-
Glass-like dynamics in confined and congested ant traffic
Nick Gravish,a Gregory Gold,a Andrew Zangwill,a Michael A.D.
Goodisman,b and Daniel I.Goldman,a,b
Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XXFirst
published on the web Xth XXXXXXXXXX 200XDOI: 10.1039/b000000x
The collective movement of animal groups often occurs in
confined spaces. As animal groups are challenged to move in
highdensity, their mobility dynamics may bear resemblance to the
flow of densely packed non-living soft materials such as
colloids,grains, or polymers. However, unlike inert soft-materials,
self-propelled collective living systems often display social
interactionswhose influence on collective mobility are only now
being explored. In this paper, we explore the mobility of
bi-directionaltraffic flow in a social insect (the fire ant
Solenopsis invicta) as we vary the diameter of confining foraging
tunnels. In all tunneldiameters, along the length of the tunnel we
observe the emergence of spatially heterogeneous regions of fast
and slow trafficthat are induced through two phenomena: physical
obstruction, because, individual ants cannot interpenetrate and
time-delayresulting from social interaction in which ants stop to
briefly antennate. Density correlation functions reveal that the
relaxationdynamics of high density traffic fluctuations scale
linearly with fluctuation size and are sensitive to tunnel
diameter. We separatethe roles of physical obstruction and social
interactions in traffic flow using cellular automata based
simulation. Social interactionbetween ants is modeled as a dwell
time (Tint ) over which interacting ants remain stationary in the
flow. Investigation over a rangeof densities and Tint reveals that
the slowing dynamics of collective motion in social living systems
are consistent with dynamicsnear a fragile glass transition in
inert soft-matter systems. In particular, flow is relatively
insensitive to density until a criticaldensity is reached. As
social interaction affinity is increased (increasing Tint ) traffic
dynamics change and resemble a strong glasstransition with density
increase. Thus, social interactions play an important role in the
mobility of collective living systems athigh density. Our
experiments and model demonstrate that the concepts of soft-matter
physics aid understanding of the mobilityof living collective
systems, and motivate further inquiry into the soft-matter dynamics
of densely confined social living systems.
1 Introduction
The natural world presents a diversity of collective biologi-cal
behaviors across scales1: these include the transport ofmotor
proteins2, the movement of cells during wound heal-ing35, the
flocking of birds6,7, and the migration of organismsacross
landscapes8. In many systems, the fluid-like patternsof moving
bands, swirls, and flocks of animals have motivateddevelopment of
hydrodynamic and statistical mechanics de-scriptions for living
systems1. These models typically incor-porate close-range repulsive
and long-range attractive interac-tions to model animal groups that
are moving in unconfinedopen spaces. For example, whirling patterns
in flocking birdshave been explained by models of neighbor
attraction basedon distance or social-network9.
Collective motion of some high density living systems (cel-
Electronic Supplementary Information (ESI) available: [details
of anysupplementary information available should be included here].
See DOI:10.1039/b000000x/a School of Physics, Georgia Institute of
Technology, Atlanta, GA 30332,USA.; E-mail: [email protected]
School of Biology, Georgia Institute of Technology, Atlanta, GA
30332, USA,
lular to human-scale collectives1013 and vehicular traffic14)can
be described as supercooled fluids and/or glasses. Whendiverse
non-living systems such as collections of grains, col-loids or
molecules1522 are cooled or increased in density, par-ticle
mobility decreases and the relaxation timescale of thesystem
increases15,23. As motion slows, heterogeneous re-gions of the
system begin to display correlated motion (orimmobility) and the
system separates into dynamically mo-bile and immobile
regions20,24. Eventually the size of immo-bile regions in the
system grows and the relaxation time ex-ceeds experimental
timescales. Many previous studies haveobserved that traffic flows
of collective systems of ants25,termites26,27, pedestrians2831, and
even inert granular sys-tems16,32 decrease in mobility as density
increases. However,the underlying behavioral mechanisms which
govern the traf-fic dynamics are largely unexplored.
Similar conditions of confined collective movement arecommon in
eusocial organisms, such as ants or termites,who often live in high
density societies with a workforce de-fined by its division of
labor between reproductive and non-reproductive colony members33.
These animals often build
111 | 1
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Foraging
arena
Camera
TunnelNest
11 cm
c) d)
f)
3 mm
e)
x
y
0 2 4 6 8 100
0.1
0
1
20 40 60
Pro
b.
N
Pro
b.
N
00 2
v (cm/s)
vfree
vtraffic
4 6
0.05
Pro
b.
