CongestioninconnedanttracNickGravish*, GregoryGold, andAndrewZangwillSchool of Physics, GeorgiaInstituteof Technology, Atlanta, GA30332, USAMichael A.D. GoodismanSchool of Biology, GeorgiaInstitute of Technology, Atlanta, GA30332, USADaniel I. GoldmanSchool of PhysicsandSchool of Biology, GeorgiaInstituteof Technology, Atlanta, GA30332, USA(Dated: July30,2014)Manysocial animals construct subterraneannests where theycollectivelylive andmove. Intheseenvironments thephysical constraints imposedbytheconned, crowdednest tunnels maypresentchallengesformobility. Tounderstandhoworganismsmitigatetracjamsinsubterraneanenvironments we studytrac inthe re ant Solenopsis invicta. We monitor foragingtrac ofreantworkersinlaboratorytunnelsof varieddiameter2, 3, 4, 6mm. Weobservethathead-oninteractions inbi-directional tracoccurredfrequentlybetweenoncomingants andoftenledtosustainedtracjams. Tracjamspersistedforatimethatincreasedlinearlywiththenumberofparticipatingants. Theslopeoftracjamdurationversusgroupsizeincreasedastunneldiameterdecreasedanddivergedat aminimumdiameter, Dc=1.47mm. Theseresults giveameasureoftracowsensitivityasafunctionof tunnel diameterandpredicttheminimumtunnel diameterwithinwhichtwo-waytrac canoccur inre ant nests. Further, we compare the trac owsensitivityobservedinexperimentwiththerangeoftunnelsizesconstructedinnatureandobservethatreantsalwaysconstructtunnelsoutsidetherangeof tracjamsensitivity. Tounderstandhow important behavioral rules are in controlling trac ow within tunnels we construct a minimalnullmodelforbi-directionaltracinconnedspace. Ourminimalmodelreproducesexperimentalobservations and indicates that ant-tunnel trac dynamics are due in part to the physical constraintsofsubterraneanlife.INTRODUCTIONTrac within crowded spaces is observed across arange of size scales fromintracellular [14] to inter-statetrac[57]. Motorproteinstransportcargoalongcrowdedmicrotubules[13, 8], aggregationsofcellscol-lectively migrate through the extra-cellular matrix [9, 10],andanimals navigate throughcrowdedaerial, aquatic,andterrestrialenvironments[11].Manysocialanimalsliveincentralized,enclosedenvi-ronmentssuchasantnests,termitemounds,andprairiedognests. Socialanimalsrelyontherapidmovementofresources, information, andindividuals withinthenestfor survival [12, 13]. However, inconnednest envi-ronments, theconstraintsof sensorydeprivation, phys-ical crowding, andenvironmental perturbationsall hin-dermobilityandchallengeeectivemovement. Recentexperiments have explored how individual ants [14], cock-roaches [15], andferrets [16] traverse connedspaces.However,littleisknownaboutthecollectivelocomotionof subterranean organisms in their natural environments.Social insectssuchasants[12] andtermites[17] con-struct nests in which they collectively live and move (Fig.1a). Many features of nest morphology (tunnel size,shape, branching) are conservedacross dierent nestswithinspecies[18]. Thus,anopenbiologicalquestioniswhetherthesenestmorphologiesareadaptationsforef-fectivesubterraneanlife,orsimplyrelatedtoothermor-phologicalorbehavioraltraits,suchasbody-size[19]orinteractiondynamics.Inredimportedreant(Solenopsis invicta)coloniestheapproximately3.5mmlongworkers createsubter-raneanforagingtunnelsof diameter3 14mmandupto50minlength(Fig. 1b, c). Foragingtunnelsextendhorizontallyawayfromthecentralnestandpro-videapathalongwhichworkers movebackandforthfrom the nest to the external environment. Foraging typ-ically incurs the highest mortality among ant workers [12]and thus construction of foraging tunnels are an eectivestrategy to minimize worker mortality [20]. However, for-agingtunnelsrestrictforagingtractotheconnesofthenestandthustunnel dimensionsmaylimitresourcetransport capabilities. In excavations ofS.invicta forag-ingtunnel networks, Tschinkel [21] discoveredthatthediameter of tunnels decreasedwithanincrease