Generically, in models of driven interacting particles the average speed of
the particles decreases monotonically with increasing density. We propose a
counter-example, motivated by the motion of ants in a trail, where the average
speed of the particles varies {\it non-monotonically} with their density
because of the coupling of their dynamics with another dynamical variable.
These results, in principle, can be tested experimentally.Comment: IOP style LATEX, 4 embedded EPS figures, Final published version.
  Journal Ref: J. Phys.A 35, L573 (2002

Chowdhury, Debashish

Guttal, Vishwesha

Nishinari, Katsuhiro

Schadschneider, Andreas

English

arXiv

Generically, in models of driven interacting particles, the average speed of the particles decreases monotonically with increasing density. We propose a counterexample, motivated by the motion of ants in a trail, where the average speed of the particles varies non-monotonically with their density because of the coupling of their dynamics with another dynamical variable. These results, in principle, can be tested experimentally

Publications of the IAS Fellows

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03 LETTER TO THE EDITOR
A cellular-automata model of flow in ant-trails:
non-monotonic variation of speed with density
Debashish Chowdhury1,2‡, Vishwesha Guttal1, Katsuhiro
Nishinari2,3§ and Andreas Schadschneider2‖
1 Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
2 Institute for Theoretical Physics, University of Cologne, 50923 Ko¨ln, Germany.
3 Department of Applied Mathematics and Informatics, Ryukoku University,
Shiga 520-2194, Japan.
PACS numbers: 05.60.-k, 89.40.+k, 89.75. -k
published in J. Phys. A35, L573 (2002)
Abstract. Generically, in models of driven interacting particles the average
speed of the particles decreases monotonically with increasing density. We propose
a counter-example, motivated by the motion of ants in a trail, where the average
speed of the particles varies non-monotonically with their density because of the
coupling of their dynamics with another dynamical variable. These results, in
principle, can be tested experimentally.
Particle-hopping models, formulated usually in terms of cellular automata (CA)
[1], have been used to study the spatio-temporal organization in systems of interacting
particles driven far from equilibrium [2, 3, 4, 5] which include, for example, vehicular
traffic [6, 7]. In general, the inter-particle interactions tend to hinder their motions
so that the average speed decreases monotonically with the increasing density of the
particles. In this letter we report a counter-example, motivated by the motion of
ants in a trail [8], where the average speed of the particles varies non-monotonically
with their density because of the coupling of their dynamics with another dynamical
variable.
The ants communicate with each other through a process called chemotaxis by
dropping a chemical (generically called pheromone) on the substrate as they crawl
forward [9, 10]. Although we cannot smell it, the trail pheromone sticks to the
substrate long enough for the other following sniffing ants to pick up its smell and
follow the trail. In this letter we develope a CA model which may be interpreted as
a model of unidirectional flow in an ant-trail. Rather than addressing the question of
the emergence of the ant-trail, we focus attention here on the traffic of ants on a trail
which has already been formed.
Each site of our one-dimensional ant-trail model represents a cell that can
accomodate at most one ant at a time (see fig.1). The lattice sites are labelled
‡ E-mail:debch@iitk.ac.in
§ E-mail:knishi@rins.ryukoku.ac.jp
‖ E-mail:as@thp.uni-koeln.de
Letter to the Editor 2
σ (t)
q q
σ (t)
f f
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pheromone
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ants
pheromone
Q
S(t)
ants
pheromone
S(t+1)
σ (t+1)
Figure 1. Schematic representation of typical configurations; it also illustrates
the update procedure. Top: Configuration at time t, i.e. before stage I of the
update. The non-vanishing hopping probabilities of the ants are also shown
explicitly. Middle: Configuration after one possible realisation of stage I. Two
ants have moved compared to the top part of the figure. Also indicated are
the pheromones that may evaporate in stage II of the update scheme. Bottom:
Configuration after one possible realization of stage II. Two pheromones have
evaporated and one pheromone has been created due to the motion of an ant.
by the index i (i = 1, 2, ..., L); L being the length of the lattice. We associate
two binary variables Si and σi with each site i where Si takes the value 0 or 1
depending on whether the cell is empty or occupied by an ant. Similarly, σi = 1
if the cell i contains pheromone; otherwise, σi = 0. Thus, we have two subsets of
dynamical variables in this model, namely, {S(t)} ≡ (S1(t), S2(t), ..., Si(t), ..., SL(t))
and {σ(t)} ≡ (σ1(t), σ2(t), ..., σi(t), ..., σL(t)). The instantaneous state (i.e., the
configuration) of the system at any time is specified completely by the set ({S}, {σ}).
We assume that the ant does not move backward; its forward-hopping probabil-
ity, however, is higher if it smells pheromone ahead of it. The state of the system is
updated at each time step in two stages. At the end of stage I we obtain the subset
{S(t+ 1)} at the time step t+ 1 using the full information ({S(t)}, {σ(t)}) at time t.
At the end of the stage II we obtain the subset {σ(t+1)} at the time step t+1 using
the subsets {S(t+ 1)} and {σ(t)}.
Stage I: The subset {S} (i.e., the positions of the ants) is updated in parallel according
to the following rules:
If Si(t) = 1, i.e., the cell i is occupied by an ant at the time step t, then the ant hops
forward to the next cell i+ 1 with
probability =