1 mma) b)
Foraging tunnel network
Mound
5 m
Fig. 1 a) Fire ant workers in a tunnel. b) Underground
foragingtunnel network. Reproduced from34. c) Schematic of
experimentalsetup. Below, tunnel diameters and characteristic ant
size. d) Imagesfrom 2 mm (top) and 6 mm (bottom) tunnel
experiments. Tunneldensity, (x, t) is shown above each image. e)
Probability density ofnumber of ants in a tunnel. Inset shows same
plot on a logarithmicy-scale. f) Probability density of velocity in
traffic and free flowconditions.
complex nests composed of networks of tunnels within whichthey
live and move (See3335 and Fig. 1a,b). The nests en-sure constant
interactions which allow for transfer of infor-mation, materials
(food & water), and colony members them-selves (movement of
brood or dead ants). Little is known,however, about how macroscopic
living systems which havecomplex body shapes, close-range
interactions, and eusocialbehavior will respond during high density
collective motionin subterranean environments12,14,3638.
Ants, such as the fire ant Solenopsis invicta move within
thenarrow crowded tunnels that comprise their nests. Moreover,fire
ants build complex nests in a variety of soils39,40 and con-struct
underground foraging tunnels which can stretch up to50 meters in
total length (See Fig. 1b34). Thus the effectiveflow of traffic and
transport of resources over long distances(and within the more
topologically complex nest) are neces-sary for the survival of the
colony. Unlike the traffic condi-tions confronted by surface
foraging ants such as leaf-cutter
and army ants4144, traffic within the nest is subject to
confine-ment (ants cannot move off of a crowded trail).
Investigationof surface-based ant traffic has discovered that ants
regulatetheir traffic flow on narrow trails through head-on contact
andpheromone based feedback mechanisms25,42,45,46. Thus, sim-ilar
to the traffic mitigating behaviors during above ground antforaging
such as lane-formation44 and platoons42,47, collec-tive locomotion
behaviors specific to subterranean traffic mayalso have developed
to prevent jamming and clogging.
Interactions among nest-mates in eusocial societies are
es-sential to enable information processing and resource
trans-port. For instance, the collective foraging behaviors of
seedharvester ants are determined by individuals counting thenumber
of nest-mate interactions at the nest-entrance48. Addi-tionally,
the head-on encounters of leaf-cutter ants on surfaceforaging
trails facilitate resource exchange and help to bene-ficially break
up traffic flows25. However, it is unclear howsimilar behaviors and
similar interactions may influence col-lective mobility within the
dense nest environment33,49, and itis further unclear how
soft-matter physics may inform collec-tive animal behavior in these
environments.
Inspired by these ideas, and by the seeming simplicity ofthe one
dimensional confined traffic in linear tunnels, we usea combination
of laboratory controlled studies of confinedfire ant traffic and a
kinetically constrained cellular automatamodel to discover
principles of discrete, flowing, living sys-tems. We show that like
in other systems, at high densitycollective ant flow can be
described by the physics of glass-forming soft materials and that
such an analogy might helpprovide guiding insight for the discovery
of principles for ac-tive matter flow. Our cellular automata model
reveals thatwhile ants can be modeled as repulsive hard spheres, we
mustinclude localizing interactions to capture their collective
be-haviors. These localizing interactions are function similarlyto
attractive interactions in non-living glass-formers. Throughour
soft-matter motivated description of ant traffic we developa
sensitivity hypothesis to categorize the modes of cloggingand means
by which ant collectives may avoid it. Under-standing the influence
of these mechanisms on the mobility ofcollective animal groups has
broad implications for biologicalgroup-behavior and soft-matter
physics.
2 Methods
2.1 Experiment
Fire ant colonies (Solenopsis invicta) were collected in
Geor-gia, USA in 2011-2012. Colonies were separated from soil
us-ing the water drip method and were housed in plastic bins
thatcontained an enclosed nest area made from petri dishes and
anopen foraging arena. We provided insects to the colonies asfood
and water ad libitum. We monitored unperturbed traffic
2 | 111
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105
Position (cm)
0
40
Tim
e (s)
10
30
a) b) c)
d)
e)e)
f)
i
iii
ii
00 4
D (mm)8
1.0
Tin
t (s)
Tint
(x,t)
Fig. 2 Space-time diagram of ant-traffic and ant-ant
interactions. a) Image sequence of ants moving bi-directionally in
a 6 mm tunnel. Imagesare separated by 14 ms and tunnel length is 10
cm. b) Spatio-temporal evolution of the one-dimensional tunnel
density, (x, t). Curves areoffset vertically in time downwards. c)
Spatio-temporal evolution of tunnel density during an interaction
between two ants. Red line highlightsthe trajectory before and
after interaction, and dashed line shows the trajectory if no
interaction had occurred. d) Characteristic interactiontimes in
four different tunnel diameters. Dashed line is Tint = 0.450.26 s.
e) Ant-ant interaction in a 4 mm tunnel showing antennation.Time
between frames is 0.25 s.