inthedistancefromthenestcenter. Thissuggeststhattracdemands may play a role in re ant nest construction anddesign.Laboratoryandeldexperiments haveshedlight onthe mechanisms by which ants manage trac on ant sur-faceforagingtrails(see[19] forareview). Forexample,laneformationduringbi-directional tracof thearmyant Ecitonburchelli [22], trail wideningbyleaf cutterants (Attacolombica) [19, 23], and priority rules for pass-ing[24]areexamplesofself-organizedprocessthatmin-imizetracjamsalongtrails. Inbi-directionalforaging2L (mm)PP0.300.205 0 10 151 mma)c)b)Foraging tunnel networkMound5 m Foraging VerticalFIG. 1. a)Imagesofreantswithinexcavatednesttunnelsinaquasi-2Denvironment (see [26] for details). b) Over-headviewofanaturalsubterraneanforagingtunnelnetworkconstructedbyanS. invictacolony. DatareproducedfromMarkin[27]. c)Sizedistributionsof anttunnel diameterinlaboratoryvertical tunnels andnatural horizontal foragingtunnels. VerticaltunneldatacollectedbyGravishetal. [14]andhorizontal tunnel datacollectedbyTschinkel [21]. Bot-tomshowsbodylengthdistributionof S. invictaworkersre-producedfrom[14].tracofAttacolombica, workersdonotformlanesandinsteadfrequentlypausetoengageinhead-onencoun-ters. Counter-intuitively, the rate of trac ow was max-imal inAttacolombicatrac whenhead-onencounterfrequency was maximum [25]. To explain this it was sug-gestedthatantinteractionsmayhaveapositiveimpactontracbybreakinguptheowandinhibitingjamsofgroupsmovinginthesamedirection.Tactile interactions areanimportant modeof infor-mationacquisitionandresourcetransferinthenesten-vironment and along foraging tunnels. For instance, suchinteractionsallowreantstoidentifyinvaders[12, 13],exchange food or water, or be recruited to a new foragingsite [2830]. While tactile interactions along surface trailsmayinhibit trac jams bybreakingupthe ow[25],insubterraneantunnels workers are connedinspaceand the tactile interactions will inhibit ow. Thus tracandtactileinteractions areat odds: workersthat stoptointeractpresentphysicalobstaclestotheowoftraf-cwhichmaycausetemporarytracjams. Resourcesandinformationmovewiththeowoftracwithinthenest,andthussmoothtracowislikelyimportantforsuccessfulsubterraneanlife.Inthisstudywetesttheabilityofreantcoloniestomaintainbi-directionalforagingtracwithintunnelsofnatural and reduced diameter. From a biological perspec-tive, weseektounderstandhowantscollectivelymovewithinsubterraneantunnels andhowdiameter aectsthis motion. Fromaphysical perspective, we seektounderstandhowsimplerulesofcollectivemotionwithinconnementrelatetotheoveralltracowofantsinacolony. Toaddressthesequestionsweperformbiologi-cal experiments and computer simulations to explore thedynamicsofreantforagingtracinconnedtunnels.METHODSExperimentFireantcolonies(Solenopsisinvicta)werecollectedinGeorgiain2011-2012. Colonieswereseparatedfromsoilusingthewaterdripmethodandwerehousedinplasticbins that contained an enclosed nest area made from petridishesandanopenforagingarena. Weprovidedinsectstothecoloniesasfoodandwateradlibitum.Wemonitoredunperturbedtracbetweenalabora-torynest andanopenforagingarena(Fig. 2a). Theforagingarenaconsistedofa27cm17cmplasticbinwithFluoncoatedwallsinwhichweplacedaconstantsupply of water and food. A lamp placed above the arenailluminated and heated the foraging zone. A plastic tubeconnectedtheforagingarenatoaglasstunnel orientedhorizontallywithvarieddiameter (D=2, 3, 4, 6 mm)andlengthof 11cm(Fig. 2b). Theglass tunnel wasconnectedtoanenclosedplasticnestwhichwaspaintedblackandcontainedamoistplaster-of-parisoor.Fiveseparategroups of 500-2000worker reants ofbody length 3.50.5 were removed from their host colonyandplacedintheforagingarena. Weexcludedqueens,males, and brood from the group. The ve worker groupsweredrawnfromthreecoloniesandweremonitoredinindependent experiments over the course of three months.Withinseveral hoursof beingrelocatedtotheforagingarena the workers migrated to the nest and maintained acontinuousowofbi-directional tracfromthenestto3ForagingarenaCameraTunnelNest11 cma)b)3 mmc)xyFIG. 