- Email: [email protected]
Collective behaviors of two-component swarms
ARTICLE IN PRESS Journal of Theoretical Biology 261 (2009) 494–500
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Collective behaviors of two-component swarms Sang Koo You a,, Dae Hyuk Kwon b, Yong-ik Park b, Sun Myong Kim b, Myung-Hoon Chung c, Chul Koo Kim d a
Natural Science Research Institute, Yonsei University, Seoul 120-749, Republic of Korea Department of Physics, Yonsei University, Wonju 220-710, Republic of Korea c College of Science and Technology, Hong-Ik University, Chochiwon 339-800, Republic of Korea d Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Republic of Korea b
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 July 2009 Received in revised form 21 August 2009 Accepted 21 August 2009 Available online 27 August 2009
We present a particle-based simulation study on two-component swarms where there exist two different types of groups in a swarm. Effects of different parameters between the two groups are studied systematically based on Langevin’s equation. It is shown that the mass difference can introduce a protective behavior for the lighter members of the swarm in a vortex state. When the self-propelling strength is allowed to differ between two groups, it is observed that the swarm becomes spatially segregated and finally separated into two components at a certain critical value. We also investigate effects of different preferences for shelters on their collective decision making. In particular, it is found that the probability of selecting a shelter from the other varies sigmoidally as a function of the number ratio. The model is shown to describe the dynamics of the shelter choosing process of the cockroach– robot mixed group satisfactorily. It raises the possibility that the present model can be applied to the problems of pest control and fishing using robots and decoys. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Collective motion Swarm dynamics Multi-component swarm Collective decision making
1. Introduction Flocking motion is a ubiquitous phenomenon in nature ranging from the colony organization of microorganisms and flocking of fish and birds, to swarming of small insects. Understanding active collective motion and swarm intelligence in these biological systems is one of the important questions in nonequilibrium statistical physics. Like equilibrium condensed matter systems, swarms exhibit distinct phases, such as crystalline ordered, vortex, and disordered states. The important connection between the nonequilibrium dynamics and the equilibrium phase transitions was first recognized by Vicsek et al. (1995). This question was further investigated using a continuum theory to reveal the deeper connection between seemingly different two cases (Toner and Tu, 1995). A recent review of the works in this line over the past decade is given by Toner et al. (2005). More recently, the question of the instability of the phases as a function of the number of swarm members has been addressed (D’Orsogna et al., 2006) and the detailed nature of the dynamic phase transition have been investigated in detail to reveal the delicate role of the noise (Gre goire and Chate, 2004; Aldana et al., 2007; Nagey et al., 2007). However, still important questions remain, such as the cause of transitions in flocks, the shapes and surface
Corresponding author. Tel.: + 82 2 2123 2611; fax: + 82 2 313 8892.
E-mail address: [email protected] (S.K. You). 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.08.025
effects of a finite flock, seemly intelligent responses to food and shelters, and the phases and dynamics of multi-component flocks (Toner et al., 2005). In nature, no individual member of a flock is identical and wide variations in size, mass, and speed are expected. In order to understand this problem, we introduce a simple two-component swarm and employ a particle based simulation approach. Although there exist various different approaches to study the motion of the swarm (Vicsek et al., 1995; Levine et al., 2000; Couzin et al., 2002; Krause and Ruxton, 2002; Schweitzer, 2003; Erdmann et al., 2005; Ame et al., 2006; Chen and Leung, 2006; Levine et al., 2006; Sumpter, 2006; Mach and Schweitzer, 2007; Li and Wang, 2007; McInnes, 2007; Halloy et al., 2007; Mailleux et al., 2008; Schmickl et al., 2009), common essential ingredients are mutual attractions, self-propelling forces, and responses to environmental attractions such as food and shelters. Here, we choose a particle-based simulation model, which employs simple forms of the essential ingredients (Chen and Leung, 2006).
2. A particle-based model for two-component swarms We consider a system consisting of N particles which is governed by the equation of motion, mi
d~ vi ¼ g~ F i þ~ xi; v i þ ai V^ i þ ~ dt
ð1Þ
ARTICLE IN PRESS S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
where mi and ~ v i are the mass and velocity of ith particle, respectively. The four terms in the right-hand side represent v i is various forces influencing particle’s motion. The first term g~ the drag force, arising from the frictional nature of the medium or from speed limitation of living creatures and manufactured robots. g is the drag coefficient. The second term ai V^ i accounts for the self-propelling force including additional tendency to align with its nearby particles. ai represents the propelling strength and ~ i which is defined as V^ i is an unit vector parallel to V ~i ¼ V
N X
~ v j expðRji =ra Þ:
3. Simulation results for two-component swarms
Here, Rji ¼ j~ rj ~ r i j is the distance between ith and jth particles. ra is a parameter controlling the range of alignment in which propelling direction is averaged with an exponential factor. It is known that if ra increases, the vortex state become gradually unstable and transforms into crystal state moving to the same direction all together (Chen and Leung, 2006). The third term includes all other forces except noise, including mutual interaction forces between member particles and external forces from environmental potentials such as food and shelters. Here, we consider only two kinds of forces for mutual interactions and individual preference, which we call body force ð~ F bi Þ and shelter force ð~ F si Þ, respectively. For the body forces, two types of expressions are usually used. One is the Morse potential type (D’Orsogna et al., 2006; Levine et al., 2000) and the other the Lennard–Jones potential type (Chen and Leung, 2006). We use the second type force as in Chen and Leung (2006). Shelter force can be represented in various ways. Here, we adopt the approach that assumes no individual particle intelligence in searching for shelters. Thus, we assume that the shelter force is inwardly attractive only inside of the shelters. Thus, the member particles of the swarm will be attracted to the shelter, only when it hits the shelter by chance. We assume the shelters to have disk shapes for simplicity. Then, the third term ~ F i is defined as ~ F bi þ ~ F si ; Fi ¼ ~ N X
R^ ji ½1 ðRji =rb ÞexpðRji =rb Þ;
ð3Þ
jai
~ F si ¼
In reality, swarms are not homogeneous. They may vary in mass, size, and speed. Also, each particle may have different selfpropelling strength, body force, and response to environment. Although we consider only two-component swarm problem in this paper, we note that the formalism given by Eqs. (1)–(4) can be applied to any multi-component swarm problems, if necessary. For a two-component flock, we assume that there are Nð1Þ type1 particles and Nð2Þ type-2 particles in the swarm. The type-1(2) particles are characterized by mass, mð1Þ ðmð2Þ Þ, the self-propelling strength, að1Þ ðað2Þ Þ.