111 | 3
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between a laboratory nest and an open foraging arena (Fig. 2aand
SI Video 1,2). The foraging arena consisted of a 27 cm 17 cm
plastic bin with Fluon coated walls in which we placeda constant
supply of water and food. A lamp placed above thearena illuminated
and heated the foraging zone. A plastic tubeconnected the foraging
arena to a glass tunnel oriented hori-zontally with varied diameter
(D = 2,3,4,6 mm) and length,L, of 11 cm (Fig. 1c). The glass tunnel
was connected to anenclosed plastic nest which was painted black
and contained amoist plaster-of-paris floor.
Five separate groups of 500-2000 worker fire ants of bodylength
3.50.5 mm were removed from their host colony andplaced in the
foraging arena. We excluded queens, males, andbrood from the group.
The five worker groups were drawnfrom three colonies and each group
was monitored in inde-pendent experiments over the course of three
months. Withinseveral hours of relocation to the foraging arena,
the workersmigrated to the nest and maintained a continuous flow of
bi-directional traffic from the nest to the foraging arena and
back.We monitored the foraging traffic of worker groups for
24-72hours within each of the four tunnel sizes. The order of
tun-nel presentation to the worker group was randomized
acrossdifferent group trials.
We recorded video sequences of traffic; recordings were40
seconds in duration at a rate of 100 Hz, and resolutionof 100 x
1328 pixels (120 pixels was equal to 1 cm). Af-ter the collection
of each video we performed post-processingin Matlab which consisted
of dividing each video frame bya stationary background image and
thresholding the resultantimage to generate a binary image
time-series, where It(x,y) isthe experimental image at time t.
Following image processinga new video was captured. The time
interval between succes-sive videos was 2 minutes. In summary, our
experimentconsisted of over 10,000 videos each with 4,000 images
oftrafficking ants.
We analyzed spatio-temporal traffic dynamics as a
one-dimensional flow of density, (x, t) along the length of the
tun-nel, x (Fig. 1d). (x, t) is defined as (x, t) = y It(x,y).
Wedefine the number of ants within a tunnel as N(t)=Cx(x, t)where x
is the sum over the entire length of the tunnel, and Cis a
normalization constant chosen such that N(t) = 1 whenone ant is in
the tunnel (see histogram in Fig. 1e). Thelongitudinal average
tunnel density is defined as G = N/L.Often, when two or more ants
approached each other, theystopped and briefly paused before moving
along the directionthey were heading. To quantify this interaction
behavior wehand-tracked ants during bouts of antennae contact and
mea-sured duration of these interactions (see Fig 2c-e). We
definethe interaction time, Tint , as the time from first antennae
con-tact between two ants to the time when they have passed
andtheir petioles are aligned.
Deff
a)
b)
x (cm) 90
0
12
Tim
e (s
)
Fig. 3 Schematic of simulation geometry and space-time results.
a)Simulated ants move left or right along square lattice with
diagonalconnectivity and move around each other through diagonal
steps. b)Space-time plot of simulated trajectories where height at
each rowindicates longitudinal density.
2.2 Simulation
To gain insight into the behavioral and physical contributionsto
the observed traffic flow, we implemented a 2D cellular au-tomata
model in which we input the rules of interaction andstudied the
resultant ant traffic (Fig. 3). In our model, antsoccupied lattice
sites on a square-lattice grid and moved bi-directionally along the
tunnel length with either straight ordiagonal steps along the
direction of travel. We consideredant-ant interactions as causing a
short time-delay to the partic-ipating individuals motions. This
mode of interaction is fun-damentally different from previous
modeling approaches ofcollective motion, such as those of bird
flocks or fish schools,in which interactions are modeled as some
combination ofattractive and repulsive potentials1. Ants entered
the tunnelfrom the left or right at random and advanced along a
direc-
4 | 111
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tion of motion towards the opposite tunnel end. Ants
advancedforward by one lattice site during each iteration; however
onlya single ant occupied a lattice site at one time. Two ants
adja-cent to each other in the head-on direction were permitted
tomove only by jumping diagonally to an open lattice site
withprobability p.
By varying p we could vary the interaction time, Tint , in
thesimulation to explore how head-on encounters influenced
traf-fic. The probability for a head-on encounter to end is givenby
the combined probability of either ant jumping past eachother, 2p
p2. The statistics of head-on encounters followa Poisson process
and thus the interaction time between twoants in the simulation was
determined from the median valueof the exponential distribution
function. The median interac-tion time of ants in the simulation
with time-step dt = 0.175 sis[ln(2)/
(2p p2)]dt. We fixed the length of the simu-
lated tunnel to match that of the experiment with tunnel lengthl
= 31L and grid-length of one bodylength. The time step ofthe
simulation (dt = 0.175 s) was chosen such that the free-speed was 2
cm/s, matching experiment. We varied the widthof the tunnel from De
f f = 3 100 lattice spaces. In simula-tions studying the
fragile-strong glass formation we performedsimulations in De f f =
10 tunnels with periodic entrance andexit boundary conditions to
fix average density.
