A cellular-automata model of flow in ant trails: non-monotonic variation of speed with density

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1 INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSA: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 35 (22) L573 L577 PII: S35-447(2) LETTER TO THE EDITOR A cellular-automata model o low in ant trails: non-monotonic variation o speed with density Debashish Chowdhury 1,2,Vishwesha Guttal 1,Katsuhiro Nishinari 2,3 and Andreas Schadschneider 2 1 Department o Physics, Indian Institute o Technology, Kanpur 2816, India 2 Institute or Theoretical Physics, University o Cologne, 5923 Köln, Germany 3 Department o Applied Mathematics and Inormatics, Ryukoku University, Shiga , Japan debch@iitk.ac.in, knishi@rins.ryukoku.ac.jp and as@thp.uni-koeln.de Received 3 September 22 Published 1 October 22 Online at stacks.iop.org/jphysa/35/l573 Abstract Generically, in models o driven interacting particles, the average speed o the particles decreases monotonically with increasing density. We propose a counterexample, motivated by the motion o in a trail, where the average speed o the particles varies non-monotonically with their density because o thecoupling o their dynamics with another dynamical variable. These results, in principle, can be tested experimentally. PACS numbers: b, 5.6. k, k, k Particle-hopping models, ormulated usually in terms o cellular automata (CA) [1], have been used to study the spatio-temporal organization in systems o interacting particles driven ar rom equilibrium [2 5] which include, or example, vehicular traic [6, 7]. In general, the inter-particle interactions tend to hinder their motions so that the average speed decreases monotonically with the increasing density o the particles. In this letter we report acounterexample, motivated by the motion o in a trail [8], where the average speed o the particles varies non-monotonically with their density because o the coupling o their dynamics with another dynamical variable. The communicate with each other through aprocess called chemotaxis by dropping achemical (generically called a ) onthesubstrate as they crawl orward [9, 1]. Although we cannot smell it, the trail sticks to the substrate long enough or the other ollowing sniing topickupits smell and ollow the trail. In this letter we develop acamodel which may be interpreted as a model o unidirectional low in an ant trail. Rather than addressing the question o the emergence o the ant trail, we ocus our attention here on the traic o on a trail which has already been ormed. Each site o our one-dimensional ant-trail model represents a cell that can accomodate at most one ant at a time (see igure 1). The lattice sites are labelled by the index i /2/ $3. 22 IOP Publishing Ltd Printed in the UK L573

2 L574 Letter to the Editor q Q q S(t) σ(t) S(t+1) σ(t) S(t+1) σ(t+1) Figure 1. Schematic representation o typical conigurations; it also illustrates the update procedure. Top: Coniguration at time t, i.e.beore stage I o the update. The non-vanishing hopping probabilities o the are also shown explicitly. Middle: Coniguration ater one possible realization o stage I. Two have moved compared to the top part o the igure. Also indicated are the s that may evaporate in stage II o the update scheme. Bottom: Coniguration ater one possible realization o stage II. Two s have evaporated and one has been created due to the motion o an ant. (i = 1, 2,...,L); L being the length o the lattice. We associate two binary variables S i and σ i with each site i where S i takes the value or 1 depending on whether the cell is empty or occupied by an ant. Similarly, σ i = 1ithecell i contains ; otherwise, σ i =. Thus, we have two subsets o dynamical variables in this model, namely, {S(t)} (S 1 (t), S 2 (t),...,s i (t),...,s L (t)) and {σ(t)} (σ 1 (t), σ 2 (t),...,σ i (t),...,σ L (t)). The instantaneous state (i.e., the coniguration) o the system at any time is speciied completely by the set ({S}, {σ }). We assume that the ant does not move backward;its orward-hopping probability,however, is higher i it smells ahead o it. The state o the system is updated at each time step in two stages. Attheendostage I, we obtain the subset {S(t +1)} at the time step t +1 using the ull inormation ({S(t)}, {σ(t)}) at time t. Attheendostage II, we obtain the subset {σ(t +1)} at the time step t +1usingthesubsets {S(t +1)} and {σ(t)}. Stage I: Thesubset {S} (i.e., the positions o the ) is updated in parallel according to the ollowing rules: I S i (t) = 1, i.e., the cell i is occupied by an ant at the time step t, then the ant hops orward to the next cell i +1with Q i S i+1 (t) = butσ i+1 (t) = 1 probability = q i S i+1 (t) = andσ i+1 (t) = (1) i S i+1 (t) = 1. where, to be consistent with real ant trails, we assume q<q. Stage II: Thesubset {σ } (i.e., the presence or absence o s) is updated in parallel according to the ollowing rules: I σ i (t) = 1, i.e., the cell i contains at the time step t, thenitcontains also in the next time step, i.e., σ i (t +1) = 1, with