ð2Þ
j
~ F bi ¼ b
495
8P < sia R^ ica expððRica Rs Þ=rs Þ
if Rica rRs ;
:
otherwise:
a
0
ð4Þ
Here, b and rb represent the interaction strength and the range between particles, respectively. R^ ji is an unit vector directed to the ith particle from the jth particle. We assume that the interaction forces are all equivalent and independent on their subgroups. sia is the ath shelter force strength. Rs and rs are the radius and the force range of shelters, respectively. The last term in Eq. (1) is the stochastic force which models the rapidly fluctuating environmental conditions. For simplicity, the noise force is assumed to have no temporal correlation and is taken from a uniform distribution over ½w; w both in x-, ydirections. It should be again noted that the present model is one of many similar models. Using this and other types of similar models, the collective motions of a single component swarm have been studied intensively (Toner and Tu, 1995; Chen and Leung, 2006; Erdmann et al., 2005; Schmickl et al., 2009). It was found that there exists basically three different dynamics phases in this type of model: crystal, disordered, and vortex, although different models may yield more diverse states (Toner et al., 2005; Gre goire and Chate, 2004).
We only consider the model of two-component swarms moving in two spatial dimensions. In the following subsections, we investigate three different problems of the present model which are characterized by parameters assuming two different masses, propelling strengths and shelter force strengths, respectively. For other parameters, g ¼ 1, b ¼ 1, and rb ¼ 1 are chosen for all three cases. 3.1. Two components with different masses As a simple example of the two-component swarms, we consider a system with N ¼ 100 particles of which Nð1Þ ¼ 50 particles have masses, mð1Þ ¼ 1, while remaining N ð2Þ ¼ 50 particles have masses, mð2Þ ¼ 0:5. Here, shelter forces are not considered. All other parameters are assumed to be the same for the two components. Since the mass does not influence the steady state speed, the crystal and the disordered state are not influenced by the mass difference. However, the vortex state is expected to be different, since the rotation can be influenced by the mass. In order to investigate the effect of the mass difference, we first investigated the vortex states of the two components separately and then studied the motion together. In order to realize the vortex states, we choose ai ¼ 1, ra ¼ 0:001, sia ¼ 0, and w ¼ 2 (Chen and Leung, 2006). In Fig. 1, we show a snapshot of a steady rotating mixed swarm, which clearly shows that most of the lighter members of the swarm are rotating inside of the heavier ones. Fig. 2 shows the density profiles for each component, both when the two components move together and separately. It is interesting to observe that the density profiles change drastically when the two components are mixed. The average radius of the vortex for the lighter component is shown to shrink, when they are mixed with the heavier component, whereas that of the heavier components becomes larger. It can be shown that this collective behavior of swarm originates from the body force. In Eq. (1), ~ v i and V^ i are approximately parallel to each other by the vortex symmetry and only ~ F bi has a role of centripetal force. All members of the swarm have the same averaged speed a=g independent of their masses. As a result, the ith particle moves around a circle with radius r under the centripetal force Fbi ¼ ðmi =rÞða=gÞ2 . Therefore, if a particle with mass mð1Þ is added to a steadily rotating vortex of mass mð2Þ , it is pushed outward from the vortex center by the force ððmð1Þ mð2Þ Þ=rÞða=gÞ2 . Therefore, heavier(lighter) mass particles are located outside(inside) as shown in Fig. 2. Real animal groups evolved to have social interactions that in turn show various dynamic behavioral patterns. The protective behavior of rotating fish flocks of hiding younger ones inside of adult members of the flock would be also an outgrowth of social evolution. The present simulation result shows seemingly protective behavior originated from the physical many-body effect with mass difference. Therefore, we conjecture that the evolution to protective behavior might be promoted and enhanced by the physical origin.
ARTICLE IN PRESS 496
S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5 −1.5
−1
−0.5
0
0.5
1
1.5
Fig. 1. A snapshot of a steady rotation of a mixed swarm with two different mass components. The lighter ones are represented by open circles while the heavier ones are represented by dark circles. They are moving in the directions of line segments from circles with speeds proportional to the lengths. Note that most of the lighter ones are rotating inside of the heavier ones. Two groups are characterized by N ð1Þ ¼ N ð2Þ ¼ 50, mð1Þ ¼ 1, mð2Þ ¼ 0:5 respectively. Other parameters used for simulations are g ¼ 1, ai ¼ 1, ra ¼ 0:001, b ¼ 1, rb ¼ 1, sia ¼ 0, and w ¼ 2.
−1.5 −1.5
−1
−0.5
0
0.5
1
1.5
Fig. 3. A snapshot of a steady crystal state of mixed swarm with two different propelling strengths. Weakly propelling ones are represented by open circles while strong ones are represented by filled circles. Note that the strongly propelling ones are moving in front of the weaker ones. Two groups are characterized by N ð1Þ ¼ N ð2Þ ¼ 50, að1Þ ¼ 1, að2Þ ¼ 0:5, respectively. Other parameters used for simulations are g ¼ 1, mð1Þ ¼ mð2Þ ¼ 1, ra ¼ 1, b ¼ 1, rb ¼ 1, sia ¼ 0, and w ¼ 2.
segregation between the two components and would such segregation eventually split the flock into fast and slow moving flocks? In order to answer such questions, we provide the different self-propelling strengthes instead of masses as in the previous model, namely, mð1Þ ¼ mð2Þ ¼ 1 and að1Þ 4að2Þ . The other parameters are the same as in the previous model except ra ¼ 1 so that the vortex state is absent. A snapshot in a crystal state is shown in Fig. 3, where að1Þ ¼ 1 and að2Þ ¼ 0:2 are used as an example. The strongly propelling group locates ahead of the weaker group and leads the other. The swarm velocity v can be obtained as follows. When we write the average body force for group 1(2) as Fbð1Þ ðFbð2Þ Þ, particles of groups 1 and 2 satisfy the relation gv það1Þ þ Fbð1Þ ¼ 0 and gv þ að2Þ þ Fbð2Þ ¼ 0, respectively, because there is no acceleration. They also satisfy another equation, Nð1Þ Fbð1Þ þ Nð2Þ Fbð2Þ ¼ 0, since the total body force in a swarm should be zero. From the above equations, we obtain the average swarm velocity v as v ¼ ðNð1Þ =ðN gÞÞ Da þ að2Þ =g; Fig. 2. The number density rðrÞ of each component in the vortex of Fig. 1. Here, r is the distance from a vortex center. rðrÞ is averaged over 10 000 times iterations. For comparison, densities of homogeneous swarms with two difference masses are also shown; dotted(dashed) line is for the lighter(heavier) swarm only.