2. Experimental apparatus to monitor trac in dierentdiametertunnels. a)Experimentshowingnestandforagingarenaseparatedby11cmlongglasstubeof diameter D[2, 3, 4, 6] mm. b) Size relationship between laboratory tunneldiameterandtypicalworkerbodysize. c)Imageoftwoantsin a 2 mm diameter tunnel (top) and a 6 mm tunnel (bottom).Plotsaboveeachimageshowataxedtime.theforagingarenaandback. Wemonitoredtheforagingtracof worker groups for 24-72hours withineachofthe four tunnel sizes. The order of tunnel presentation totheworkergroupwasrandomizedacrossdierentgrouptrials.Werecordedvideosequencesof trac, 40secondsinduration at a rate of 100 Hz, and resolution of 100 x 1328pixels(120pixelswasequal to1cm). Afterthecollec-tionofeachvideoweperformedpost-processinginMat-lab which consisted of dividing each video frame by a sta-tionary background image and thresholding the resultantimage togenerate abinaryimage time-series, It(x, y),containingonlyants. Followingimageprocessinganewvideowascaptured. Thetimeinterval betweensucces-sivevideoswas2minutes. Ourexperimentconsistedofover10,000videoseachwith4,000framesoftrac.We analyzed spatio-temporal trac dynamics as a one-dimensional owof density, (x, t) alongthelengthofthetunnel, x(Fig. 2candFig. 3). (x, t)isdenedas(x, t)=

y It(x, y)whereIt(x, y)istheexperimentalimageattimet. Wedenethenumberofantswithinatunnelasn(t) =1C

x (x, t)where

xisthesumovertheentirelengthofthetunnel,andCisanormalizationconstantson(t) = 1whenoneantisinthetunnel.We analyzedthe spatial clustering of ants that oc-curredwithinthetunnel byndingspatiallyconnectedregionswhere(x, t)isnon-zero. Foreachclusterofsizen (binned in increments of 0.5) we computed the densitycorrelationfunctionQn() =(x, t0)(x, t0 + ) (x, t0)2(x, t0)2 (x, t0)2(1)(introducedin[31]). Qn() is afunctionwhichvariesfrom 0 to 1 with time interval,,and measures how cor-related an ants position is over time. Ants in trac jamsmoveslowlyandthusQn()remainslarge(highcorre-lation) for longer time durations thanwhencomparedtofree-ow. Thebrackets...arethespatio-temporalaverageofthefunctionovertheclustersizeandtimein-terval [0, 13s]. For the calculation of Qn() we evalu-atedeveryfourthvideoframetospeedupcomputation.Thecorrelationfunctioninterval waschosentobelongenoughsuchthatQ() = 0forlongtimeintervals.Totrackant motionwithinthe tunnel we usedtwotechniques. Todeterminethespeed-densityrelationshipwemanuallytrackedthedistanceants traveledduring1.2secondtimeintervalswhichcorrespondedtoafree-owtravel distanceofapproximately7bodylenghts. Inadditionwe measuredthe local densityof ants within2bodylengthsof thetrackedant at thebeginningofmotion. Wealsousedanautomatedtracker whichlo-catedall antclustersbetweenadjacentframesandcor-relatedsimilarclustersamongtheframesbyminimizingthe traveled distance and cluster size. From the distanceeachclustermovedwedeterminedthespeedoftheclus-ter.To study the interaction behavior of ants we handtrackedantsduringantennaecontactandmeasureddu-ration. Wedenetheinteractiontime,Tint,asthetimefrom rst antennae contact between two ants to the timewhentheyhavepassedandtherepetiolesarealigned.SimulationWe seek to understand how and if ant trac in tunnelsdepends on the details of ant-ant interactions. To explorethisquestionweimplementeda2Dcellularautomatainwhichweinputtherulesof interactionandstudiedtheresultant ant trac (Fig. 4). In our model ants occupiedlatticesitesandmovedbi-directionallyalongthetunnellength. Ants enteredthetunnel fromtheleft or rightatrandomandadvancedalongadirectionofmotionto-wardstheoppositetunnel end. Antsadvancedforwardbyonelatticesiteduringeachiteration; howeveronlyasingleant occupiedalatticesiteat atime. Twoants410 5Position (cm)10 5Position (cm)0 40 Time (s)10 30 0 40 Time (s)10 30 b)c)d)a)FIG. 3. Space-time representation of ant-trac in dierent diameter tunnels. a) Image sequence of ants moving bi-directionallyin a 6 mm tunnel. Images are separated by 14 ms and tunnel length is 10 cm. b) Plot of(x, t) with individual times separatedvertically. Imagesequenceistakenfromthehighlightedsectioninthemiddle. c)Imagesequenceof tracowfrom2mmtunnel. d)Plotof(x, t)ina2mmtunnelwithindividualtimesseparatedvertically.Deffa)b)x (cm) 9 0012Time (s)FIG. 4. Simulation of bi-directional tunnel trac. a)Schematic of cellular automata simulation. Orange ants moveleftandblackantsmoveright. b)Space-timeplotof(x, t).adjacent to each other in the head-on direction were per-mitted to move only by jumping to an open lateral latticesitewithprobabilityp.ByvaryingpwecouldvaryTintinsimulationtoex-plore how head-on encounters inuence trac. The prob-ability for a head-on encounter to end is given by the com-binedprobabilityofeitherantjumpingpasteachother,2p p2. Thestatistics of head-onencounters followaPoissonprocess andthus theinteractiontimebetweentwoantsinsimulationwasdeterminedfromthemedianvalueoftheexponential distributionfunction. Theme-dian interaction time of ants in simulation with time-stepdt is

ln(2)/

2p p2

dt. We xed the length of the sim-ulatedtunneltomatchthatoftheexperimentwithtun-nel lengthl =31Landgrid-lengthof onebodylength.The time step of the simulation (dt =0.175 s) waschosensuchthat thefree-speedwas 2cm/s, matchingexperiment. We variedthe widthof the tunnel fromDeff= 3 100latticespaces.Tocomputeow-densitycurvesinsimulationwemea-sured the distance traveled by each ant over 7 time-steps(a1.225s time interval chosentomatchexperiment).Similartoexperiment, thelocal densitywascalculatedas thenumber of ants withina2latticespacingdis-tanceaheadandbehindthe focalant. We computedthemeanvalueofantspeedindensityintervalsof1ant/cmand scaled the simulation speed to match the experimentatasinglevalue,thefreeowcondition(1ant/cm). Wescaled the simulation for comparison because our simula-tionparametersweresettomatchtheaveragefree-owspeed of ants in experiment but speed-density results arebasedonmaximumsspeedsachieved(theupperboundcurvesinexperiment).5024Speed (cm/s)0 10 0 102 mm6 mmDensity (ants/cm)0 10 0 5 10FIG.5. Fundamentaldiagramofbi-directionaltracowintunnels. Tunneldiameterincreasesfromlefttoright. Solidlinesareestimatesof theboundingcurvesforthespeed-densityrelationship. Dashedlineinleftpanel isforcomparisonbetween2 mm and 6 mm tunnels. Black and white circles are results from simulation for simulated tunnel diameters ofDeff= 2, 3, 4, 6fromlefttoright.RESULTSExperimental resultsAntinteractiontimeAntsmovedbi-directionallythroughtheforagingtun-nel (Fig. 3). Often, twoantsmeetinghead-onbrushedtheir antennae against one anothers head and body (Fig.2b and SI Movies 1,2). We measured the interaction time,Tint, between inbound and outbound ants in the four tun-nel diameter treatments and found that Tint diered onlyinthesmallesttunnel(SeeSIFig. 2). InD = 3 6mmtunnelsTint= 0.45 0.26s,inthe2mmtunnelinterac-tiontimeswereskewedtolongerdurationswithameanvalueofTint= 1.13 1.30.From automated tracking of the velocity of ant clusterswithinthetunnel, wecomputedspeeddistributionsforscenarios inwhichants movedintrac, andinwhichthey moved freely through the tunnel. We observedthat the free speed distribution was roughly gaussian dis-tributedwithvfree =1.930.63cm/s. The speeddistributionintracwasskewedtotherightwiththemajorityofspeedsnearzero(SeeSI).Weobservedthattheseinstancesof lowspeedcorrespondedtosituationswhereantennationoccurredinthetunnel.BulktracowInall tunnels, ant interactions impactedthe owoftracnearbycreatinglocal trac-jams(Fig. 3). Todetermine if trac owwas relatedtoant densitywemeasuredthespeed-densityrelationshipwithinthefourtunnels fromexperiment. Weobservedthat withinalltunnel