3 Results & Discussion
3.1 Traffic and interactions
We observed fire ant foraging traffic within tunnels of
diam-eter ranging from 2 - 6 mm over periods of days for eachtunnel
diameter. Ants maintained bi-directional traffic in alltunnel
diameters. From automated tracking of the velocity ofant
aggregations within the tunnel we computed speed distri-butions for
scenarios in which ants moved in traffic, and inwhich they moved
freely through the tunnel. We observedthat the free speed
distribution (when only a single ant waspresent in the tunnel) was
roughly Gaussian distributed withv f ree= 1.930.63 cm/s. The speed
distribution when morethan one ant was in the tunnel was skewed to
the right with themajority of speeds near zero (Fig. 1f).
The motion of an individual ant was interspersed with boutsof
free running and stationary interactions with other ants (Fig.2a,b)
as they moved to or from the nest site. U-turns wereinfrequent
(approximately less than 10% of trajectories) andwere not
considered in our analysis. Stationary aggregationsof ants often
nucleated and persisted within the tunnels asshown in the
space-time image of Figure 2b. A fundamen-tal observation of ant
traffic is that independent of the avail-able space to maneuver,
ants routinely halted locomotion tointeract with each other (Fig.
2c-e). Ants antennated duringhead-on encounters which is a primary
mechanism of tactile
b)
a)
c)
d) e)
PP
0 20
10-3
10-1
10-5
10-7
P
10-3
10-5
10-7
10-3
10-1
10-5
10-7
Vacancy time (s)
40
L / vfree
Sim. Sim.
Exp.Exp.
10-2 100
Residence time (s)
102
Vacancy timeResidence time
(xi,t)
0 10 30 40Time (s)
Fig. 4 Probability distribution of the site vacancy times
inexperiment and simulation. a) Example of (x, t) at fixed location
inspace (xi) over time. Annotations highlight vacancy and
residencytimes. b) Vacancy time distribution in experiment.
Colorscorrespond to varied tunnel diameter, D, with 2 mm black, 3
mmblue, 4 mm red, 6 mm orange. Dashed line shows exponential
decaywith time constant of 8 s. c) Probability distribution of the
siteoccupancy time in experiment. Colors same as in (b). Dashed
line isa power-law with exponent of -3. d) Vacancy time
distribution insimulation. Diameter increasing from D = 310 from
light to dark.e) Residence time in simulation. Dashed line is same
as in (c).
information acquisition33. We measured the interaction
timebetween ants (Tint ) and observed that in tunnel diameters
largerthan 2 mm Tint = 0.45 0.26 s (Fig. 2d). In contrast, the2 mm
tunnel interaction times were skewed to longer dura-tions, with a
mean value of Tint = 1.13 1.30 s. This differ-ence is likely due to
the reduced lateral space which alteredthe duration over which ants
crossed paths.
From a behavioral perspective, halting to interact with
otherants in the tunnel serves an important biological function
thatenables social-insects to acquire information about
foragingresources50, colony state27, and to detect intruders.
However,these stationary interactions within the tunnel have the
poten-tial to cause large-scale aggregations and traffic jams,
whichcan be detrimental to resource flow. To address the decreasein
mobility associated with local density fluctuations we nextmeasured
the mobility of individuals within the tunnel.
111 | 5
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3.2 Traffic statistics
We characterized the mobility statistics in experiment
andsimulation through measurement of the occupancy and va-cancy
time of sites within the tunnel (Fig. 4a). The vacancytime is the
time interval that a site along the tunnel length,x, is unoccupied
(i.e. (x, t) = 0). The occupancy time isthe duration of time that
the site is occupied by an ant (i.e.(x, t) 6= 0). Vacancy and
occupancy time distributions forthe four tunnels were insensitive
to tunnel diameter (Fig. 4b,c). Vacancy distributions were broadly
distributed with an ex-ponential tail (Fig. 4b) indicative of a
Poisson waiting pro-cess for site occupations. However, the
occupancy time dis-tributions did not display an exponential decay
and insteadwere better described with a power law tail (Fig. 4c)
with aroughly 3 scaling exponent. The distribution of occupancytime
among all tunnels had similar peak locations at a timeof
bodylengthv f ree
0.35cm1.9cm/s 0.18 sthe occupation time of freely
moving ants.We did not observe differences in occupancy and
vacancy
time distributions despite differences in ant-ant
interactiontimes, likely due to variation in traffic density as a
functionof tunnel diameter. Local density alters traffic flow
speeds dif-ferently in different tunnel diameters. We measured the
re-lationship between individual speed and local density
(Fig.SI1,2) which is typically used to characterize vehicular
andpedestrian traffic12. We found that in all cases speed
mono-tonically decreased with increasing density, and that the
trafficflux (product of speed and density) displayed a maximum
atintermediate density called the carrying capacity. The upperlimit
of traffic flow curves differed as a function of diameterwith the 2
mm diameter flow-speed curves decreasing fastestwith increasing
density. The carrying capacity increased ap-proximately linearly
with tunnel diameter (Fig. SI2) and thiswas similar to results from
pedestrian traffic flow51.