Q if Si+1(t) = 0 but σi+1(t) = 1,
q if Si+1(t) = 0 and σi+1(t) = 0,
0 if Si+1(t) = 1.
(1)
Letter to the Editor 3
where, to be consistent with real ant-trails, we assume q < Q.
Stage II: The subset {σ} (i.e., the presence or absence of pheromones) is updated in
parallel according to the following rules:
If σi(t) = 1, i.e., the cell i contains pheromone at the time step t, then it contains
pheromone also in the next time step, i.e., σi(t+ 1) = 1, with
probability =
{
1 if Si(t+ 1) = 1 at the end of stage I,
1− f if Si(t+ 1) = 0 at the end of stage I.
(2)
where f is the pheromone evaporation probability per unit time.
On the other hand, if σi(t) = 0, i.e., the cell i does not contain pheromone at the time
step t, then
σi(t+ 1) = 1 iff Si(t+ 1) = 1 at the end of stage I. (3)
In certain limits our model reduces to the Nagel-Schreckenberg (NS) model ¶ [11]
which is the minimal particle-hopping model for vehicular traffic on freeways. The
most important quantity of interest in the context of flow properties of the traffic
models is the fundamental diagram, i.e., the flux-versus-density relation, where flux is
the product of the density and the average speed. For a hopping probability qNS at a
given density c the exact flux F (c) in the NS model is given by [6, 12]
FNS(c) =
1
2
[
1−
√
1 − 4 qNS c(1− c)
]
. (4)
which reduces to FNS(c) = min(c, 1− c) in the deterministic limit qNS = 1.
Note that in the two special cases f = 0 and f = 1 the ant-trail model becomes
identical to the NS model with qNS = Q and qNS = q, respectively. Extensions
of the NS model have been used not only to capture different aspects of vehicular
traffic [6] but also to simulate pedestrian dynamics [7, 13, 14]. In a closely related CA
model for pedestrian dynamics [14] the floor fields, albeit virtual, are analogs of the
pheromone fields {σ} in the ant-trail model. However, in the pedestrian model there
is no exclusion principle for the floor field.
The ant-trail model we propose here is also closely related to the bus route model
(BRM) [15, 16]. The variables S and σ in the ant-trail model are the analogs of the
variables representing the presence (or absence) of bus and passengers, respectively,
in the BRM. Because of the periodic boundary conditions, the number of ants and
buses are conserved while the pheromone and passengers are not conserved. However,
unlike the BRM, the pheromones are not dropped independently from outside, but
by the ants themselves. Another crucial difference between these two models is that
in the bus-route model Q < q (as the buses must slow down to pick up the waiting
passengers) whereas in our ant-trail model Q > q (because an ant is more likely to
move forward if it smells pheromone ahead of it).
In fig.2 we show the fundamental diagrams obtained by extensive computer
simulations of the ant-trail model for several values of f . The most unusual features
of the fundamental diagrams shown in fig.2 are that, over an intermediate range of
¶ By the term ’NS model’ in this letter we shall always mean the NS model with maximum allowed
speed unity, so that each particle can move forward, by one lattice spacing, with probability qNS if
the lattice site immediately in front is empty.
Letter to the Editor 4
0 0.2 0.4 0.6 0.8 1
Density
0
0.1
0.2
Fl
ux
(b)
0 0.2 0.4 0.6 0.8 1
Density
0
0.2
0.4
Fl
ux
(a)
Figure 2. The flux of ants plotted against their densities for the parameters
(a) Q = 1, q = 0.25 and (b) Q = 0.75, q = 0.25. The discrete data points
corresponding to f = 0.0001(▽), 0.0005(3), 0.001(◦), 0.005(•), 0.01(△), 0.05(2),
0.10(×), 0.25(+), 0.50(∗) have been obtained from computer simulations; the
dotted lines connecting these data points merely serve as the guide to the eye.
The two continuous solid curves at the top and bottom correspond to the flux
in the NS model for qNS = 1.0 and qNS = 0.25, respectively, in (a) and for
qNS = 0.75 and qNS = 0.25, respectively, in (b).
