文档介绍：该【expanding spatial domains and transient scaling regimes in populations with local cyclic competition p. p. avelino资料 】是由【赖大文档】上传分享，文档一共【8】页，该文档可以免费在线阅读，需要了解更多关于【expanding spatial domains and transient scaling regimes in populations with local cyclic competition p. p. avelino资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。PHYSICALREVIEWE99,052310(2019),1,,1,3,,ísicaeCiênciasdoEspa?o,UniversidadedoPorto,CAUP,RuadasEstrelas,PT4150-762Porto,Portugal2DepartamentodeFísicaeAstronomia,FaculdadedeCiências,UniversidadedoPorto,RuadoCampoAlegre687,PT4169-007Porto,Portugalologia,UniversidadeFederaldoRioGrandedoNorteCaixaPostal1524,59072-970Natal,RN,Brazil4InstituteforBiodiversityandEcosystemDynamics,UniversityofAmsterdam,SciencePark904,1098XHAmsterdam,herlands5DepartamentodeFísica,UniversidadeEstadualdeMaringá,,87020-900Maringá,PR,Brazil6DepartamentodeFísicaTeóricaeExperimental,UniversidadeFederaldoRioGrandedoNorte,59078-970Natal,RN,Brazil(Received18November2018;published24May2019)Weinvestigateasix-speciesclassofMay-petingspatialdomains,eachoneinhabitedbythreespecieswiththeirowninternalcyclicrock-paper--?ndthatasthree-speciesdomainsshrink,thereisanincreasingprobabilityofextinctionoftwoofthespeciesinhabitingthedomain,withtheconsequentcreationofone--,withasloweraveragegrowthrateofthecharacteristiclengthscaleLofthespatialdomainswithtimet,takesplacebeforethetransitiontoastandardL∝t1/2scalinglaw,:.[15,16].Weshallinvestigateindetailaparticularpropertyofthissubclassofmodelsas-Thereisampleevidencethatnonhierarchicalinteractionssociatedwiththeformationofsingle-speciesspatialdomainsbetweenindividualsofdifferentspeciesplayacrucialroleduringthe?nalstagesofthecollapseofthree--domains,andtheirsubsequentgrowthiftheirinitialsizeistion,reproduction,-scalinglawdescribingthetimeevolutionofthecharacteristictitionmodels,,andMayandLeonard[1–3].-calledrock-paper-[4,5](see[6–10]umericalsimulationofmodelswithadditionalinteractions).Despiteitssimplicity,,bothana-posedofthreespecieswithlyticallyandnumerically,anovelfeaturepresentinthesecyclicselectioninteractions[4,11,12](see[13,14]forrecentsimulations:upiedbyareviews).,acriticalinitialradius,beyondIn[15,16],abroadfamilyofspatialstochasticMay-whichsphericallysymmetricspatialdomainsareexpectedLeonardmodelswithanarbitrarynumberofspecieshasbeentoexpand,umericalintroduced,thusgeneralizingthestandardrock-paper-scissorssimulations,paredwiththeanalyticalmodel(seealso[17–21]).-speciesplexspatialstructures,whichmayincludespatialdomainsontheevolutionofthepopulationisstudiedspiralswithanarbitrarynumberofarms[16],,withaparticularemphasisonthescalingofthe([22–24]),andstringswithorwithoutjunctions[25,26].,thescalinglawsgoverningthedynamicsofsuchsystemsmayalsobeverydiverse[16,23](seealso[27–29]).InthispaperweshallconsideraparticularsubclassoftheInthispaperweconsiderthepopulationdynamics,intwomoregeneralfamilyofMay-Leonardmodelswithanarbitraryspatialdimensions,foraparticularsix-speciessubclassofthenumberofspecies(N)introducedin[15,16].InthesemodelsmoregeneralfamilyofMay-Leonardmodelswithanarbitraryindividualsofvariousspeciesaredistributedonasquare2470-0045/2019/99(5)/052310(8)052310-1?2019AmericanPhysicalSocietyAVELINO,MENEZES,DEOLIVEIRA,ANDPEREIRAPHYSICALREVIEWE99,052310(2019)1passiveisanemptysiteandapredationinteractionispicked),(ourtimeunit).-4tions,whereateachsiteanindividualofanyofthesixspeciesoranemptysitewasselectedwithauniformdis-:bidirectionalpredationcreteprobabilityof1/7atthebeginningofthesimulation.