The experimental results were well captured by the cellu-lar
automata simulation in both the density-speed relationship(Fig.
SI1,2) and vacancy and occupancy time distributions(Fig. 4d,e).
Similarity between simulation and experimentsuggest that this
simple kinetically constrained model consist-ing of excluded volume
and finite time-delay interactions issufficient to understand the
basic statistics of motion withinthe tunnel.
3.3 Spatio-temporal dynamics of aggregations
To quantify the spatial and temporal structure of ant
aggrega-tions we located all connected non-empty regions in the
space-time density which we define as aggregations.
Spatiallyconnected regions of (x, t) correspond to aggregation
sizes ofn=
x fx0 (x, t) where x0,x f are the initial and final positions
of
the connected aggregation respectively (Fig. 6a). The number
(x
,t)
P
100
10-2
10-4
10-6
L (ants/BL)0 2 4 6 8
Fig. 5 Probability distribution of aggregation density in
experiment.Colors are tunnel diameters, D = 6 mm purple, 4 mm
yellow, 3 mmred, 2 mm blue. Solid lines are gaussian fits to the
low densityregime highlighting the exponential tails in
aggregation. Inset showsthe onset of an aggregation as three ants
approach the central regionof the tunnel from left and right, with
time increasing down thefigure.
of ants in an aggregation divided by the longitudinal-lengthof
the aggregation represents the density of that aggregation,L =
n/L.
The distribution of L across all aggregations was similarbetween
the four tunnel diameters in experiment (Fig. 5). Theshape of P(L)
was gaussian at small L with an exponentialtail for all four tunnel
diameters. The peaks at L 5 corre-spond to a characteristic
aggregation density of approximately1.75 ants/bodylength which are
likely instances of single antsor tandem aggregations. The
exponential tail in P(L) rep-resents the presence of large
contiguous aggregations in thetraffic flow. The distribution tails
suggest that at smaller tunneldiameter there is a smaller
probability to observe high-densityaggregations. Similar phenomena
have been observed in thelength distribution hard spheres (or rods)
moving collectivelyin long strings upon transition to the glass
transition52,53.
To gain insight into the temporal fluctuations of the
tunneldensity, we used a measure of mobility typically applied
tonon-biological dense soft-matter systems5456. For each
ag-gregation of size n (binned in increments of 0.5) we computedthe
density overlap correlation function.
Qn() =(x, t0)(x, t0+ )(x, t0)2
(x, t0)2(x, t0)2(1)
which has previously been used to study colloids54 and gran-ular
materials56. Qn() is a function which varies from 0 to
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x (cm)0 1 2 10
-1 101 (s)
0
1
Qn(
)
0
1
Qn(
)
n
n
Experiment
Simulation
b)a)
c)
100
n
102
G
0 8
n
Fig. 6 Ant traffic jam dynamics investigated with the
densitycorrelation function. a) Image of four ants approaching each
other ina 6 mm tunnel. Below is shown the spatiotemporal evolution
of(x, t) with a jam shown by the black line. b) Correlation
functionof traffic jam duration versus time for the 2 mm tunnel
inexperiment. c) Correlation function for simulation. Curves
ofdifferent color correspond to different size traffic jams
withincreasing n from 2 - 15 shown by the arrow from left to right
forboth (b) and (c). Inset in (b) experiment shows scaling of n
versusthe global tunnel density (in units of ants/bodylength).
1 measures the spatial overlap of an ants position as a
func-tion of time lag. If ant aggregations decay slowly, the
spatialoverlap is high and thus Qn() remains large for longer
timedurations than when compared to free-flow. The brackets ...are
the spatio-temporal mean of the function over the aggre-gation size
and time interval [0,13s]. For the calculationof Qn() we evaluated
every fourth video frame to speed upcomputation. The correlation
function interval was chosen tobe long enough such that Q() = 0 for
long time intervals.
The correlation function Qn() associated with traffic
ag-gregations of size n measures how quickly high density
traffic-fluctuations returned to steady flow. The Qn() curves
de-creased from 1 to 0 over time scale which varied with tun-nel
diameter and jam size (Fig. 6). We measured by fittingan empirical
stretched exponential function, Qn() = e[(
)
]
where is a fit parameter of order unity. For fixed tunnel
di-ameter (D), curves of larger n were shifted to the right
indicat-ing that increased with aggregation size. Comparing
Qn()curves of similar n across D indicates that increased
withdecreasing D.
