Consensus of self-driven agents with avoidance of collisions

0.63 MB·English

by  Liqian Peng

Consensus of self-driven agents with avoidance of collisions
Liqian Peng1, Yang Zhao1, Baomei Tian1, Jue Zhang1, Bing-Hong Wang1,2, Hai-Tao Zhang3,4, and Tao Zhou1,2,5∗
1Department of Modern Physics and Nonlinear Science Center,
University of Science and Technology of China, Hefei 230026, P. R. China
2Research Center for Complex System Science, University of
Shanghai for Science and Technology, Shanghai 200093, P. R. China
3 Department of Control Science and Technology,
Huazhong University of Science and Technology, Wuhan 430077, P.R. China
4 Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K.
5Department of Physics, University of Fribourg, Chemin du Muse 3, CH-1700 Fribourg, Switzerland
9 (Dated: January 23, 2009)
0
0 In recent years, many efforts have been addressed on collision avoidance of collectively moving
2 agents. Inthispaper,weproposeamodifiedversionoftheVicsekmodelwithadaptivespeed,which
can guarantee the absence of collisions. However, this strategy leads to an aggregated state with
n
slowly moving agents. We therefore further introduce a certain repulsion, which results in both
a
J faster consensus and longer safe distance among agents, and thus provides a powerful mechanism
for collective motions in biological and technological multi-agent systems.
3
2
PACSnumbers: 89.75,-k,05.45.Xt

]
n
a I. INTRODUCTION
-
a
t
a Oneofthemostmarvelousandubiquitousphenomena
d in nature is collective motion, a kind of motion that can
s. be observed at almost every scale: from bird flocks and
c fishschoolsatamacroscopicleveltobacteria,individual
i
s cells and even molecular motors at a microscopic level
y [1,2,3,4,5,6,7,8,9]. Althoughinmostcasesagentsdo
h not share global information and often travel in the ab-
p
sence ofleadersorexternalforces,collectivemotionmay
[
stilloccur. Analogousbehaviorsarereportedinengineer-

3 ing systems also, such as groups of autonomous mobile
v robotsandairvehicles[10,11,12,13,14,15,16](seealso
9
a newly reported swarm model that may connect granu-
7
3 lar materials and agent-based models [17]). In order to FIG.1: Illustrationofthecurrentmodelwithadaptivespeed,
0 uncovertheunderlyingmechanismleadingtotheconsen- where r denotes the horizon radius of an agent, di denotes
. sus of collective population, Vicsek et al. [18] proposed the distance of the ith agent and its nearest neighbor, vmax
1 a model with self-driven agents to mimic the biological denotes the possibly maximal velocity, a denotes the size of
0 swarm,whichdisplaysanoveltypeofkineticphasetran- an agent. Accordingly, 2a corresponds to the least distance
8
0 sition. From then on, the nature of the nonequilibrium of two agents.
: phase transition of collective motion attracted more and
v
more attentions [19, 20, 21, 22, 23, 24]. Due to simplic-
i
X ity and efficiency, many modified versions of the Vicsek
Recently, much attention has focused on how to keep
r model were proposed. For example, some new methods distances among agents. A common way is to introduce
a witheffectiveleadershipwereintroduced[15,25,26],and
attractionand/orrepulsion[15,16,17,33,34,35,36,37,
new moving protocols with adaptive speed to accelerate
38, 39, 40]. However, any kinds of repulsions alone can
the consensus were designed [27, 28]; meanwhile some
not sufficiently avoid collision at all times, because it is
scholarshavestudied the consensusofcollective motions
entirely possible that in a high-density area, two agents
via low-cost communication [29] and predictive mecha-
are compelled to collide for the purpose of avoidance of
nism [30, 31, 32], all of which can greatly enhance the
collision with a third agent. In this article, we propose
global convergence.
a swarmmodel with adaptive speed to completely elimi-
natecollisions. Inaplane,eachagentadjustsitsdirection
as the average direction of its neighbors while resets its
speed according to the minimal distance from its neigh-
bors. Thefartheranagentisawayfromitsnearestneigh-
bor,thehigherspeedithas. Thisstrategycancompletely
∗Electronicaddress: [email protected] avoidcollisions,however,itresultsinanaggregatedstate
2
where the agents move very slow in average. Therefore,
we further introduce a repulsion that can break down
the aggregation of agents, and thus sharply speeds up 5
the globalconvergenceandenlargesthe averagedistance
among agents. 4
3
II. MODEL WITH ADAPTIVE SPEED