sizes, theupperboundof speedvs. densityde-creasedwithincreasingdensity(Fig. 5). Comparisonoftheupperboundcurvesofspeedvs. densitybetweenlargeandsmall diametertunnelsillustratedthatspeeddecreasedwithincreasingdensitymorerapidlyinlargertunnelsthaninsmaller(Fig. 5 left,solidline2 mm,anddashedline6mm).Afundamental measure of trac throughput is thetrac ow, F, which is the product of speed and density(Fig. 6a) [32]. Fis theproduct of speedanddensityand is the number of ants that pass a point in space overtime within the tunnel. At zero density, no ants are in thetunnel and F= 0. At large density ants become jammedtogetherandspeedapproacheszero,inwhichcaseagainF=0. ThustheowismaximizedwithavalueFmaxatanintermediatedensitycalledthecarryingcapacity.Flow curves constructed from the upper bound curves ofspeed vs. density increased in size with increasing tunneldiameter (Fig. 6a). Themaximumowachievableintunnels of varied diameter increased with tunnel diameter(Fig. 6c).Spatio-temporal tracuctuationsTo gain insight into the process of formation and break-up of trac jams, we measured spatial and temporal uc-tuations of the tunnel density, (x, t). Temporal statisticsweredeterminedbyevaluating(x, t)atdierentxedpointsinspaceovertime. Spatial statisticsweredeter-minedbyevaluating(x, t)inspaceatxedtimes. Thetime interval between ants crossing a point in spacethewait timeapproximately followed an exponential distri-bution(Fig. 7a)withatimeconstantthatwassimilaramongthefourtunnels. Anexponential waittimedis-tributionindicatesthatmotionwithinthetunnelcanbeconsideredasarandomprocess.We dene the occupation time as the time an ant occu-pies a point in space (Fig. 7c). The distribution of occu-pationtimesamongalltunnelsweresimilarinthattheywerepeakedatatimeofbodylengthvfree0.35cm1.9cm/s 0.18stheoccupationtimeof freelymovingantsandhada60 5 10Density (ants/cm)F (ants/s)F (ants/s)15015a)c)b)d)D0 20 40 60Density (ants/cm)030Experiment0 2 4 6 804812D (mm)Fmax (ants/s)ExperimentSimulationSimulation0 2 4 6 804812DeffFmax (ants/s)FIG. 6. Flowvs. densityintunnelsofvaryingD. a-b)Flowcurves in experiment (a) and simulation (b). Experiment owareofincreasingdiameterasshownbyarrow. Dashedboxin(b) shows the axis range of (a). c-d) Maximum ow vs. tunneldiameterinexperiment(c)andsimulation(d).powerlawtail withaslope3forall fourtunnel di-ameters(Fig. 7c).Finally, we measuredthe typical size (linear dimen-sion) of interaction clusters. The cluster size of ant inter-actionswaspeakedatavalueofasingleantbodylengthandhadanexponential tail withslopethatvariedwithtunnel diameter. The magnitude of the exponentiallength constant increased with Dindicating that themeanclustersizeincreasedastunnelsizeddecreased.TracjamdurationsWeobservedthat thetimedurationof similar sizedjams (clusters) was sensitive toD(Fig. 5). Inspiredbystudiesof mobilityinnon-biological soft-mattersys-tems like granular materials [31, 37, 38], we measuredthecorrelationfunctionfordensityuctuations, Qn(),associatedwithtracjams of sizen. Qn() measureshowquicklyhighdensitytrac-uctuationsreturnedtosteadyow. WeobservedthatQn()curvesdecreasedfrom 1 to 0 over a characteristic time scale which var-iedover dierent tunnel diameters andjamsizes (Fig.8a, 9a). For xedtunnel diameter, curves of larger nwere shiftedto the right indicating that increasedwithcluster size. Comparing Qn() curves of similarnacrossDindicatesthatincreasedwithdecreasingD. We measuredby tting anempirical functionQn() =e[( )]whereis at parameter of orderunity.The correlation time of ant aggregations, , increasedapproximatelylinearlywithincreasingn(Fig. 9a) forall tunnel sizes. Theslopeof asafunctionof nisaa)b) P0 2010-310-110-510-7P10-310-110-510-7Wait time (s)0 20 40 40Wait time (s)c)d) 10-2100Occupation time (s)L / vfree10-2100102102Occupation time (s)Experiment SimulationFIG. 