependingonimbalancedﬂowswith lowprobabilitiesoverallasintragroupcompetitionisintensiﬁed.

Intrinsically,ifpopulationoutflowsaremore frequentthanin- flows, all groups will not be ableto coexist, andwe found that, byScoex,thephenomenonoccursevenifintragroupcompetitionis moderateorstrong.ThelandscapeofScoexprovidethatcoexistence can be sensitively affected by population flow considering intragroup competition, and in particular, strong (robust) coexistence can be only promoted forthe weak level of inflow and outflow overall.

In the absence of population flow either balanced or imbalanced, the system which can be definedby (4) can only exhibit threedistinct phaseswithout showingmultistabilitiesamongdis- tinctsurvival states,andtheoverallfeaturearesimilarlyobtained tothecaseofbalancedflow.However, thesurvival statesofthree groupscan be strongly changedunder imbalancedflow of populations by exhibitingvarious survival states, multistability among singlegroupsurvivals,andoscillatorybehaviorsforcoexistencede- pendingonrelativisticrelationsbetweenflowmagnitudes.

4. Analysis

We now provide theoriesbased on linear stabilityanalysis to supportournumericalfindings. Withintwoperspectivesofpopu- lationfloweitherbalancedorimbalancedcase,thelinearstability analysisforfixedpointsaresimilarly obtainedforall cases.Thus, we provide the analysisin detail for thecase ofbalanced popu- lationflow,andfocusontheemergence ofmultistability andsur- vivaloftwointeractinggroupsforimbalancedflow.

4.1. Balancedpopulationﬂow

Under the uniform consideration of each inflow and outflow rate to yield balanced flow, direct calculations dx/dt=dy/dt= dz/dt=0canyieldthatEq.(5)canpossessthreedistinct survival stateswhichcanbeclassifiedbythenumberofsurvivinggroups.

Proposition 1. Under the condition

β

+1>

δ

^with

β

,

δ

>0, Eq.

(5)possesses three types of ﬁxed points which canbe deﬁned with additionalconditions:

(7)

(a) P₁forthesurvivalofonlyonegroup:

2

( β

⁻

δ

⁺¹

)

p+2 ,0,0

, (9)

0,2

( β

−

δ

+1

)

p+2 ,0

, (10)

0,0,2

( β

⁻

δ

⁺¹

)

p+2

, (11)

for2(

β

−

δ

)≤p,

(b) P₂forsurvivalsofanytwointeractinggroups:

(

^x^˜,y˜,0

)

=

2p

( β

⁻

δ

⁺¹

)

p²+4p−4 ,2

(

^p−2

)( β

⁻

δ

⁺¹

)

p²+4p−4 ,0

, (12)

(

⁰^,^y^˜^,^z^˜

)

⁼

0,2p

( β

−

δ

+1

)

p²+4p−4 ,2

(

^p−2

)( β

−

δ

+1

)

p²+4p−4

, (13)

(

^x^˜,0,z˜

)

=

2

(

^p−2

)( β

⁻

δ

⁺¹

)

p²+4p−4 ,0,2p

( β

⁻

δ

⁺¹

)

p²+4p−4

, (14)

whicharedeﬁnedbyp>2,and (c)P₃forcoexistenceofallgroups:

(

^x^∗^,^y^∗^,^z^∗

)

⁼²

( β

⁻

δ

⁺¹

)

p+8

(

¹^,¹^,¹

)

^, ⁽¹⁵⁾

for6(

β

−

δ

)−2≤p.

Proof. Since we easily obtain ﬁxed pointsof forms above by direct calculations,we hereonlyprove theexistence conditionof eachﬁxedpoint.

Wefirstinvestigatethestabilityoffixedpoints(9)–(11).Accord- ing to the formofthe fixed points, we mayeasily findthe exis- tenceconditionwhichsatisfiesthefollowings:

0≤2

( β

−

δ

+1

)

p+2 ≤1

⇒2

( β

⁻

δ

⁺¹

)

^≤^p⁺²

⇒2

( β

−

δ )

≤p,

wheretheborderdeﬁnedbyp=2(

β

−

δ

)^.

For the ﬁxed points (12)–(14) of P₂, the components among three pointsare circulant,andthus,duetothesymmetry, we in- vestigatethestabilityofP₂byemployingtheﬁxedpoint(12):

(

^x^˜^,^y^˜^,⁰

)

⁼

2p

( β

−

δ

+1

)

p²+4p−4 ,2

(

^p−2

)( β

−

δ

+1

)

p²+4p−4 ,0

.