3 Letter to the Editor L575 { 1 i Si (t +1) = 1attheendostageI probability = (2) 1 i S i (t +1) = attheendostagei where is the evaporation probability per unit time. On the other hand, i σ i (t) =, i.e., the cell i does not contain at the time step t,then σ i (t +1) = 1 i S i (t +1) = 1attheendostageI. (3) In certain limits our model reduces to the Nagel Schreckenberg (NS) model 1 [11] which is the minimal particle-hopping model or vehicular traic on reeways. The most important quantity o interest in the context o low properties o the traic models is the undamental diagram,i.e., the lux-versus-density relation, where lux is the product o the density and the average speed. For a hopping probability q NS at a given density c the exact lux F(c) in the NS model is given by [6, 12] [ F NS (c) = ] 1 4q NS c(1 c). (4) which reduces to F NS (c) = min(c, 1 c) in the deterministic limit q NS = 1. Note that in the two special cases = and = 1theant-trail model becomes identical to the NS model with q NS = Q and q NS = q, respectively. Extensions o the NS model have been used not only to capture dierent aspects o vehicular traic [6] but also to simulate pedestrian dynamics [7, 13, 14]. In a closely related CA model or pedestrian dynamics [14] the loor ields, albeit virtual,are analogous to the ields {σ } in the ant-trail model. However, in the pedestrian model there is no exclusion principle or the loor ield. 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 analogues o the variables representing the presence (or absence) o bus and passengers, respectively, in the BRM. Because o the periodic boundary conditions, the numbers o and buses are conserved while the and passengers are not conserved. However, unlike the BRM, the s are not dropped independently rom outside, but by the themselves. Another crucial dierencebetweenthese two modelsis that in the bus-routemodelq <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 orward i it smells ahead o it). In igure 2 we show theundamental diagrams obtained by extensive computer simulations o the ant-trail model or several values o.the most unusual eatures o the undamental diagrams shown in igure 2 are that, over an intermediate range o values o (e.g., in igure 2, =.5,.1,.5,.1) the lux in thelow-density limit c isveryclose to that or the NS model with q NS = q whereas in the high-density limit c 1theluxorthesame is almost identical to that or the NS model with q NS = Q. These unusual eatures o the undamentaldiagramsarise rom the non-monotonic variation o the average velocity with the density o the (see igure 3). The presence o the essentially introduces an eective hopping probability q e (c), whichdepends on the ant density c. The particle-hole symmetry (and hence the symmetry o the undamental diagram about c = 1/2) observed in the special limits = and = 1, is broken by the c-dependent eective hopping probability or all < <1 leading to a peak at c>1/2. Furthermore, the analysis o correlation unctions reveals some interesting clustering properties which will be studied in detail in a uture publication [17]. 1 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 orward, by one lattice spacing, with probability q NS i the lattice site immediately in ront is empty.