3.2. Two components with different self-propelling strengths In a swarm existing in nature, we expect that younger members of the flock have smaller self-propelling strengths as well as lighter masses. In fact, a smaller self-propelling strength implies a smaller mean speed for those members of the flock. In this section, we study a two-component swarm model in which fast and slow members are mixed. Again, here we do not consider any environmental forces. In the crystal state, there is no mass dependence (Chen and Leung, 2006). Thus we focus only on the difference of self-propelling strengths. This question has been raised as a future problem by Toner et al. (2005). Here, we address the following questions: Would there be a large scale spatial
ð5Þ
where Da ¼ að1Þ að2Þ is the difference of self-propelling strengthes between the two groups. The linear relationship between v and Da is confirmed in Fig. 4. where N ð1Þ ¼ Nð2Þ ¼ 50, g ¼ 1, and að2Þ ¼ 0 are chosen for the simulation. Then, Eq. (5) yields v ¼ Da=2, which is well satisfied by the simulation data below the separate phase in Fig. 4. Thus, the results show that swarms will move together even when there are substantial differences in the propelling strengths between individual members, although fast moving members or leaders locate themselves in front. When the difference in the selfpropelling strengths is substantial, spatial segregation between the two components is expected. It is found that this separation takes place abruptly at a certain critical value Dac as shown in Fig. 4. Since the body forces are expressed as Fbð1Þ ¼ ðNð2Þ =NÞ Da;
Fbð2Þ ¼ ðN ð1Þ =NÞ Da;
ð6Þ
the body forces increase linearly to Da and reach maximum values at a certain spacial distribution of particles. Thus the maximum body ð1Þ ð2Þ ¼ bað1Þ and Fb;max ¼ bað2Þ , where að1Þ forces can be written as Fb;max
ARTICLE IN PRESS S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
497
5
5 1
1 0
0 2
2 −5 −5
0 0
5
5
−5 −5
0 10
5 1
1 0 Fig. 4. The velocity of a swarm as a function of propelling strength difference, Dað ¼ að1Þ að2Þ Þ, with the fixed að2Þ ¼ 0. The relation is linear up to the critical value Dac beyond which the separation between fast and slow groups occurs. Other parameters are the same as in Fig. 3. Inset: The linear relationship between Dac and b.
and a are constants to be determined from the spatial distribution just before the separation by Eq. (3). They also satisfy the relation, Nð1Þ að1Þ ¼ N ð2Þ að2Þ . As a result, the critical value Dac has a linear relationship with b as ð2Þ
Dac ¼ ðNað2Þ =Nð1Þ Þb:
0 2
−5 −5
5
2
0 50
5
−5 −5
0 154
5
Fig. 5. Snapshots of a selection process for the final shelter. The numbers under snapshots represent the time steps. Two big circles represent two different shelters 1 and 2. Shelter 2 is finally chosen in this process, although 13(7) particles marked by open(filled) circles prefer shelter 1(2).
ð7Þ
This linear relationship is confirmed in the inset of Fig. 4. After the separation, two groups move with velocities, að1Þ =g and að2Þ =g, independently. In this subsection, we have shown that a strong body force can maintain a swarm to move steadily as a whole, in spite of differences in propelling strengths of individual members. Also, it is shown in the simulation that the strong ones occupy leading positions in moving. 3.3. Two components with different responses to environment In this subsection, we investigate a two-component swarm which consists of two groups with different responses to environment. This type of model studies is applicable to various problems, such as collective decision making, collective intelligence, pest control through robots, and leader-follow problems in flocks (Couzin et al., 2005; Ame et al., 2006; Couzin, 2007; Halloy et al., 2007; Mailleux et al., 2008; Schmickl et al., 2009; Couzin, 2009). In the present case, the particular environment is the existence of two different types of shelters. We focus on the collective behavior of a swarm where elements are divided into two groups, one of which prefers the shelter 1 while the other shelter 2. The present model can be readily applied to the problem of mixed swarm of cockroaches and robots studied in Halloy et al. (2007) in that robots are artificially programmed to have the opposite preference for shelters against cockroaches. Our interest in the present problem is to understand the dynamics and conditions of selection processes of particular shelters. Detailed understanding of this problem may have huge implications on pest control and fishing techniques using robots and decoys. In the present model, 20 elements are programmed to be confined in a regular square with each side length of 10. The discshaped shelters 1 and 2 with radii of 0.8 are centered at (2,2) and ð2; 2Þ, respectively. Particles of 1 ri rN ð1Þ and N ð1Þ þ1 r ir N belong to groups 1 and 2, respectively. The self-propelling strength ai is assumed to be ai ¼ 1 outside of shelters and ai ¼ 0 inside. For the shelter force strength sia in Eq. (4), we assign four
ð1Þ ð2Þ ð2Þ parameter values, sð1Þ 1 , s2 , s1 , and s2 , where subscripts and superscripts correspond to indices of shelters and groups, respectively. The larger sðaa0 Þ means the larger preference. The shelter force range is set to be rs ¼ 0:1, and the noise parameter is chosen w ¼ 7 so that particles behave like real insects and can escape from the shelters by chance. m ¼ 0:5, b ¼ 0:5, and rb ¼ 0:3 are used. Other parameters are same as in the previous models. This choice of parameters implies that we adopt the assumption that a swarm has no individual intelligence of detecting a shelter or food. Only when they hit a shelter by chance, they are attracted to it. However, it will be shown that the overall swarm motion appears as if the swarm as an entity possesses intellectual capability to detect a shelter of their choice. Fig. 5 shows a simulation of selection process of the swarm as time step increases. Initially, 20 particles are located randomly in a regular square with a segment length 0.5 centered at ð1:5; 1:5Þ and with random velocities of x- and y-components ranged by ð1Þ ½0:5; 0:5. Group 1 has 13 particles with sð1Þ 1 ¼ 0:3 and s2 ¼ 0:2 ð2Þ ¼ 0:2 and s while group 2 has seven particles with sð2Þ 1 2 ¼ 0:3. Therefore, group 1(2) is more attracted to shelter 1(2) than to shelter 2(1). Once the simulation starts, particles move stochastically due to the noise force until they hit a shelter and are attracted inside. The probability to step out of a shelter is dependent on the following competing factors. One is the stochastic noise force which allows a particle to escape from the shelter. This force is opposed by the attraction measured by the parameter sðaa0 Þ. Here, we note that the body force also plays an important role. If the number of particles in one particular shelter becomes larger, the probability to step out is reduced due to the body force. Therefore, the selection process is a result of competition between individual and collective responses. Although it is more likely for the swarm to congregate in shelter 1 than in shelter 2, it is also possible for the swarm to choose the shelter 2 as can be seen in Fig. 5. By performing 200 simulations, we have obtained P1 , the probability for the swarm to completely congregate in shelter 1 as
ARTICLE IN PRESS 498
S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500 ð1Þ ð1Þ ð1Þ ð2Þ as e1 ¼ ðsð1Þ þsð2Þ þ sð2Þ 1 N 1 N Þ in shelter 1 and e2 ¼ ðs2 N 2 Nð2Þ Þ in shelter 2 using a dimensionless energy scale. Although the self-propelling dynamics of the present problem is of highly nonequilibrium, the final state is very simple and quasi-static because all members remain within a shelter. If we construct an ensemble only with the final states, it will be similar to two-state ensemble in equilibrium statistical mechanics. Furthermore, while particles feel the above potential energies from shelters, they are also influenced by the noise in Eq. (1) whose effect is similar to thermal noise to scatter particles. Therefore, we assume the probability of the shelter selection could be approximately determined from the Boltzmann-like factor, expðbea Þ as in equilibrium statistics. Here 1=b represents the random noise of the swarm movements, and later will be determined from data. P1 , the probability for the swarm to be all congregated in shelter 1 could be approximated by 1=ðexpðbðe1 e2 ÞÞ þ 1Þ. Then, finally we have
Fig. 6. P1 , the probability of selecting shelter 1, as a function of N ð1Þ under the ð2Þ ð1Þ ð2Þ obtained averaging 200 conditions, N ¼ N ð1Þ þ N ð2Þ ¼ 20, sð1Þ 1 ¼ s2 , and s2 ¼ s1 simulations. Solid lines are functions of data fit using Eq. (8) with b values in parentheses.