The size of ant aggregations, n, grew approximately
expo-nentially with increasing global density, G, within all
tunnels(Inset Fig. 6b). The rate of increase in ant aggregations
wassimilar across all tunnels as a function of G.
3.4 Effects of tunnel diameter
The correlation curves in Fig. 6 are similar to many
otherobservations of slowing relaxation in soft-matter glassy
sys-tems24,55,57 where heterogeneous mobile and immobile re-gions
form. The relaxation time of ant aggregations in experi-ment
increased approximately linearly with increasing n for alltunnel
sizes (Fig. 7). We contrast these results to those of drygranular
materials56 and colloids54 in which is more sen-sitive to
heterogeneity size, scaling approximately as n3.That is,
aggregations of ants relax more quickly than wouldbe expected of
inert soft-matter. Our observed linear depen-dence of on
heterogeneity size suggests at different relax-ation processes that
may occur in active systems such as theone studied here.
The slope of as a function of n and can be also interpretedas a
measure of how sensitive traffic flow is to the formation ofjams
from intermittent density fluctuations. We fit lines to vs. n and
observed that the normalized slope of (n) (nor-malized such that
limD = 1 ) increased with decreas-ing tunnel diameter (Fig. 8a). We
fit vs. n with a functionof the form A
(DDC) + 1 (which was chosen to describe sim-ulation results).
Aggregation time susceptibility diverged ata tunnel diameter of
1.47 0.2 mm in experiment, which isapproximately two-times the 0.8
mm mean head width of afire-ant worker (Fig. 8a). The analysis of
aggregation timesusceptibility thus predicts that traffic-flow is
not possible inthe limit that two ants cannot pass each other in a
tunnel, aresult that is surely to be the case in natural
tunnels.
The divergence of indicates that as tunnel diameter de-creases
traffic flow becomes increasingly sensitive to tunneldensity. The
system behavior at large and small D is consis-tent with
predictions for dense and dilute cases: in the smallD (dense)
regime interactions are enforced among all ants be-cause of spatial
limitations and thus additional workers neces-sarily decrease
mobility, in the large D (dilute) regime the ad-dition of workers
does not significantly affect relaxation time.In the intermediate
regime we see that the effect of social-interaction influences the
relaxation dynamics of worker ag-gregations and that this process
is sensitive to tunnel diameter.In simulation we are able to
explore the effect of ant-ant in-teraction behavior on the
divergence of traffic susceptibility(Fig. 8b). We find that
increasing Tint in simulation results ina more gradual increase of
versus De f f . Thus, these re-sults suggest that tunnel diameter
in combination with innatebehaviors (individual mobility and
interaction times) affect thespatio-temporal traffic dynamics.
We can address the question of whether fire ant traffic is
sus-ceptible to traffic jams in natural nests by examining the
Under our space-time representation of traffic, the width of an
aggregation ofn ants scales linearly with n, independent of
interactions and thus we normal-ize the slope, , to 1 to remove
this effect.
111 | 7
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Experiment
Simulation
0 5 10
n
0 5 10
n
*(s)
*(s)
1
0
2
1
0
2
a)
b)
Fig. 7 Traffic jam correlation time as a function of n for
increasingtunnel diameter (arrow) in experiment (a) and simulation
(b). Colorscorrespond to tunnel diameter increasing from large to
small asshown by arrow.
curve in relation to the size of tunnels in natural nests
(Fig.8a). From measurements of the cross-sectional area of fireant
foraging tunnels58, we estimate the effective diameter ofhorizontal
foraging tunnels in nature to be, D = 7.81.9 mm(tunnels are
elliptical in cross-section with eccentricity of 2,See34). Vertical
nest entrance tunnels in natural nests werereported in the range of
3-4 mm in diameter and a laboratoryx-ray study found that vertical
tunnels were D = 3.7 0.8,See59. is a diverging function with
decreasing diame-ter and predicts that bi-directional traffic
should be impossiblein tunnels smaller than Dc = 1.47 mm. This
minimum tun-nel diameter is consistent with a previous locomotion
studyof climbing in tunnels in which velocity dropped to zero
nearthis tunnel size59. However, over the range of natural
tunneldiameters is relatively flat (Fig. 8a). Thus fire ants
buildtunnels that allow for efficient movement and flow under
nor-mal conditions.