2
We consider each agent as an inelastic ball with ra-
dius a, limited in a square shaped cell of linear size L
1
with periodic boundary conditions. Initially, each agent
israndomlydistributedinthesquare,withmovingdirec-
tion randomly distributed in [−π,π). At each time step, 0
0 1 2 3 4 5
the position of the ith agent is updated as:

(a)
~x (t+1)=~x (t)+~v (t), (1)
i i i
and its direction is updated as:
5
θ (t+1)=hθ (t)i +∆θ , (2)
i i r i
4
where ∆θ denotes the thermal noise which is a random
i
number uniformly distributed in the interval [−η,η] (In
3
the main context, we only consider the noise-free case,

namely η = 0. A brief discussion about the effect of
2
noise is presented in the last section), hθ (t)i denotes
i r
the average direction of the agents within the horizon
radius r of the ith agent (including the ith agent itself), 1
which reads:
0
tan[hθ (t)i ]=hv sinθ (t)i /hv cosθ (t)i . (3) 0 1 2 3 4 5
i r i i r i i r

(b)
In natural swarms, the speed of each agent is alter-
able, that is, agent may adjust not only its moving di-
rection, but also its absolute velocity. In the common FIG. 2: Illustrations of locations and velocities in the initial
sense, to avoid collisions with other agents, an agent in configuration(a),andatthe500thtimestep(b). Theparam-
a high-density group should adopt lower speed. Taking eters are set as L=5, N=300, r=1, vmax=0.03 and a=0.01.
urban traffic as an example, the speed of an automobile The length and direction of an arrow represent the absolute
valueand direction of thecorresponding agent’s velocity.
is very low in the near-jammed situation, whereas it is
generally of high speed when sparse automobiles taking
up the road. Accordingly, we set the speed of the ith
of each agent (labeled by i) no more than (d −2a)/2 is
i
agentnotmorethanv =(d −2a)/2,whered is the ge-
i i i the only way guaranteeing the absence of collisions.
ographicaldistance between two centers of the ith agent
Accordingly, the absolute velocity of each agent is up-
and its nearest neighbor (see the illustration shown in
dated with the following rule:
Fig. 1). No matter how the ith agent and its nearest
neighbor choose their directions in the next time step, di−2a
the restriction can guarantee the distance between them vi(t+1)=Min vmax, 2 . (4)
(cid:18) (cid:19)
nolessthan2aandthereforeavoidcollision. Infact,this
Clearly,whenthedistancebetweenanagentanditsnear-
restriction is not only sufficient, but actually necessary.
estneighborislongerthan2v +2a,itsfollowingspeed
As the direction θ (t+1) of each agent in the next time max
i canachievethe maximum;otherwise,itsspeedislimited
stepisdeterminedbytheaveragedirectionwithinitsown
as (d −2a)/2.
horizon radius, it is impossible for an agent to know all i
Moreover,inordertoquantifytheconsensusofmoving
the information of its neighbors, especially the moving
directions, an order parameter [18] is introduced as:
directions of its neighbors in the next time step. There-
fore, in order to avoid collision, it is obliged to take into N ~v
account the worst circumstance, that is, two neighbors i=1 i
V = , 0≤V ≤1, (5)
mutuallyapproach. Inthisworstcase,keepingthespeed a (cid:12)(cid:12)PNi=1vi(cid:12)(cid:12) a
(cid:12) (cid:12)
P
3
Especiallywhen V =0, allagentssharethe same speed.
b

1.0
0.8
a

V
Initial Velocity 0.6
Average Velocity u=0
Repulsion Force
0.4 u=0.01
Resultant Force
Direction Vector u=0.1
Final Velocity
0.2
0 10 20 30 40 50 60 70
a Time step