7. Probabilitydistributionfunctions for ant stoppingandmovingfromexperimentandsimulation. a)Histogramofwaittimefromthefourtunnelsinexperiment. b)Distri-butionof wait times fromvariedtunnel diameters insimu-lation. Dierentcolorcurvescorrespondtovarieddiametersimulations whichincrease fromupper tolower curves. c)Distributionof site-occupationtimefromexperiment. Notelog-logaxes. Dashedlineiscurveformt3. d)Distributionof site-occupation times from simulation. Dashed line is sameasinc)forcomparison.measureofhowsensitivetracowistotheformationof jams fromintermittent densityuctuations. We tlinestovs. nandobservedthatthenormalizedslopeof (n)(normalizedsuchthatlimD=1)in-creasedwithdecreasingtunneldiameter(Fig. 10a). Wetvs. nwithafunctionof theformA(DDC) + 1(chosentomatchsimulationresultswhichwediscussinthenextsection). Thistshowsthattracsensitivitydivergesat atunnel diameter of 1.47 0.2mmtunneldiameterwithinwhichbi-directionaltracoccurs.SimulationofanttracWait time(Fig. 7b) andoccupationtime(Fig. 7d)probabilitydistributions fromour simulations qualita-tively matched the experimental results across a range ofD. Waittimedistributionsfromthesimulationshadamixed shape that was not well described by either an ex-ponential or a power law distribution. However the rangeofvaluesobservedinexperimentandsimulationwereinaccord. Occupationtimeprobabilitydistributions(Fig.7d) fromthesimulations matchedtheexperiment welland were roughly power-law in shape with a slope 3,similartoexperiment.7We performed similar measurements of the density cor-relationfunctioninour cellular automata simulations(Fig. 8b). Thesimulations reproducedfeatures of theexperiment: ((n)increasedwithn(Fig. 9b), anddiverged with decreasing tunnel diameter (Fig. 10b). Toquantifytheeects of tunnel diameter andinteractiontime on trac ow we t curves from simulation andexperiment to the empirical function =A(DDc)+1(solidlinesinFig. 10b).DISCUSSIONPhysical aspectsofconnedtracWestudiedreantforagingtracwithintunnelsofdiameterrangingfrom2-6mm. Weobservedthatantcolonies were able to maintain bi-directional trac in alltunneldiameters(SIMovie2)andtunneldiameteronlyappearedtonegativelyimpacttracowinthe2mmtunnel. Head-onencountersfrequentlyoccurredwithinthetracowandresultedintracjamsthatblockedowlocally. Head-onencounters consistedof antenna-tion, aprimarymechanismoftactileinformationacqui-sitionbyants[12]. WeobservedthatTintwaslargerin2mmtunnelscomparedtotheotherdiameters. Thisislikelyduetothereducedlateralspacewhichalteredthedurationoverwhichantscrossedpaths.Wecapturedthebasicfeaturesof bi-directional traf-cowandtracjams that occur withintubular anttunnel trac through a simple cellular automata simula-tion. Thetracdynamicsinthecellularautomataareconsistent withmanyof our experimental observations(Fig. 5-10)givenonlythreevariableparameters. Traf-cowinthesimulationatvariedDeffexhibitedsimi-larspeed-densityscalingwhenthefree-speedwassettomatchthatoftheexperiment(SeemethodsandFig. 5).The upper bound curves of speed vs. density in ant traf-cwerequalitativelysimilartomeasurementsfromve-hicular,andpedestriantrac[32,33],andfromsurfaceforagingant trac [34]. The density-owrelationshipinsimulationacrossvariedtunneldiametersexhibitedasimilarshapeforlowerdensitiesasinexperiment(Fig.6b), however a plateau in ow was observed for high den-sities in simulation that was not observed in experiment.Flow-densitycurvespredictedthatFmaxshouldalsoin-creasewithDeff(Fig. 6d) andweobservedasimilarpositivetrendinexperiment.One goal of our simulations was to determine how tac-tileinteractionsorpossiblytracjamavoidancestrate-giesmayfactorintotheresultanttracdynamicsoftheantcolony. Anothergoalwastodeterminehowthecon-straintsof movementinconnedtunnelslimitthetraf-cowof theant colony. Theagreement betweenthesimulationandexperimentindicatesthattracowinsubterraneanenvironmentsislargelyconstrainedbythe10-1101 (s)01Qn(t)01Qn(t)nnExperimentSimulationa)b)FIG. 