For p∈R+, the sign of denominator p²+4p−4 changes either negative for 0<p<−2+2√

2 or positive for p>−2+2√ 2. For the parameter p on 0<p<−2+2√

2, since the value x˜should be positive, we obtain the condition p>0 and

β

−

δ

+1<0.At the same time, since y˜ should be also positive, we obtain p>2 which contradictsto theassumption 0≤p<−2+2√

2.Thus, the point (12) isnot deﬁned for0<p<−2+2√

2. However, for p>

−2+2√

2, both coordinatesbecome positive if

β

−

δ

+1>0 and p>2.Thusweobtaintheexistenceconditionof(12)as

p>2,

β

−

δ

+1>0. (16)

We ﬁnally investigate the existence condition of the interior ﬁxed point P₃ (15), which naturally yields existence conditions

β

−

δ

+1>0 and p≥6(

β

−

δ

)−2 since the total sum of three speciesshouldnotexceed1.

The stabilityof each ﬁxed point can be investigated by linear stabilityanalysis.Here,wedeﬁne

f₁=x

1− 1+ p 2

x−y−2z+

β

⁻

δ

,

f2=y

1−2x− 1+ p 2

y−z+

β

−

δ

, f3=z

1−x−2y− 1+ p 2

z+

β

⁻

δ

,

andtheJacobianmatrixJoftheequationsystemis

J=

⎡

⎣

∂^f1(^x,y,z)

∂^x −x −2x

−2y ^∂^f²⁽_∂^x_y^,^y^,^z⁾ −y

−z −2z ^∂^f³⁽_∂^x,y,z_z ⁾

⎤

⎦

, (17)

where

∂

^f¹

(

x,y,z

)

∂

^x ⁼¹⁻

⁽

^p⁺²

⁾

^x⁻^y⁻²^z⁺

⁽ β

⁻

δ )

,

∂

^f2

(

^x,^y,^z

)

∂

^y ⁼¹⁻²^x⁻

⁽

^p⁺²

⁾

^y⁻^z⁺

⁽ β

⁻

δ )

,

∂

^f³

(

x,y,z

)

∂

^z ⁼¹⁻^x⁻²^y⁻

⁽

^p⁺²

⁾

^z⁺

⁽ β

−

δ )

.

FromtheJacobianmatrix(17),weobtainassociatedeigenvalues fortheﬁxedpoints(9)–(11)ofP₁as:

λ

1=

δ

−

β

−1,

λ

2= p

( β

−

δ

+1

)

p+2 ,

λ

³⁼

⁽

^p⁻²

⁾⁽ β

⁻

δ

⁺¹

)

p+2 , (18)

whereallpointsofP₁havesameeigenvalues.

Since

β

−

δ

+1>0, we ﬁnd that

λ

1 in (18) is negative, and

λ

² ^is ^positive ^for ^p∈R+. Thus, the ﬁxed points(9)–(11) are al- waysunstable,buttheycanconstituteheterocliniccyclesinwhich are asymptotically stable when

λ

3 is negative, i.e., p<2. To be concrete,theheterocliniccyclecanbe asymptoticallystablewhen p<1asproveninTheorem1.

Theorem1. Threeabsorbingﬁxedpoints(9)–(11)canconstitutehet- erocliniccyclesinwhichareasymptoticallystableforp<1whenthe pointsaredeﬁned.

Proof. Toidentifytheexactconditionforthestabilityofthehete- rocliniccycles,weneedtocalculatethesaddlevalueV:[50]

V= 3

i=1

V_i= 3

i=1

−

λ

^s1

λ

^u

i

, (19)

whereeigenvalues satisfy

λ

u>0>

λ

s₁>

λ

s₂.Since

β

+1>

δ

, we have

λ

³⁻

λ

¹⁼

⁽

^p⁻²

⁾⁽ β

⁻

δ

⁺¹

)

p+2 −

( δ

⁻

β

⁻¹

)

= p−2 p+2+1

( β

⁻

δ

⁺¹

)

= 2p

p+2

( β

⁻

δ

⁺¹

)

>0,

for p<2,which yields

λ

3>

λ

1,andhenceV_i ateachpoint ofP₁ canbecalculatedby

V_i=−

λ

^s1

λ

^u ⁼⁻

λ

3

λ

²

since

λ

^u=

λ

2whichisalwayspositive forp>0.Accordingtothe deﬁnitionofV (19),weobtainV asaform:

V= 3

i=1

Vi= 3

i=1

−

λ

3

λ

2

=

−

λ

3

λ

2

3

= 2−p p

³

,

and V >1 for p<1, which implies the heteroclinic cycles are asymptotically stable. Otherwise, we obtain V≤1 and thus the heterocliniccyclesareunstable(seeFig.8).