values of f (for example, f = 0.0005, 0.001, 0.005, 0.01 in fig.2) the flux in the low-
density limit c → 0 is very close to that for the NS model with qNS = q whereas in
the high-density limit c→ 1 the flux for the same f is almost identical to that for the
NS model with qNS = Q. These unusual features of the fundamental diagrams arise
from the non-monotonic variation of the average velocity with the density of the ants
(see fig.3).
The presence of the pheromone essentially introduces an effective hopping
probability qeff(c), which depends on the ant density c. The particle-hole symmetry
(and hence the symmetry of the fundamental diagram about c = 1/2) observed in
the special limits f = 0 and f = 1, are broken by the c-dependent effective hopping
Letter to the Editor 5
0.1 0.3 0.5 0.7 0.9
Density
0
0.2
0.4
0.6
0.8
1
Ve
lo
ci
ty
(b)
0.1 0.3 0.5 0.7 0.9
Density
0
0.2
0.4
0.6
0.8
1
Ve
lo
ci
ty
(a)
Figure 3. The average velocity of ants plotted against their densities for the
parameters (a) Q = 1, q = 0.25 and (b) Q = 0.75, q = 0.25. Same symbols in the
figs.2 and 3 correspond to the same values of the parameter f .
probability for all 0 < f < 1 leading to a peak at c > 1/2. Furthermore, the analysis
of correlation functions reveals some interesting clustering properties which will be
studied in detail in a future publication [17].
The qualitative features of the c-dependence of qeff can be reproduced by an
analytical argument based on a mean-field approximation (MFA) [17]. In this MFA,
let us assume that all the ants move with the mean velocity 〈V 〉 which depends on
the density c of the ants as well as on f ; although, to begin with, the nature of these
dependences are not known we’ll obtain these self-consistently.
Let us consider a pair of ants having a gap of n sites in between. The
probability that the site immediately in front of the following ant contains pheromone
is (1 − f)n/〈V 〉 since n〈V 〉 is the average time since the pheromone has been dropped.
Therefore, in the MFA the effective hopping probability is given by
qeff = Q(1− f)
n/〈V 〉 + q{1− (1− f)n/〈V 〉}. (5)
We replace n by the corresponding exact global mean separation 〈n〉 = 1c − 1 between
Letter to the Editor 6
successive ants. Moreover, since 〈V 〉 is identical to qeff , we get the equation(
qeff − q
Q− q
)qeff
= (1− f)
1
c
−1 (6)
which is to be solved self-consistently for qeff as a function of c for a given f . Note
that the equation (6) implies that, for given f , limc→0 qeff = q; this reflects the fact
that, in the low-density regime, the pheromone dropped by an ant gets enough time to
completely evaporate before the following ant comes close enough to smell it. Equation
(6) also implies limc→1 qeff = Q; this captures the sufficiently high density situations
where the ants are too close to miss the smell of the pheromone dropped by the leading
ant unless the pheromone evaporation probability f is very high. Similarly, from (6)
we get, for given c, limf→1 qeff = q and limf→0 qeff = Q which are also consistent with
intuitive expectations.
In view of the fact [10] that the lifetime of pheromones can be as long as thirty
to sixty minutes, the interesting regime of f (≪ 1), where the average velocity varies
non-monotonically with the ant density, seems to be experimentally accessible. We
hope that the non-trivial predictions of this minimal model of ant-trail will stimulate
experimental measurement of the ant flux as a function of the ant density for different
rates of pheromone evaporation by using different varieties of ants [8].
Acknowledgment
We thank B. Ho¨lldobler for drawing our attention to the reference [8].
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0 0.2 0.4 0.6 0.8 1
Density
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A cellular-automata model of flow in ant-trails: non-monotonic variation of speed with density

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