Fig. 7. P1 , the probability of selecting shelter 1, as a function of N ð1Þ under the conditions N ¼ N ð1Þ þ N ð2Þ ¼ 20 and asymmetric sðaa0 Þ ’s. Solid lines are functions of data fit using Eq. (8) with b values in parentheses.
a function of particle number Nð1Þ with fixed total number N ¼ 20. In Fig. 6, we show that P1 has symmetric distributions as expected ð2Þ when two groups have exactly opposite preferences; sð1Þ 1 ¼ s2 and ð2Þ ¼ s . Fig. 7 treats a more general case where two groups have sð1Þ 2 1 asymmetric preferences. Here, we note that when the attraction of shelter 1 for group 1 is much larger than that of shelter 2 for group 2, then the swarm can be lured to shelter 1 with relatively small numbers of robots or decoys. In principle, the probability can be analyzed using the master equation. However, this approach involves the lengthy computing procedure due to highly correlated motion between particles and uncertainties, when the total number is rather small. Alternatively, we adopt an intuitive approach to depict underlying physics as follows. The potential energy for a particle belonging to the group a in shelter a0 will be proportional to the parameter sðaa0 Þ in Eq. (4). Once the swarm is congregated in one of shelters, the center of mass does not move much. Thus, the total kinetic energy can be safely neglected as well as the internal interaction in the swarm dynamics. Finally, the total energies can be written
P1 ¼ 1=ðexp½bððDð1Þ þ Dð2Þ ÞNð1Þ Dð2Þ NÞþ 1Þ;
ð8Þ
ð1Þ ð2Þ where Dð1Þ sð1Þ and Dð2Þ sð2Þ are defined as the 1 s2 2 s1 differences of preferences. In Figs. 6 and 7, we show that the simulation data can be fitted by the formula, Eq. (8) using the listed values for b. In Fig. 7, b is the same for all three curves, since P1 has same fixed values at Nð1Þ ¼ 0. Although further rigorous studies are needed for the strict validity of the above derivation, we expect that Eq. (8) would provide a simple picture for the problems like collective decision making for swarms. In order to check the usefulness of the present two-component swarm formalism to the particular problem of pest control using robots, we compare our simulation results with the recent experimental results reported in Halloy et al. (2007). They carried out experimental study on collective selection of shelters by a mixed group of four socially integrated autonomous robots and 12 cockroaches. Autonomous robots were perceived by cockroaches as congeners and used as interactive decoys. Therefore, the present model of two-component swarm with different responses to shelters maps exactly to this problem. When there exist two types of shelters, light (shelter 1) and dark (shelter 2), it is known that cockroaches (particle group 2) prefer the dark shelter over the light one. However, the robots (particle group 1) can be programmed to favor either light or dark shelter depending on experimental design. It was found that by programming the robots to favor light shelter over dark one, the whole group can be lured to the light shelter with high probability. In order to apply the two-component swarm formalism to this problem, we use Eq. (8) to determine the relations between b, sð2Þ 1 (preference of the (preference of the dark shelter by light shelter by cockroach), sð2Þ 2 ð1Þ cockroach), sð1Þ 1 (preference of the light shelter by robot), and s2 (preference of the dark shelter by robot). From Eq. (8), we obtain ð2Þ bDð2Þ ¼ bðsð2Þ 2 s1 Þ ¼ 0:062 for the cockroach-only-group choosing the dark shelter with 73% probability, and bDð1Þ ¼ bðsð1Þ 1 sð1Þ 2 Þ ¼ 0:298 for the mixed group (four robots and 12 cockroaches) choosing the light shelter to become 61% (Fig. 8(a) and Halloy et ð1Þ al., 2007). We use fixed values of sð2Þ 1 ¼ s2 ¼ 0:2 without loss of generality, since only the preference differences appear in the calculation. Here, we note that the parameters cannot be determined uniquely due to lack of experimental data. Thus, we choose a set of parameters that produce the probabilities of shelter selections close enough to the experimental values (Fig. 8(b)). We note that this process corresponds to the interpolation of the probability curve, P1 using two points. In principle, one more experimental point will allow to fix the parameter values uniquely. Even with this uncertainty, ranges for allowed values are not large. Chosen parameters are b ¼ 1:05, ð2Þ sð1Þ 1 ¼ 0:484, and s2 ¼ 0:259. Since, we have all the parameters necessary for the simulations, we can describe any detailed
ARTICLE IN PRESS S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
499
results are in good agreement with the actual experiments, thus indicating that the chosen parameters are suitable to describe the detailed dynamics of the mixed group. In Halloy et al. (2007), the experiments were simulated using a different mathematical model. In their mathematical model, the shelter choosing probability was calculated from the individual rates for entering and quitting shelters. Although both methods give satisfactory description of the experiments, we believe that the present twocomponent swarm method provides a more direct and useful tool to analyze the pest control problem, because it can be used to visualize real time spatial dynamics of the total group in addition to the final shelter selection probability. Also the parameters used in the present model is easier to control, because they are directly related to the physical properties of swarms.
4. Conclusions
Fig. 8. (a) Experimental selection probabilities for the light and the dark shelters (Halloy et al., 2007). (b) Simulation results. (Dark bars: cockroach-only-group, gray bars: mixed group (four robots and 12 cockroaches).)
In this paper, we present a particle-based-simulation study on the model of two-component swarms. When two components differ only in masses, it is shown that the heavier particles rotate outside while the lighter ones inside. This seemingly protective behavior of living flocks could emerge from a simply dynamic many-particle nature without any intelligent trait of each particle. When the self-propelling strengthes are allowed to differ for the two components, strongly propelling group lead the swarm motion in front. It is shown that there exists a linear relationship between the swarm velocity and the difference of the selfpropelling strengths Da as given in Eq. (5). When the strength difference reaches a critical value Dac , the swarm splits into two groups moving with different velocities. Thus, the picture emerging from the present analysis suggests that stronger members of a group not only lead the swarm, but also assist the weaker members to maintain the same speed through the body force, although there is no intelligent intention is involved. The case when two components of the swarm have different responses to shelters is also studied. As an application of the model, a control of cockroaches using robots is studied. It is found that the present formalism of the two-component swarm can describe the dynamics of the sheltering process of the mixed group of cockroaches and robots satisfactorily. It raises a possibility of interesting applications of the present model in the problems of pest control and fishing using robots and decoys. Although the present studies have been confined to swarms having only two kinds of components, the extension to multicomponent swarms is straightforward.