0.7
FIG.3: (Coloronline)Illustrationofthemotionprotocolwith
0.6
repulsive effect.
0.5 u=0
b0.4

V
0.3
5
0.2 u=0.1
u=0.01
4 0.1
0.0
3
0 150 300 450 600 750 900

b Time Step
2

1 0.032
0.028
0 ed 0.030
0 1 2 3 4 5 e0.024
p u=0.1 0.029
S
e 0.020 u=0.01 0.028
FIG. 4: The distribution of positions and velocities at 500th g 0.027
time step in the scattering model. The parameters are set a0.016 0 100200 300400500
r
as L=5, N=300, r=1, vmax=0.03 and a=0.01. The length ve0.012
and direction of an arrow represent the absolute value and A
u=0
direction of thecorresponding agent’s velocity. 0.008
0 300 600 900 1200 1500 1800
where vi = |~vi|. A larger value of Va indicates better c Time Step
consensus. Since the speed in this model is no longer
constant, it is necessary to introduce another order pa- FIG. 5: (Color online) Va, Vb, and the average speed versus
timestepsunderdifferent repulsion strengthes. Theparame-
rameter V to evaluate the consensus of the absolute ve-
b ters are set as L=5, N=300, r=1, vmax = 0.03 and a=0.01.
locity, as:
Allthedatacomeformtheaverageresultsof500independent
h∆v2i runs.
V = , V ≥0, (6)
b b
v
p Numerical results reveal that after the direction con-
where v =hv i is the average absolute velocity of all the sensus, speed still varies. Figures 2(a) and 2(b) respec-
i
2
agents, and ∆v is the variance of the absolute velocity. tively illustrate the locations and velocities of all the
Apparently,asmallerV correspondstobetterconsensus. agents in the initial configuration and at the 500th time
b
4
step. After a certain time period from the beginning, Accordingly, we set the form of repulsion force as:
the positionsofagentsarenotuniformly distributedand
aTnheargeignretghaetiaovnerapgheensopmeeednionnaahpipgeha-drsen(ssietye aFriega.is2m(bu)c)h. f~ij = u×exp(−1−ri1j/r0)× ~rriijj rij <r0 (7)
( 0 rij ≥r0
slower than that in a low-density area. This aggregated
state can be understood as follows: agents in a high-
where u is a free parameter. Since the mass of an agent
density region agglomerate together and mutually move
playsnoroleinthepresentmodel,wesupposethe repul-
in a low speed, thus they can seldom disperse apart.
sive effect (caused by the repulsion) can directly affect
Meanwhiletheytakeupthewayoftheirsubsequentpeers
the velocity vector in the next time step (see Fig. 3 for
whosespeedishigher,makingthehigh-densityareacon-
an illustration).
gregate more agents, and in turn achieving even higher
density and slower speed (of course, on the other hand,
the density is limited by the size of agents, a). More-