(solidarrows)andcyclicpredation(dashedarrows).ThepredationFigure2presentsresultsobtainedfromasinglerealizationinteractionprobabilityofspeciesiwithspeciesi±1,andi±3isofa20002latticenumericalsimulationofourmodelas-equaltop(solidarrows).Thepredationinteractionprobabilityofsumingp=,p=p,m=,r=+PRSspeciesiwithspeciesi2isequaltopPRS(dashedarrows).Exceptandlowerpanelsshowsnapshotsofthespatialpatternsatforthelabelingofthedifferentspecies,this?gureisinvariantundervariousinstantsoftime:t0=0,t1=16600,t2=37555,t3=rotationbyanangleof2kπ/6,wherekisaninteger,thusleadingto55420,t4=89870,t5=97360,t6=103784,t7=197600,=211161,andt9==Ii/N,(1)ijThedifferentspeciesarelabeledbythenumber(or),with=i,j=1,...,N,andemptysitesshallbedenotedby?.Thefortheentiretimespanofthesimulation:,andthecolorsthenumberofemptysitesbyI?.Thepossibleinteractionsareasfollows:[30]→i?,Thesnapshotatt=t0=,essentiallytwotypesofspatialdomainsappear—aspatialdomainbeingde?nedasa→,iiconnectedspatialpatchdominatedbyindividualsbelongingtoandreproductionasinglepartnership(either{1,3,5}or{2,4,6}).ursbecausemutualpredationtakesplacebetweenthegroupsofi?→ii,specieswhererepresentseitheranindividualofanyspeciesoran{1,3,5},{2,4,6}.,weshallconsidermodelswithsixspecies(N=6).Asaresult,twoenemypartnershipsareformed,asshowninurwithprob-,respectively(thesameforallspecies),andenemiesiftheyhavebidirectionalpredationinteractionsbe-ordingtotweenthem—thespeciesconnectedbydouble-({1,3,5}and{2,4,6}shallarrowsrepresentthepredationinteractionprobabilitiespandbereferredtoasenemypartnershipsintheremainderofthepPRS,respectively).Notethatthemobility,reproduction,,-,theyinteractwithacyclicpredationingthespeciesweusemodulararithmetic,wherenumbersrule(),creatingalocalrock-paper-wraparounduponreaching1orN(=jmodN,wheremodspatialdomains,theydonotcrosstheboundariesduetothedenotesthemodulooperation).Exceptforthelabelingofthemutualpredationbetweenthemembersofdistinctpartner-differentspecies,-speciesof2kπ/6,wherekisaninteger,,asdescribedindetailInourmodel,ateachsimulationstep,thealgorithmran-in[15],withtheirvelocitybeingroughlyproportionaltotheirupiedsitetobetheactiveone,,However,?nalstagesofthecollapseofthree-speciesindividualattheactiveposition—inthispaperweusethevonspatialdomainscanleadtotheformationofspatialdomainsNeumannneighborhood(orfour-upiedbyasinglespecies,whichmaythenbeabletoacentralcell(theactiveone),thesnapshotstakenatt=t1andt=(.,ifthet2depictsmallsingle-speciesspatialdomainsofindividuals052310-2EXPANDINGSPATIALDOMAINSANDTRANSIENT…PHYSICALREVIEWE99,052310(2019)(withp=,pPRS=p,m=,r=),,respectively(notethatnosingle-species(III)thegrowthofone-speciesspatialdomains,iftheirinitialspatialdomainispresentinthesnapshottakenatt=t3).,outsideasingle-speciesspatialdomain,indi-Forlarget,thecharacteristicsizeLofthespatialdomainsvidualshavetodealwithacyclicpredationamongpartnersincreasesand,asaconsequence,-single-speciesspatialdomainsdecreases(arigorousde?nitionquence,individualsfromthesingle-).However,oncetheyemerge,-isresponsiblefortheexpansionofthesingle-speciesspatialshotstakenatt=t5,t=t6,andt=t8,thesesingle-speciesdomains,asshowninthesnapshotstakenatt=t4andt=e,withacharacteristic(weshallquantify,inthefollowingsection,parabletotheoneofthethree-speciesspatialsphericallysymmetricsingle--speciesareastoexpand).Asasingle-speciesspatialdomaingrowslarger,,forsmallttheaveragesizeofthespatialbyaspiralwavefront,asdepictedinthesnapshottakenatdomainsistinyandsingle-speciesspatialdomainsarepresentt=-speciesspatialdomainand,consequently,,,,are(I)thecurvaturedominateddynamicsofinterfacessepa-takenafter1000,2000,3000,and4000generations,showaratingspatialdomainswiththree-speciesenemypartnerships;fastdecreaseofthenumberofsingle-speciesspatialdomains.(II)thespiralwavefrontsinsidethespatialdomainsandtheThelargerthree-speciesspatialdomainsare,thelongerinterferencebetweenthem,whichplayacrucialroleinthetheytaketocollapse(thecollapsetimetcbeingroughlypro-2upiedbyasinglespecies;portionaltotheirinitialareatc∝Lforacurvaturedominated052310-3AVELINO,MENEZES,DEOLIVEIRA,ANDPEREIRAPHYSICALREVIEWE99,052310(2019)?,[15]).Hence,oneexpectsthenumberofcollapsesNotethatinone-speciesspatialdomainsthecorresponding2?4perunitareaperunittimetoscalewith1/(Ltc)∝L,>0,then?>0(thein-teampredationintheouterthree-speciesspa-one-speciesspatialdomainsareformedattheendstagestialdomainre