Acknowledgment This work was partially supported by the Korea Science and Engineering Foundation (R01-2006-000-10083-0). References Fig. 9. Fraction of population under the light shelter for the cockroach-only-group (a) and for the mixed group (b) as a function of time. They are averaged only with the light shelter selection processes. Dots are the experimental data from Halloy et al. (2007). The line represents averaged values from 200 simulations and time is rescaled to the experiment time scale.
dynamics of the cockroach–robot mixed group. We have simulated the dynamic shelter choosing processes of the cockroach-only-group and the mixed group (four robots and 12 cockroaches). The results are compared with the experimental results of Halloy et al. (2007) (Fig. 9). We note that the simulated
Aldana, M., Dossetti, V., Huepe, C., Kenkre, V.M., Larralde, H., 2007. Phase transitions in systems of self-propelled agents and related network models. Phys. Rev. Lett. 98, 095702(4). Ame , J.-M., Halloy, J., Rivault, C., Detrain, C., Deneubourg, J.-L., 2006. Collegial decision making based on social amplification leads to optimal group formation. Proc. Natl. Acad. Sci. USA 103, 5835–5840. Chen, H.-Y., Leung, K., 2006. Rotating states of self-propelling particles in two dimensions. Phys. Rev. E 73, 056107(9). Couzin, I.D., Krause, J., James, R., Ruxton, G.D., Franks, N.R., 2002. Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11. Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A., 2005. Leadership and decisionmaking in animal groups on the move. Nature 433, 513–516. Couzin, I.D., 2007. Collective minds. Nature 445, 715.
ARTICLE IN PRESS 500
S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
Couzin, I.D., 2009. Collective cognition in animal groups. Trends Cogntive Sci. 13, 36–43. D’Orsogna, M.R., Chuang, Y.L., Bertozzi, A.L., Chayes, L.S., 2006. Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302(4). Erdmann, U., Ebeling, W., Mikhailov, A.S., 2005. Noise-induced transition from translational to rotational motion of swarms. Phys. Rev. E 71, 051904(7). Gre goire, G., Chate, H., 2004. Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702(4). Halloy, J., Sempo, G., Caprari, G., Rivault, C., Asadpour, M., et al., 2007. Social integration of robots into groups of cockroaches to control self-organized choices. Science 318, 1155–1158. Krause, J., Ruxton, G.D., 2002. Living in Groups. Oxford University Press, USA. Levine, H., Rappel, W.-J., Cohen, T., 2000. Self-organization in systems of selfpropelled particles. Phys. Rev. E 63, 017101(4). Levine, H., Ben-Jacob, E., Cohen, I., Rappel, W.-J., 2006. Swarming patterns in microorganisms. In: 45th IEEE Conference on Decision and Control, 2006, pp. 5073–5077. Li, W., Wang, X., 2007. Adaptive velocity strategy for swarm aggregation. Phys. Rev. E 75, 021917(7). Mach, R., Schweitzer, F., 2007. Modeling vortex swarming in Daphnia. Bull. Math. Biol. 69, 539–562 and references therein.
Mailleux, A.-C., Furey, R., Saffre, F., Krafft, B., Deneubourg, J.-L., 2008. How nonnestmates affect the cohesion of swarming groups in social spiders. Insect. Soc. 55, 355–359. McInnes, C.R., 2007. Vortex formation in swarms of interacting particles. Phys. Rev. E 75, 032904(3). Nagey, M., Daruka, I., Vicsek, T., 2007. New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion. Physica A 373, 445–454. Schmickl, T., Thenius, R., Moeslinger, C., Radspieler, G., Kernbach, S., et al., 2009. Get in touch: cooperative decision making based on robot-to-robot collisions. Auton. Agent Multi-Agent Syst. 18, 133–155. Schweitzer, F., 2003. Brownian Agents and Active Particles. Springer, Berlin. Sumpter, D.J.T., 2006. The principles of collective animal behavior. Philos. Trans. R. Soc. B 29, 5–22. Toner, J., Tu, Y., 1995. Long-range order in a two-dimensional dynamical XY model: How birds fly together. Phys. Rev. Lett. 75, 4326–4329. Toner, J., Tu, Y., Ramaswamy, S., 2005. Hydrodynamics and phases of flocks. Ann. Phys. 318, 170–244. ´ A., Ben-Jacob, E., Cohen, I., Shochet, O., 1995. Novel type of phase Vicsek, T., Czirok, transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229.
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Collective behaviors of two-component swarms Sang Koo You a,, Dae Hyuk Kwon b, Yong-ik Park b, Sun Myong Kim b, Myung-Hoon Chung c, Chul Koo Kim d a
Natural Science Research Institute, Yonsei University, Seoul 120-749, Republic of Korea Department of Physics, Yonsei University, Wonju 220-710, Republic of Korea c College of Science and Technology, Hong-Ik University, Chochiwon 339-800, Republic of Korea d Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Republic of Korea b
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 July 2009 Received in revised form 21 August 2009 Accepted 21 August 2009 Available online 27 August 2009
We present a particle-based simulation study on two-component swarms where there exist two different types of groups in a swarm. Effects of different parameters between the two groups are studied systematically based on Langevin’s equation. It is shown that the mass difference can introduce a protective behavior for the lighter members of the swarm in a vortex state. When the self-propelling strength is allowed to differ between two groups, it is observed that the swarm becomes spatially segregated and finally separated into two components at a certain critical value. We also investigate effects of different preferences for shelters on their collective decision making. In particular, it is found that the probability of selecting a shelter from the other varies sigmoidally as a function of the number ratio. The model is shown to describe the dynamics of the shelter choosing process of the cockroach– robot mixed group satisfactorily. It raises the possibility that the present model can be applied to the problems of pest control and fishing using robots and decoys. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Collective motion Swarm dynamics Multi-component swarm Collective decision making
1. Introduction Flocking motion is a ubiquitous phenomenon in nature ranging from the colony organization of microorganisms and flocking of fish and birds, to swarming of small insects. Understanding active collective motion and swarm intelligence in these biological systems is one of the important questions in nonequilibrium statistical physics. Like equilibrium condensed matter systems, swarms exhibit distinct phases, such as crystalline ordered, vortex, and disordered states. The important connection between the nonequilibrium dynamics and the equilibrium phase transitions was first recognized by Vicsek et al. (1995). This question was further investigated using a continuum theory to reveal the deeper connection between seemingly different two cases (Toner and Tu, 1995). A recent review of the works in this line over the past decade is given by Toner et al. (2005). More recently, the question of the instability of the phases as a function of the number of swarm members has been addressed (D’Orsogna et al., 2006) and the detailed nature of the dynamic phase transition have been investigated in detail to reveal the delicate role of the noise (Gre goire and Chate, 2004; Aldana et al., 2007; Nagey et al., 2007). However, still important questions remain, such as the cause of transitions in flocks, the shapes and surface
Corresponding author. Tel.: + 82 2 2123 2611; fax: + 82 2 313 8892.