Our observations of traffic regulation differ from those ofabove
ground foraging traffic in the ant Leptogenys proces-sionalis in
which the absence of a jammed-phase has beenobserved42, and has
been attributed to specific interaction
0 5 10 150
1
2
3
D (mm)
Foraging
tunnels
Entrance
tunnels
0 10 20 30
Deff
(mm)
0
1
2
3
*
Experiment
Simulation
Tint
a)
b)
Fig. 8 Sensitivity of the slope of versus n (Fig. 7) as a
functionof tunnel diameter. a) Results from experiment. Red curve
is fitfunction A
(DDc) +1 described in text. Red points with bars showdiameter of
foraging and entrance tunnels (Foraging tunnels from58,entrance
tunnels in lab from59 and field from 34). b) Results insimulation.
Different points and curves correspond to simulationswith different
interaction times. Arrow indicates increasing Tint .
behaviors like lane-formation and platoons25,44. Thus wehave
found that in addition to behavioral adaptations whichsome ant
species posses to mitigate traffic jams along trails(See47 for
review), environmental modification may also en-able smooth traffic
flow. A similar result was observed in theconstruction of termite
tunnels, whose width increased lin-early with the number of termite
workers queued up at thedigging site27. Thus, it is likely that the
size and shape oftunnel diameters in eusocial insect nests are an
expression ofthe organisms extended phenotype60. However, to
determineif tunnel morphology emerges from traffic-feedback or
innatefeedforward digging behavior in fire ants further
experimentson tunnel construction in heavy traffic are needed.
3.5 Sociality leads to strong glass formers
We now cast our results in the framework of
glass-transitionphysics23 to discuss how traffic fluctuations and
jams are anal-
8 | 111
-
ogous to the slowing and arrest of fragile glassy systems.
Thelanguage and concepts of glass-transition physics provides
aconvenient framework to explore collective behaviors of com-plex
living systems with slowing dynamics. This approachhas proven
useful in framing observations of cellular-scalecollective
behaviors4,10,61, however has not been applied tomacroscopic
collective systems. The main motivating featureof glass-transition
physics is to determine not only the pointof collective motion
arrest, but the dynamics of the approachto arrest as well.
We observed increasing length and time-scales associatedwith
aggregations during traffic (Fig. 7) and hypothesized
thataggregations were the result of repulsive physical
interactionsdue to crowding within the tunnel, and short time
delays be-tween interacting ants as they paused to antennate. The
prob-ability distribution of spatial aggregations displayed an
expo-nential tail (Fig. 5), which has been previously observed
dur-ing the formation of heterogeneous immobile regions in
glassformation52,53.
A primary interest in glassy materials is the dynamics ofthe
approach to the glass transition. The sensitivity of sys-tem
relaxation time ( in this case) with increasing den-sity (or
decreasing temperature for molecular and polymericglasses)
distinguishes two extremes of glassy behavior fromeach other. In a
strong glassy system, increases exponen-tially (referred to as
Arrhenius scaling) with increasing den-sity on approach to the
glass transition. In a fragile glassysystem, however, increases
slowly (slower than exponen-tial, i.e. non-Arrhenius) over a broad
range of densities untila critical density is reached at which
point increases veryrapidly (super-exponential)57.
We may thus ask the question what type of glass transitionis
experienced by trafficking ants? Fragile glass-like behaviorof the
ants in simulation has the physical interpretation thattraffic may
flow smoothly as density increases up to a specificdensity, at
which point the traffic comes to an abrupt slow-down or complete
halt (Fig. 9a-b). Further increases in densityresult in
super-exponential slow-down in relaxation dynamicsand the system
very rapidly comes to a halt. Thus, display-ing traffic flow
similar to fragile glassy phenomena impliesthat as long as the
arrest density is avoided, traffic will flowsmoothly. In the case
of strong glassy behavior, increasingdensity across a wide range of
densities results in an exponen-tial increase in traffic flow
time-scales (Fig. 9c). In the caseof strong-glassy arrest dynamics
the ants would be subject torapid slowing of traffic over a wide
range of densities. Wemeasured system relaxation dynamics as a
function of social-interaction time, from no social-interaction
time (Tint = 0 s)to an extremely large social interaction time
(Tint = 60 s, wellabove the observed interaction time of Tint =
0.45 0.26 s.).We found that by increasing the social-interaction
times, fromasocial to highly social, the relaxation time-scale
versus den-
sity curves crossed over from fragile (sharply increasing at
acritical density) to strong (varying smoothly with
increasingdensity) relaxation dynamics as shown in Fig. 9c.