over, the motions of agents are similar to the laminar
flow in hydromechanics: when V gets close to 1, each 0.035
a
agentis movingalonga line with the same directionand
0.030
will never diverge from its final track. Thus, different
layers present various flowing speeds. 0.025
In the current model, the nearest distance among n0.020
mi
agents in high-density areas is very close to 2a, making d
0.015
the involved agents move in a very low speed (close to
0); in the meantime the nearest distances among agents 0.010
in low-density areas are usually more than 2a+2v ,
max 0.005
accordinglythe involvedagentscanachievea highspeed
(close to v ). Consequently, the absolute velocities of 0.000
max 0 200 400 600 800 1000
allagents in the whole systemcan be in a highdiversity. N
Onlyinalow-densitylayercantheagentsmaintainhigh-
speedmovementinacomparativelongterm. Asamatter FIG.6: Minimalgeographicaldistancebetweenpairsofagent
of fact, the swarm never get speed consensus even with inthestablestateofthestandardVicsekmodel,dmin,versus
identical direction, as shown later in Fig. 5(b). For the the number of agents, N. Each data point is the average
purpose of making all the agents achieve the consensus of 1000 independent runs. The restriction to avoid collisions
with higher speed, it is necessary to introduce a certain with agent size a=0.01 corresponds to dmin >0.02.
repulsion to avoid agglomeration. In addition, denoting
r −2a as the safe distance between the ith andthe jth After introducing such a repulsion, the moving direc-
ij
agents, where r denotes the geographical distance be- tion of each agent is determined not only by the av-
ij
tween the ith and the jth agents. In real applications of erage direction within its horizon radius, but also by
unmanned air vehicles and auto-robots, the longer safe the repulsive effect. The synthesis of repulsive effect f~
i
distance is favorable. Therefore we hope a properly de- (f~ = N f~ , determined by Eq. (7)) and the average
i j=1 ji
signed moving protocol with repulsion could make the velocity ~v (whose direction and magnitude are respec-
i
safe distance longer. tivelyPdetermined by Eq. (3) and Eq. (4)) is set as the
following moving direction of the agent (see Fig. 3). On
the other hand, the absolute velocity should also follow
the Eq. (4). Numerical simulations, as shown in Fig.
4, indicate that this new protocol can effectively scatter
III. SCATTERING MODEL
the aggregated agents (take Fig. 2(b) as an example for
comparison). Actually, under this protocol, each agent
Based on the strategy with adaptive speed mentioned canholdacertaindistance(muchlongerthanthesystem
above, in this section, we introduce a repulsion to en- mentionedinSectionII)withitsneighbors,andtherefore
large the safe distances among agents. We assume: (1) achieves its maximal speed, v .
max
the direction of the repulsion should be along the line We alsoinvestigatethe effects ofrepulsionstrengthby
of two agents, and (2) the magnitude of the repulsion adjusting the parameter u. Figure 5(a) shows that the
should decrease with the increase of distance between convergence of moving direction is not sensitive to the
two agents. Moreover, as long as the distance between repulsion strength. However, a larger value of u corre-
twoagentsisover2v +2a,nomatterhowtheychoose sponds to a shorter time for the system to achieve the
max
theirdirectionsandspeeds,collisionwillnotoccurinthe consensus of speed, as well as a higher average speed in
following steps. Considering this, the repulsion in our the steady state (see Fig. 5(b) and Fig. 5(c)). Consid-
model should be a short-distance force and work only ering Eq. (4), the larger average speed actually implies
when the distance is shorter than r0 (r0 = 2vmax+2a). thatthe averagedistance betweenagentsis longer. Note
5
that,whenu=0,V cannotapproach0,andtheaverage IV. CONCLUSION AND DISCUSSION
b
speed is very low.
As long as we consider the sizes of agents, it is not
only possible but actually necessary to propose a proto-
colto avoidcollisionsamongthem. Although the Vicsek
model[18] hasachieveda greatsuccessin mimicking the
self-drivenswarm,itcannotguaranteetheabsenceofcol-
1.0 Vicsek Model lisions. We report in Fig. 6 a simple simulation of the
Scattering Model
noise-free Vicsek model neglecting the sizes of agents.
0.8
Asthegrowingofthepopulation,inthe stablestate,the
minimal geographical distance between pairs of agents
0.6
a decreases quickly. If the size of agent is set as a = 0.01,
V
then the minimal distance to avoid collisions must be
0.4
larger than 2a= 0.02. That is to say, the standard Vic-
sek model can only hold less than 200 agents with size
0.2
0.01inan5×5square. Incomparison,thecurrentmodel
with adaptive speed can hold thousands of such agents.
0.0
0 1 2 3 4 5 6 However, numerical simulations showed an aggrega-
tion phenomenon in the current model, which impedes
the convergence of speed. To overcome this blemish, we
FIG. 7: (Color online) Comparison of order parameter Va introduce a repulsion to scatter the aggregated agents.
in the Vicsek model and the scattering model under noisy The simulation results are exciting: Eachagentcan hole
environment. The parameters are set as L=5, N=300, r=1
a certain personal space; what is more, they can quickly
andvmax=0.03. Inscatteringmodel,u=0.01anda=0.01. All achieve speed consensus and move in a very high speed.
thedata come form theaverage over500 independentruns.
Numerical results also indicate that the stronger the re-
pulsive effect is, the less convergent time it takes to
achieve the consensus. In section II, we have already
proved that even two neighbors mutually approach, the
adaptivestrategycanstillavoidpossiblecollision. There-
fore, in any event, collision will never occur in the scat-