E-mail address: [email protected] (S.K. You). 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.08.025
effects of a finite flock, seemly intelligent responses to food and shelters, and the phases and dynamics of multi-component flocks (Toner et al., 2005). In nature, no individual member of a flock is identical and wide variations in size, mass, and speed are expected. In order to understand this problem, we introduce a simple two-component swarm and employ a particle based simulation approach. Although there exist various different approaches to study the motion of the swarm (Vicsek et al., 1995; Levine et al., 2000; Couzin et al., 2002; Krause and Ruxton, 2002; Schweitzer, 2003; Erdmann et al., 2005; Ame et al., 2006; Chen and Leung, 2006; Levine et al., 2006; Sumpter, 2006; Mach and Schweitzer, 2007; Li and Wang, 2007; McInnes, 2007; Halloy et al., 2007; Mailleux et al., 2008; Schmickl et al., 2009), common essential ingredients are mutual attractions, self-propelling forces, and responses to environmental attractions such as food and shelters. Here, we choose a particle-based simulation model, which employs simple forms of the essential ingredients (Chen and Leung, 2006).
2. A particle-based model for two-component swarms We consider a system consisting of N particles which is governed by the equation of motion, mi
d~ vi ¼ g~ F i þ~ xi; v i þ ai V^ i þ ~ dt
ð1Þ
ARTICLE IN PRESS S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
where mi and ~ v i are the mass and velocity of ith particle, respectively. The four terms in the right-hand side represent v i is various forces influencing particle’s motion. The first term g~ the drag force, arising from the frictional nature of the medium or from speed limitation of living creatures and manufactured robots. g is the drag coefficient. The second term ai V^ i accounts for the self-propelling force including additional tendency to align with its nearby particles. ai represents the propelling strength and ~ i which is defined as V^ i is an unit vector parallel to V ~i ¼ V
N X
~ v j expðRji =ra Þ:
3. Simulation results for two-component swarms
Here, Rji ¼ j~ rj ~ r i j is the distance between ith and jth particles. ra is a parameter controlling the range of alignment in which propelling direction is averaged with an exponential factor. It is known that if ra increases, the vortex state become gradually unstable and transforms into crystal state moving to the same direction all together (Chen and Leung, 2006). The third term includes all other forces except noise, including mutual interaction forces between member particles and external forces from environmental potentials such as food and shelters. Here, we consider only two kinds of forces for mutual interactions and individual preference, which we call body force ð~ F bi Þ and shelter force ð~ F si Þ, respectively. For the body forces, two types of expressions are usually used. One is the Morse potential type (D’Orsogna et al., 2006; Levine et al., 2000) and the other the Lennard–Jones potential type (Chen and Leung, 2006). We use the second type force as in Chen and Leung (2006). Shelter force can be represented in various ways. Here, we adopt the approach that assumes no individual particle intelligence in searching for shelters. Thus, we assume that the shelter force is inwardly attractive only inside of the shelters. Thus, the member particles of the swarm will be attracted to the shelter, only when it hits the shelter by chance. We assume the shelters to have disk shapes for simplicity. Then, the third term ~ F i is defined as ~ F bi þ ~ F si ; Fi ¼ ~ N X
R^ ji ½1 ðRji =rb ÞexpðRji =rb Þ;
ð3Þ
jai
~ F si ¼
In reality, swarms are not homogeneous. They may vary in mass, size, and speed. Also, each particle may have different selfpropelling strength, body force, and response to environment. Although we consider only two-component swarm problem in this paper, we note that the formalism given by Eqs. (1)–(4) can be applied to any multi-component swarm problems, if necessary. For a two-component flock, we assume that there are Nð1Þ type1 particles and Nð2Þ type-2 particles in the swarm. The type-1(2) particles are characterized by mass, mð1Þ ðmð2Þ Þ, the self-propelling strength, að1Þ ðað2Þ Þ.
ð2Þ
j
~ F bi ¼ b
495
8P < sia R^ ica expððRica Rs Þ=rs Þ
if Rica rRs ;
:
otherwise:
a
0
ð4Þ
Here, b and rb represent the interaction strength and the range between particles, respectively. R^ ji is an unit vector directed to the ith particle from the jth particle. We assume that the interaction forces are all equivalent and independent on their subgroups. sia is the ath shelter force strength. Rs and rs are the radius and the force range of shelters, respectively. The last term in Eq. (1) is the stochastic force which models the rapidly fluctuating environmental conditions. For simplicity, the noise force is assumed to have no temporal correlation and is taken from a uniform distribution over ½w; w both in x-, ydirections. It should be again noted that the present model is one of many similar models. Using this and other types of similar models, the collective motions of a single component swarm have been studied intensively (Toner and Tu, 1995; Chen and Leung, 2006; Erdmann et al., 2005; Schmickl et al., 2009). It was found that there exists basically three different dynamics phases in this type of model: crystal, disordered, and vortex, although different models may yield more diverse states (Toner et al., 2005; Gre goire and Chate, 2004).
We only consider the model of two-component swarms moving in two spatial dimensions. In the following subsections, we investigate three different problems of the present model which are characterized by parameters assuming two different masses, propelling strengths and shelter force strengths, respectively. For other parameters, g ¼ 1, b ¼ 1, and rb ¼ 1 are chosen for all three cases. 3.1. Two components with different masses As a simple example of the two-component swarms, we consider a system with N ¼ 100 particles of which Nð1Þ ¼ 50 particles have masses, mð1Þ ¼ 1, while remaining N ð2Þ ¼ 50 particles have masses, mð2Þ ¼ 0:5. Here, shelter forces are not considered. All other parameters are assumed to be the same for the two components. Since the mass does not influence the steady state speed, the crystal and the disordered state are not influenced by the mass difference. However, the vortex state is expected to be different, since the rotation can be influenced by the mass. In order to investigate the effect of the mass difference, we first investigated the vortex states of the two components separately and then studied the motion together. In order to realize the vortex states, we choose ai ¼ 1, ra ¼ 0:001, sia ¼ 0, and w ¼ 2 (Chen and Leung, 2006). In Fig. 1, we show a snapshot of a steady rotating mixed swarm, which clearly shows that most of the lighter members of the swarm are rotating inside of the heavier ones. Fig. 2 shows the density profiles for each component, both when the two components move together and separately. It is interesting to observe that the density profiles change drastically when the two components are mixed. The average radius of the vortex for the lighter component is shown to shrink, when they are mixed with the heavier component, whereas that of the heavier components becomes larger. It can be shown that this collective behavior of swarm originates from the body force. In Eq. (1), ~ v i and V^ i are approximately parallel to each other by the vortex symmetry and only ~ F bi has a role of centripetal force. All members of the swarm have the same averaged speed a=g independent of their masses. As a result, the ith particle moves around a circle with radius r under the centripetal force Fbi ¼ ðmi =rÞða=gÞ2 . Therefore, if a particle with mass mð1Þ is added to a steadily rotating vortex of mass mð2Þ , it is pushed outward from the vortex center by the force ððmð1Þ mð2Þ Þ=rÞða=gÞ2 . Therefore, heavier(lighter) mass particles are located outside(inside) as shown in Fig. 2. Real animal groups evolved to have social interactions that in turn show various dynamic behavioral patterns. The protective behavior of rotating fish flocks of hiding younger ones inside of adult members of the flock would be also an outgrowth of social evolution. The present simulation result shows seemingly protective behavior originated from the physical many-body effect with mass difference. Therefore, we conjecture that the evolution to protective behavior might be promoted and enhanced by the physical origin.