By making an analogy with non-living high density sys-tems, we
can motivate why ant traffic displays featuresof a fragile
glass-transition. In supercooled liquids whichform fragile glasses,
molecules interact weakly and non-directionally, while for strong
glasses the interactions arethrough large-scale networks of strong
covalent bonds57. Incolloidal and hard-sphere systems a crossover
from fragileto strong glass formation is observed as the stiffness
of theparticles is decreased21,62. Hard-sphere systems with
short-range interactions like granular materials exhibit fragile
be-havior62, similar to the ants. The fragility of hard-sphere
sys-tems is thought to originate from the fact that
single-particlerearrangements at high-density require the
corresponding mo-tion of large regions surrounding the particles
(dynamical het-erogeneities), whereas for soft particles,
rearrangements canbe accommodated through the compression of nearby
parti-cles. Thus hard-sphere systems can support minimally
obtru-sive particle rearrangements until a critical density is
reachedat which point large volumes of the material must be movedto
support rearrangements.
Since the collective behaviors observed in ant traffic aredriven
by local interactions the fragile nature of collective anttraffic
arrest is consistent with the observations of arrest ininert
systems that interact through close-range potentials (i.e.granular
materials and hard colloids). In the case of ant traf-fic, fragile
dynamics may be advantageous for these organismssince smooth
traffic flow is assured over a broad range of den-sities as long as
the critical arrest density is avoided. Further-more, remaining
insensitive to traffic fluctuations over a rangeof densities may
allow for behavioral mechanisms to avoidcritical tunnel densities.
For instance, ants may be able toavoid critical tunnel densities by
counting antennation eventsin a tunnel and basing decisions of
motion on the frequencyof these events. Similar
interaction-counting based behavioralmechanisms have been shown to
regulate decisions to foragein seed harvester ants48.
The social-nature of the flow dynamics suggests a newmechanism
for the transition from strong to fragile glassy be-havior in
driven systems. The discrete pauses that accompanyant-ant
interactions within the tunnel are similar to the finitecontact
time and loss of energy that occurs in granular
inelasticcollapse63,64 which leads to clustering in dissipative
granularsystems. We may expect to observe similar phenomena in
ac-tive matter particulate systems with tunable
particle-particlecorrelation times. Thus, through the study of
collective mo-tion and collective behavior in complex living
systems likesocial-insects, we may usher in a new form of active
matter,the smart-particle or smarticle, which can tunably avoid
orseek jamming phenomena based on the scenario.
111 | 9
-
103
102
101
100
0 1 / g
(s
)
Tint
x
Fragile/
g = 0.69 /
g = 0.82 /
g = 1 /
g = 1.5
Strong
Tim
e (s
)
Tim
e (s
)
x x x
a) b)
c)
875
0
800
0 10(x,t)
0
Fig. 9 Strong and fragile behavior of simulated traffic. a)
Space-time traffic flow for Tint = 0.1 s simulations at two
densities near the arresttransition. Colors in both a and b
represent magnitude of (x, t). b) Space-time traffic flow for Tint
= 60.7 s simulations at two densities below(left) and above (right)
the arrest transition. c) Relaxation time versus density plot
illustrating the glass transition dynamics. Note thelogarithmic
vertical axis. Density has been normalized the glass transition
densities. Interactions times from upper to lower are
Tint=60.7,12.2, 6.1 1.2, 0.1 s. The observed biological interaction
time in tunnels 3 mm and larger was Tint = 0.450.26 s (See Fig.
2d).
4 Conclusion
We find that social-interactions and physical obstruction
dur-ing foraging traffic of fire ant colonies lead to traffic
aggre-gations and jams. Our experiments and model revealed thatthe
social interactions among worker ants in a tunnel lead
totraffic-arrest dynamics similar to that of a hard-sphere,
fragileglassy system. The fragile nature of ant traffic arrest is
dueto the short, finite-time delays on motion induced by
social-interactions. We find through simulation that if
time-delaysof social-interaction increase, traffic flow dynamics
change tothat of a strong glassy system over which traffic jam
timessteadily increase. Having the character of a fragile glassy
sys-tem may be advantageous for a collective animal group as
theagents would be relatively insensitive to jamming over a
broadrange of traffic densities. Only when the density increases
be-
yond a threshold would the system jam, and thus organismscould
regulate traffic to stay below this threshold.
The diversity of collective behavior in nature is
extremelybroad, and thus searching for universal descriptions of
col-lective living systems poses many challenges. However,
thelanguage and tools of soft-matter and glass-transition
physicshave proven useful in providing a framework in which to
dis-cuss and understand collective living systems4,36,65. While
wehave described how collective motion of fire ants arrests,
thereare many open questions as to how other collective living
sys-tems with potentially long-range interactions (like bird
flocksor fish schools) might arrest. The insight we have gained
intocollective locomotion in crowded environments may be usedfor
the design of artificial robotics systems to collectively nav-igate
disaster zones, extra-terrestrial environments, or the nat-ural
world.
10 | 111
-
5 Acknowledgments
We acknowledge Gregg Rodriguez for help with
preliminaryexperiments. Funding for N.G., D.I.G., and M.A.D.G.
pro-vided by NSF PoLS #0957659 and #PHY-1205878.
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