4 L576 Letter to the Editor (a) (b).4.2 Flux.2 Flux Figure 2. The lux o plotted against their densities or the parameters (a) Q = 1, q =.25 and (b) Q =.75,q =.25. The discrete data points corresponding to =.1( ),.5( ),.1( ),.5( ),.1( ),.5( ),.1( ),.25(+),.5( ) have been obtained rom computer simulations; the dotted lines connecting these data points merely serve as a guide to the eye. The two continuous curves at the top and bottom correspond to the lux in the NS model or q NS = 1. andq NS =.25, respectively, in (a) andorq NS =.75 and q NS =.25, respectively, in (b). 1 (a) 1 (b).8.8 Velocity.6.4 Velocity Figure 3. The average velocity o plotted against their densities or the parameters (a) Q = 1,q =.25 and (b) Q =.75,q =.25. The same symbols in igures 2 and 3 correspond to the same values o the parameter. The qualitative eatures o the c-dependence o q e can be reproduced by an analytical argument based on a mean-ield approximation (MFA) [17]. In this MFA, let us assume that all the move with the mean velocity V which depends on the density c o the as well as on ;although, to begin with, the nature o these dependences are not known, we will obtain these sel-consistently. Let us consider a pair o having a gap o n sites in between. The probability that the site immediately in ront o the ollowing ant contains is (1 ) n/ V since n V is theaverage time since the has been dropped. Thereore, in the MFA the eective hopping probability is given by q e = Q(1 ) n/ V + q{1 (1 ) n/ V }. (5) We replace n by the corresponding exact global mean separation n = 1 c 1between successive. Moreover, since V is identical to q e,wegettheequation ( ) qe q qe = (1 ) 1 c 1 (6) Q q

5 Letter to the Editor L577 which is to be solved sel-consistently or q e as a unction o c or a given. Note that the equation (6) implies that, or a given, lim c q e = q; thisrelects the act that, in the low-density regime, the dropped by an ant gets enough time to completely evaporate beore the ollowing ant comes close enough to smell it. Equation (6) alsoimplies lim c 1 q e = Q; thiscaptures the suiciently high-density situations where the are too close to miss the smell o the dropped by the leading ant unless the evaporation probability is very high. Similarly, rom (6)weget,orgiven c, lim 1 q e = q and lim q e = Q which are also consistent with intuitive expectations. In view o the act [1] that the lietime o s can be as long as 3 6 min, the interesting regime o ( 1), where the average velocity varies non-monotonically with the ant density, seems to be experimentally accessible. We hope that the non-trivial predictions o this minimal model o ant trail will stimulate experimental measurement o the ant lux as aunction o the ant density or dierent rates o evaporation by using dierent varieties o [8]. Acknowledgment We thank B Hölldobler or drawing our attention to [8]. Reerences [1] Wolram S 1986 Theory and Applications o Cellular Automata (Singapore: World Scientiic) Wolram S 1994 Cellular Automata and Complexity (Reading, MA: Addison-Wesley) [2] Schmittmann B and Zia R K P 1995 Phase Transitions and Critical Phenomena vol 17, eds C Domb and J L Lebowitz (New York: Academic) [3] Schütz G 2 Phase Transitions and Critical Phenomena vol 19, eds C Domb and J L Lebowitz (New York: Academic) [4] Marro J and Dickman R 1999 Nonequilibrium Phase Transitions in Lattice Models (Cambridge: Cambridge University Press) [5] Chopard B and Droz M 1998 Cellular Automata Modelling o Physical Systems (Cambridge: Cambridge University Press) [6] Chowdhury D, Santen L and Schadschneider A 2 Phys. Rep [7] Helbing D 21 Rev. Mod. Phys [8] Burd M, Archer D, Aranwela N and Stradling D J 22 Am. Natur [9] Wilson E O 1971 The Insect Societies (Cambridge, MA: Belknap) Hölldobler B and Wilson E O 199 The Ants (Cambridge, MA: Belknap) [1] Camazine S, Deneubourg J L, Franks N R, Sneyd J, Theraulaz G and Bonabeau E 21 Sel-organization in Biological Systems (Princeton, NJ: Princeton University Press) [11] Nagel K and Schreckenberg M 1992 J. Phys. I [12] Schreckenberg M, Schadschneider A, Nagel K and Ito N 1995 Phys. Rev. E Schadschneider A and Schreckenberg M 1993 J. Phys. A 26 L679 Schadschneider A 1999 Eur. Phys. J. B [13] Helbing D, Schweitzer F, Keltsch J and Molnar P 1997 Phys. Rev. E seealso Helbing D 21 Rev. Mod. Phys [14] Burstedde C, Klauck K, Schadschneider A and Zittartz J 21 Physica A Kirchner A and Schadschneider A 22 Physica A [15] O Loan O J, Evans M R and Cates M E 1998 Phys. Rev. E [16] Chowdhury D and Desai R C 2 Eur. Phys. J. B [17] Chowdhury D, Nishinari K and Schadschneider A to be published