ARTICLE IN PRESS 496
S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5 −1.5
−1
−0.5
0
0.5
1
1.5
Fig. 1. A snapshot of a steady rotation of a mixed swarm with two different mass components. The lighter ones are represented by open circles while the heavier ones are represented by dark circles. They are moving in the directions of line segments from circles with speeds proportional to the lengths. Note that most of the lighter ones are rotating inside of the heavier ones. Two groups are characterized by N ð1Þ ¼ N ð2Þ ¼ 50, mð1Þ ¼ 1, mð2Þ ¼ 0:5 respectively. Other parameters used for simulations are g ¼ 1, ai ¼ 1, ra ¼ 0:001, b ¼ 1, rb ¼ 1, sia ¼ 0, and w ¼ 2.
−1.5 −1.5
−1
−0.5
0
0.5
1
1.5
Fig. 3. A snapshot of a steady crystal state of mixed swarm with two different propelling strengths. Weakly propelling ones are represented by open circles while strong ones are represented by filled circles. Note that the strongly propelling ones are moving in front of the weaker ones. Two groups are characterized by N ð1Þ ¼ N ð2Þ ¼ 50, að1Þ ¼ 1, að2Þ ¼ 0:5, respectively. Other parameters used for simulations are g ¼ 1, mð1Þ ¼ mð2Þ ¼ 1, ra ¼ 1, b ¼ 1, rb ¼ 1, sia ¼ 0, and w ¼ 2.
segregation between the two components and would such segregation eventually split the flock into fast and slow moving flocks? In order to answer such questions, we provide the different self-propelling strengthes instead of masses as in the previous model, namely, mð1Þ ¼ mð2Þ ¼ 1 and að1Þ 4að2Þ . The other parameters are the same as in the previous model except ra ¼ 1 so that the vortex state is absent. A snapshot in a crystal state is shown in Fig. 3, where að1Þ ¼ 1 and að2Þ ¼ 0:2 are used as an example. The strongly propelling group locates ahead of the weaker group and leads the other. The swarm velocity v can be obtained as follows. When we write the average body force for group 1(2) as Fbð1Þ ðFbð2Þ Þ, particles of groups 1 and 2 satisfy the relation gv það1Þ þ Fbð1Þ ¼ 0 and gv þ að2Þ þ Fbð2Þ ¼ 0, respectively, because there is no acceleration. They also satisfy another equation, Nð1Þ Fbð1Þ þ Nð2Þ Fbð2Þ ¼ 0, since the total body force in a swarm should be zero. From the above equations, we obtain the average swarm velocity v as v ¼ ðNð1Þ =ðN gÞÞ Da þ að2Þ =g; Fig. 2. The number density rðrÞ of each component in the vortex of Fig. 1. Here, r is the distance from a vortex center. rðrÞ is averaged over 10 000 times iterations. For comparison, densities of homogeneous swarms with two difference masses are also shown; dotted(dashed) line is for the lighter(heavier) swarm only.
3.2. Two components with different self-propelling strengths In a swarm existing in nature, we expect that younger members of the flock have smaller self-propelling strengths as well as lighter masses. In fact, a smaller self-propelling strength implies a smaller mean speed for those members of the flock. In this section, we study a two-component swarm model in which fast and slow members are mixed. Again, here we do not consider any environmental forces. In the crystal state, there is no mass dependence (Chen and Leung, 2006). Thus we focus only on the difference of self-propelling strengths. This question has been raised as a future problem by Toner et al. (2005). Here, we address the following questions: Would there be a large scale spatial
ð5Þ
where Da ¼ að1Þ að2Þ is the difference of self-propelling strengthes between the two groups. The linear relationship between v and Da is confirmed in Fig. 4. where N ð1Þ ¼ Nð2Þ ¼ 50, g ¼ 1, and að2Þ ¼ 0 are chosen for the simulation. Then, Eq. (5) yields v ¼ Da=2, which is well satisfied by the simulation data below the separate phase in Fig. 4. Thus, the results show that swarms will move together even when there are substantial differences in the propelling strengths between individual members, although fast moving members or leaders locate themselves in front. When the difference in the selfpropelling strengths is substantial, spatial segregation between the two components is expected. It is found that this separation takes place abruptly at a certain critical value Dac as shown in Fig. 4. Since the body forces are expressed as Fbð1Þ ¼ ðNð2Þ =NÞ Da;
Fbð2Þ ¼ ðN ð1Þ =NÞ Da;
ð6Þ
the body forces increase linearly to Da and reach maximum values at a certain spacial distribution of particles. Thus the maximum body ð1Þ ð2Þ ¼ bað1Þ and Fb;max ¼ bað2Þ , where að1Þ forces can be written as Fb;max
ARTICLE IN PRESS S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
497
5
5 1
1 0
0 2
2 −5 −5
0 0
5
5
−5 −5
0 10
5 1
1 0 Fig. 4. The velocity of a swarm as a function of propelling strength difference, Dað ¼ að1Þ að2Þ Þ, with the fixed að2Þ ¼ 0. The relation is linear up to the critical value Dac beyond which the separation between fast and slow groups occurs. Other parameters are the same as in Fig. 3. Inset: The linear relationship between Dac and b.
and a are constants to be determined from the spatial distribution just before the separation by Eq. (3). They also satisfy the relation, Nð1Þ að1Þ ¼ N ð2Þ að2Þ . As a result, the critical value Dac has a linear relationship with b as ð2Þ
Dac ¼ ðNað2Þ =Nð1Þ Þb:
0 2
−5 −5
5
2
0 50
5
−5 −5
0 154
5
Fig. 5. Snapshots of a selection process for the final shelter. The numbers under snapshots represent the time steps. Two big circles represent two different shelters 1 and 2. Shelter 2 is finally chosen in this process, although 13(7) particles marked by open(filled) circles prefer shelter 1(2).