tering model.
60
Present Model Va Furthermore, it is well known that the thermal noise
50 Present Model Vb canalsoplayasignificantroleindeterminingthemoving
e Vicsek Model Va directionsofagents. Thus,weneedtocheckwhetherour
m
Ti40 rule is robust in the presence of noise. The numerical
ce result indicates that even in the noisy environment, in
gen30 the stable state, the average distance and average speed
over20 arebothlargerthanthosewithouttherepulsion. Theor-
C derparameterfordirectionconsensusofcoursedecreases
10 with the increasing of noise strength, η, and it exhibits
almostthesametrendasthestandardVicsekmodel(ac-
0 tually, it is a little bit larger than the Vicsek model, see
0 200 400 600 800 1000 1200
please the simulation result shown in Fig. 7).
N
In the noise-free Vicsek model, given r and L, the
convergence is faster with more agents (i.e., larger N)
FIG. 8: (Color online) Comparison on convergence time be-
tweenthestandardVicsekmodelandthepresentmodel(i.e., since they will have more frequent communications in a
the scattering model) in the absence of noise. In the Vicsek denser circumstance. Actually, a recent numerical study
model and the present model, the convergence time for Va is [28] indicates that the convergence time scales approxi-
definedastherequiredtimestepsmakingValargerthan0.99; mately as (lnN)−1.3, that is, the larger the population
while the convergence time for Vb is defined as the required is, the shorter the convergence time is. In Fig. 8, we re-
time steps making Vb smaller than 10−3. Blue dash curve portthesimulationresultontheconvergencetimeinthe
represent the simulation result for the Vicsek model, while
noise-free Vicsek model (see the blue dash curve), where
the black squares and red circles represent the results for Va the threshold quantile is set as V = 0.99. It decreases
and Vb, respectively. The parameters are set as L=5, r=1 a
monotonously with the increasing of N, in accordance
and vmax=0.03. In the present model, u=0.1 and a =0.01.
Allthedatacomeform theaverageover5×103 independent with Ref. [28]. In contrast, in the present scattering
model, more effort should be taken to avoid collisions
runs.
in the denser circumstance. Figure 8 compares the con-
vergence time between the standard Vicsek model and
6
the scattering model in the absence of noise. One can communication is not always the most efficient manner
find that in the sparse circumstance, N ≤ 600, the con- whilepartialcommunicationmaybebetterinsomecases
vergence times of the Vicsek model and the scattering [29, 41]. In addition, the properties of the phase transi-
model are almost the same, while in the denser range, tion induced by the noise (see, for example, in Ref. [34],
the convergence time in the scattering model quickly in- Gr´egoire and Chat´e showed that a swarm model with
creases versus the slowly decreasing of that in the Vic- repulsion as well as the minimal Vicsek model suffers
sek model. The convergence time for absolute velocity a discontinuous phase transition) remains an open issue.
increases even most quickly than that for moving direc- Thoughnotthefocusinthisarticle,itworthsandetailed
tion. This result indicates a limitation of the present investigation in the future.
model, namely it can not efficiently deal with the sys-
tems with huge population. Accordingly, how to design
an efficient method to simultaneously guarantee the ab-
sence of collisions and the quick convergence is still an
Acknowledgments
open problem for us. Anyway, in the case of a = 0.01,
the standard Vicsek model can avoid the collisions only
if the number of agents is less than or about 100 (see This work is funded by the National Basic Research
Fig. 6), while the scattering model can hold about 600 Project of China (973 Program No.2006CB705500),
agentswiththesamespeedofconvergence. Wetherefore the Specialized Research Fund for the Doctoral Pro-
believe the scattering model can find applications in the gram of Higher Education of China under Grant No.
design of motion protocol for self-driven agents. 20060358065, the National Science Fund for Fostering