8. Correlationfunctionversustimeforthe2mmtun-nel inexperiment(a)andatracsimulation(b). Curvesofdierent color correspondtodierent sizetracjams withincreasingn [2 15]shownbythearrowfromlefttorightforboth(a)and(b).physics of collectiveinteractions inconnedspaces. Itisnotnecessarytoincludecomplexbehavioral rulesforinteractionstoreproducetheobservedtracdynamics.Therelevantfeaturestothisprocessarethattheoccu-pants of a tunnel exclude volume and halt to interact forshortperiodsoftime. Inthiswaythetracowofreants within their nest may be similar to the bi-directionaltrac of termites [36, 40, 41], molecular motors along mi-crotubules [1], or human pedestrians in crowded hallways[4244].Tracinnatural nestsFieldobservations of excavatedtunnel diameters innaturalforagingtunnels, foragingtunnelentrances, andlaboratoryexcavatedtunnelsshowthattunneldiameterinanest varies from3-14mmindiameter withsometunnels exceeding this range [14, 21, 27] (Fig. 1c). Exca-vatedhorizontalforagingtunnelsare6-14mmindiame-ter [21, 27] and vertical nest entrance tunnels are slightlysmallerwith3-4mmdiameter[14, 27]. Arecentstudyhasshownthat smaller tunnel diameter nest entrancesenhanceclimbingperformance[14]andadditionally,thesmallernestentrancesizemayenhancesecurity[12].Thetunnel diameterof horizontal foragingtunnels8where a majority of the colony trac occurslikely playsseveral importantrolesintracdynamics. Ourexper-imentsshowthatthecarryingcapacityof atunnel andthemaximumowof workersincreaseslinearlywithD(Fig. 6c)indicatingthatweexpecttondlargerdiame-tertunnelswheretracowishigher. Oursimulationsalsoexhibitthisresult(Fig. 6d)andfurthermorestud-iesofroadtracincitycentersshowthatvehiclesobeythe same linear relationship between ow and road width[35]. Innatural systemsthereisobservational evidencethat tunnel diameter increases in regions of higher tracinreantforagingtunnels[21], andtermiteshavebeenobservedtomodifyarticial tunnels inregions of hightrac[36]suggestingthatsocialorganismscontroltheirtunneldiameterbasedonlocaltracdemands.The simulation results have implications for tunnelconstructionenergetics. The throughput of workersand thus resource and information throughputincreaseslinearlywithD. However, foracircularcross-section tunnel, D2soil must be excavated per unit lengthandthustherelativeenergeticcosttotracowgainsfollowsasurfaceareatovolumescalinglaw. Thescal-ingofresourcethroughputperenergeticcostofthetun-nel suggests that larger tunnels arenot always better.These results present two hypothesis for the shape ofnatural foragingtunnels: 1)antsshouldconstructnon-circularcrosssectionswhichhavelargeperimeter(walk-ingpath) andsmall area(excavationvolume), 2) antsshould construct many smaller tunnels which would pro-videthesamenetthroughputasonelargertunnel butat a lower energetic cost. There is evidence that re antshave adopted both strategies to manage eective resourcetransport whileminimizingenergeticcostsof construc-tion. From excavations of foraging tunnels Tschinkel [21]determinedthat the cross-sections of re ant foragingtunnels areelliptical inshapewithaspect ratio2(themajoraxisishorizontal totheground), thusprovidingalargerperimeterforthesamecross-sectionalarea. Ad-ditionally, reant foragingtunnel networks beginas asingletunnelthatbranchesmanytimesalongitslengthintosmaller tunnels (Fig. 1b) [27]. The reductionoftunnel diameteratbranches[21] maycorrespondtoanexcavationstrategytomaximizeperimeter whilemini-mizingenergeticcostsofexcavation.Wecanaddressthequestionof whetherreanttraf-cis susceptibletotracjams byexaminingthecurveinrelationtothesizeof tunnelsinnatural nests(Fig. 10a). Frommeasurements of thecross-sectionalareaofreantforagingtunnelsin[21], weestimatetheeectivediametertobeintherange,D= 7.8 1.9mm(tunnels areelliptical incross-sectionwitheccentricityof 2 [27]). Vertical nest entrance tunnels in naturalnests