Some difficult yet important problems about the con- Talents in Basic Science under J0630319, and the Na-
servative model remain to be further explored. For ex- tionalNaturalScience FoundationofChina under Grant
ample, if the ahead ones of a group of agents need not No. 10532060. H.T.Z. acknowledges the support of the
pay attention to the following ones (that is, each agent NationalNaturalScience FoundationofChina(NNSFC)
only receives informationin a sector aheadin its moving underGrantNo. 60704041,andtheSpecializedResearch
direction rather than all the neighbors within its sight Fund for the Doctoral Program of Higher Education of
radius [41]), collisions may automatically disappear. If ChinaunderGrantNo. 20070487090. T.Z.acknowledges
the swarm needs shorter time to get convergence while theNationalNaturalScienceFoundationofChinaunder
avoidingthe collisions,itmayindicate thatthecomplete Grant No. 10635040.
[1] L. Segel, SIAMJ. Appl.Math. 32, 653 (1977). [15] N. E. Leonard and E. Fiorelli, Proceedings of the 40th
[2] A.Czir´ok, E.Ben-Jacob, I.Cohen,andT.Vicsek,Phys. IEEEConferenceonDecisionandControl3,2968(2001).
Rev.E 54, 1791 (1996). [16] Y. Liu, K. M. Passino, and M. M. Polycarpou, IEEE
[3] F. Nedelec, T. Surrey, A. Maggs, and S. Leibler, Nature Trans. Automat. Control 48, 76 (2003).
(London) 389, 305 (1997). [17] D.Grossman, I. S. Aranson, and E. Ben-Jacob, New J.
[4] J. K. Parrish and W. M. Hamner, Animal Groups in Phys. 10, 023036 (2008).
Three Dimensions (Cambridge University Press, Cam- [18] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen, and O.
bridge, England, 1997), and thereferences therein. Shochet, Phys.Rev. Lett.75, 1226 (1995).
[5] M. T. Laub and W. F. Loomis, Mol. Biol. Cell 9, 3521 [19] A. Czairok, H. E. Stanley, and T. Vicsek, J. Phys. A:
(1998). Math. Gen. 30, 1375 (1997).
[6] E. Ben-Jacob, I. Cohen, and H. Levine, Adv. Phys. 49, [20] A. Czir´ok, A.-L. Barab´asi, and T. Vicsek, Phys. Rev.
395 (2000). Lett. 82, 209 (1999).
[7] R.Kemkemer,V.Teichgr¨aber,S.Schrank,D.Kaufmann, [21] J. Toner, Y.Tu,and S.Ramaswamy,AnnalsPhys. 318,
and H. Gruler, Eur. Phys.J. E 3, 101 (2000). 170 (2005).
[8] Y. Inada and K. Kawachi, J. Theor. Biol. 214, 371 [22] M. Aldana, V. Dossetti, C. Huepe, V. M. Kenkre, and
(2002). H. Larralde, Phys.Rev. Lett. 98, 095702 (2007).
[9] I. D. Couzin and J. Krause, Adv. Study Behav. 32, 175 [23] W.Li, H.T.Zhang, M.Z.Q.Chen,and T.Zhou,Phys.
(2003). Rev. E, 77, 021920 (2008).
[10] I.PrigogineandR.Herman,KineticTheoryofVehicular [24] H.Chate,F.Ginelli,G.Gregoire,andF.Raynaud,Phys.
Traffic (American Elsevier, New York,1971). Rev. E, 77, 046113 (2008).
[11] D.Helbingand B.A.Huberman,Nature(London)396, [25] I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin,
738 (1998). Nature (London) 433, 513 (2005).
[12] D. Helbing and M. Treiber, Phys. Rev. Lett. 81, 3042 [26] S.Mu,T.Chu,andL.Wang,PhysicaA351,211(2005).
(1998). [27] W.LiandX.F.Wang,Phys.Rev.E75, 021917 (2007).
[13] A.Jadbabaie, J.Lin,andA.S.Morse, IEEETrans.Au- [28] J.Zhang,Y.Zhao,B.M.Tian,L.Q.Peng,H.T.Zhang,
tomat. Control 48, 988 (2003). B. H.Wang, and T. Zhou,Physica A (tobe published).
[14] T.Chu,L.Wang,andT.Chen,J.ControlTheoryAppl. [29] H. T. Zhang, M. Z. Q. Chen, and T. Zhou, arXiv:
1, 77 (2003). 0707.3402.
7
[30] H. T. Zhang, M. Z. Q. Chen, G. B. Stan, T. Zhou, and Hayakawa, and M. Sano, Phys. Rev. Lett. 76, 3870
J. M. Maciejowski, IEEE Circ. &. Syst. Mag. 8(3), 67 (1996).
(2008). [37] H.Levine,W.-J.Rappel,andI.Cohen,Phys,Rev.E63,
[31] H. T. Zhang, M. Z. Q. Chen, T. Zhou, and G.-B. Stan, 017101 (2000).
Europhys.Lett. 83, 40003 (2008). [38] I.D.Couzin,J.Krause,R.James, andG.D.Ruxton,J.
[32] H.T.Zhang,M.Z.Q.Chen,andT.Zhou,Phys.Rev.E Theor. Biol. 218, 1 (2002).
(tobe published). [39] V.GaziandK.M.Passino,IEEE.Trans.Automat.Con-
[33] G. Gr´egoire, H. Chat´e, and Y. Tu, Physica D 181, 157 trol. 48, 692 (2003).
(2003). [40] V. Gazi and K. M. Passino, Int. J. Control 77, 1567
[34] G. Gr´egoire and H. Chat´e, Phys. Rev. Lett. 92, 025702 (2004).
(2004). [41] B.-M. Tian, H.-X. Yang, W. Li, T. Zhou, W.-X. Wang,
[35] Z.Csahok and T.Vicsek, Phys.Rev.E 52,5297 (1995). and B.-H. Wang, arXiv: 0806.3594.
[36] N. Shimoyama, K. Sugawara, T. Mizuguchi, Y.