were reportedinthe range of 3-4 mmindiam-eter anda laboratoryx-raystudyfoundthat verticaltunnels were D=3.70.8[14]. is adivergingfunctionwithdecreasingdiameterandpredictsthatbi-ExperimentSimulation0 5 10n* (s)102* (s)102a)b)FIG.9. Tracjamcorrelationtimevs. jamsizefordierentdiameter tunnels. a) Trac-jam correlation time as a functionofn for increasingtunnel diameter(arrow) inexperiment. b)Trac-jamcorrelationtimevs. ninsimulationforrangesofDeff [2, 100].directional trac should be impossible in tunnels smallerthanDc=1.47mm. Thisminimumtunnel diameterisconsistentwithapreviouslocomotionstudyofclimbingintunnels inwhichvelocitydroppedtozeronear thistunnel size[14]. However, overtherangeofnatural tun-nel diameters is fairly at (Fig. 10a). This suggeststhatinnatural nesttunnelsreanttracisnotsensi-tivetouctuations indensity. Theseobservations aresimilartothoseof abovegroundforagingtracintheant Leptogenys processionalisinwhichanabsenceof ajammed-phasewasobserved[39]. Thuswehavefoundthat inadditiontobehavioral adaptations whichsomeant species posses tomitigate trac jams alongtrails(See[19] for review), ants mayalsomodifytheir envi-ronment toallowfor smoothtrac owandresourceacquisition.Byvaryingtheinteractiontimebetweenantsweareable tomodifythe curve inour simulation(Fig.10b). Increasing Tintresults in a shift of to the rightwhichmeansthattracowbecomesincreasinglysen-sitivetotracjamformationinlargertunnels. Thisin-crease in jam sensitivity is due to ants forming blockagesinthetunnelsforlongerperiodsof time. WendthattheincreaseinjamsensitivityislinearlyproportionaltoTint. Thissuggeststhattomaintainstabletracow90 5 10 150123D (mm) ForagingtunnelsEntrancetunnels0 10 20 30Deff (mm)0123* ExperimentSimulationTinta)b)FIG. 10. Sensitivityof theslopeof asafunctionof tun-nel diameter. a)Resultsfromexperiment. RedcurveistfunctionA(DDc)+1 describedintext. Redpoints withbars showdiameter of foragingandentrance tunnels (For-aging tunnels from [21],entrance tunnels in lab from [14] andeldfrom[27]). b) Results insimulation. Dierent pointsandcurvescorrespondtosimulationswithdierentinterac-tiontimes. ArrowindicatesincreasingTint.ifantinteractiontimeincreases,tunneldiametershouldincreaseaswell.CONCLUSIONWestudiedthebi-directionaltracofreantworkergroupstodeterminehowmodulationoftunneldiameterandantinteractionbehaviormayinuencetracow.We foundthat the resultant trac dynamics are inalargepartgovernedbythephysical mechanicsofcollec-tiveinteractionsinconnedspace. Throughsimulationwe nd that by modulating the characteristic interactiontime between worker ants, we may vary the sensitivity ofthecolonyasawholetotheformationandlongevityoftracjams.Eectivetracowandtransportofresourceswithinanant nest areimportant for thesurvival andsuccessofacolony. Therelationshipbetweenthebehaviorsandnestmorphologyofsocialorganismsisanopenquestionin organismal dynamics. Conserved features of nest mor-phology such as tunnel size or shape [18] are examples ofanorganismsextendedphenotype. Whetherfeaturesofthisextendedphenotypethenestareadaptationsforsubterraneanlifeareunknown. Ourresultshaveimpli-cationsfortheeectivedesignof jam-freesubterraneanenvironments. Observationof thesizes of natural andlaboratoryexcavatedtunnelsshowsthatreantgroupsconstruct tunnels outsidethesizerangeinwhichtraf-cowbecomessensitivetotracjams. Thusmobilitywithinnatural foragingtunnels is likelynot subject tolargescaletracjams.Our trac model consists of simple interactions in con-ned space and the resultant dynamics may be applicabletoabroadrangeof organismsthatliveandinteractinsubterraneanenvironments. The approachof generat-ing predictions from a simple single parameter computa-tionalmodelallowsforthesensitivevariationofanindi-vidual behavior(interactiontime)andexploreitsresul-tanteectonthecomplex,collectiveprocessofforagingtrac. 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