ð7Þ
This linear relationship is confirmed in the inset of Fig. 4. After the separation, two groups move with velocities, að1Þ =g and að2Þ =g, independently. In this subsection, we have shown that a strong body force can maintain a swarm to move steadily as a whole, in spite of differences in propelling strengths of individual members. Also, it is shown in the simulation that the strong ones occupy leading positions in moving. 3.3. Two components with different responses to environment In this subsection, we investigate a two-component swarm which consists of two groups with different responses to environment. This type of model studies is applicable to various problems, such as collective decision making, collective intelligence, pest control through robots, and leader-follow problems in flocks (Couzin et al., 2005; Ame et al., 2006; Couzin, 2007; Halloy et al., 2007; Mailleux et al., 2008; Schmickl et al., 2009; Couzin, 2009). In the present case, the particular environment is the existence of two different types of shelters. We focus on the collective behavior of a swarm where elements are divided into two groups, one of which prefers the shelter 1 while the other shelter 2. The present model can be readily applied to the problem of mixed swarm of cockroaches and robots studied in Halloy et al. (2007) in that robots are artificially programmed to have the opposite preference for shelters against cockroaches. Our interest in the present problem is to understand the dynamics and conditions of selection processes of particular shelters. Detailed understanding of this problem may have huge implications on pest control and fishing techniques using robots and decoys. In the present model, 20 elements are programmed to be confined in a regular square with each side length of 10. The discshaped shelters 1 and 2 with radii of 0.8 are centered at (2,2) and ð2; 2Þ, respectively. Particles of 1 ri rN ð1Þ and N ð1Þ þ1 r ir N belong to groups 1 and 2, respectively. The self-propelling strength ai is assumed to be ai ¼ 1 outside of shelters and ai ¼ 0 inside. For the shelter force strength sia in Eq. (4), we assign four
ð1Þ ð2Þ ð2Þ parameter values, sð1Þ 1 , s2 , s1 , and s2 , where subscripts and superscripts correspond to indices of shelters and groups, respectively. The larger sðaa0 Þ means the larger preference. The shelter force range is set to be rs ¼ 0:1, and the noise parameter is chosen w ¼ 7 so that particles behave like real insects and can escape from the shelters by chance. m ¼ 0:5, b ¼ 0:5, and rb ¼ 0:3 are used. Other parameters are same as in the previous models. This choice of parameters implies that we adopt the assumption that a swarm has no individual intelligence of detecting a shelter or food. Only when they hit a shelter by chance, they are attracted to it. However, it will be shown that the overall swarm motion appears as if the swarm as an entity possesses intellectual capability to detect a shelter of their choice. Fig. 5 shows a simulation of selection process of the swarm as time step increases. Initially, 20 particles are located randomly in a regular square with a segment length 0.5 centered at ð1:5; 1:5Þ and with random velocities of x- and y-components ranged by ð1Þ ½0:5; 0:5. Group 1 has 13 particles with sð1Þ 1 ¼ 0:3 and s2 ¼ 0:2 ð2Þ ¼ 0:2 and s while group 2 has seven particles with sð2Þ 1 2 ¼ 0:3. Therefore, group 1(2) is more attracted to shelter 1(2) than to shelter 2(1). Once the simulation starts, particles move stochastically due to the noise force until they hit a shelter and are attracted inside. The probability to step out of a shelter is dependent on the following competing factors. One is the stochastic noise force which allows a particle to escape from the shelter. This force is opposed by the attraction measured by the parameter sðaa0 Þ. Here, we note that the body force also plays an important role. If the number of particles in one particular shelter becomes larger, the probability to step out is reduced due to the body force. Therefore, the selection process is a result of competition between individual and collective responses. Although it is more likely for the swarm to congregate in shelter 1 than in shelter 2, it is also possible for the swarm to choose the shelter 2 as can be seen in Fig. 5. By performing 200 simulations, we have obtained P1 , the probability for the swarm to completely congregate in shelter 1 as
ARTICLE IN PRESS 498
S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500 ð1Þ ð1Þ ð1Þ ð2Þ as e1 ¼ ðsð1Þ þsð2Þ þ sð2Þ 1 N 1 N Þ in shelter 1 and e2 ¼ ðs2 N 2 Nð2Þ Þ in shelter 2 using a dimensionless energy scale. Although the self-propelling dynamics of the present problem is of highly nonequilibrium, the final state is very simple and quasi-static because all members remain within a shelter. If we construct an ensemble only with the final states, it will be similar to two-state ensemble in equilibrium statistical mechanics. Furthermore, while particles feel the above potential energies from shelters, they are also influenced by the noise in Eq. (1) whose effect is similar to thermal noise to scatter particles. Therefore, we assume the probability of the shelter selection could be approximately determined from the Boltzmann-like factor, expðbea Þ as in equilibrium statistics. Here 1=b represents the random noise of the swarm movements, and later will be determined from data. P1 , the probability for the swarm to be all congregated in shelter 1 could be approximated by 1=ðexpðbðe1 e2 ÞÞ þ 1Þ. Then, finally we have
Fig. 6. P1 , the probability of selecting shelter 1, as a function of N ð1Þ under the ð2Þ ð1Þ ð2Þ obtained averaging 200 conditions, N ¼ N ð1Þ þ N ð2Þ ¼ 20, sð1Þ 1 ¼ s2 , and s2 ¼ s1 simulations. Solid lines are functions of data fit using Eq. (8) with b values in parentheses.
Fig. 7. P1 , the probability of selecting shelter 1, as a function of N ð1Þ under the conditions N ¼ N ð1Þ þ N ð2Þ ¼ 20 and asymmetric sðaa0 Þ ’s. Solid lines are functions of data fit using Eq. (8) with b values in parentheses.
a function of particle number Nð1Þ with fixed total number N ¼ 20. In Fig. 6, we show that P1 has symmetric distributions as expected ð2Þ when two groups have exactly opposite preferences; sð1Þ 1 ¼ s2 and ð2Þ ¼ s . Fig. 7 treats a more general case where two groups have sð1Þ 2 1 asymmetric preferences. Here, we note that when the attraction of shelter 1 for group 1 is much larger than that of shelter 2 for group 2, then the swarm can be lured to shelter 1 with relatively small numbers of robots or decoys. In principle, the probability can be analyzed using the master equation. However, this approach involves the lengthy computing procedure due to highly correlated motion between particles and uncertainties, when the total number is rather small. Alternatively, we adopt an intuitive approach to depict underlying physics as follows. The potential energy for a particle belonging to the group a in shelter a0 will be proportional to the parameter sðaa0 Þ in Eq. (4). Once the swarm is congregated in one of shelters, the center of mass does not move much. Thus, the total kinetic energy can be safely neglected as well as the internal interaction in the swarm dynamics. Finally, the total energies can be written
P1 ¼ 1=ðexp½bððDð1Þ þ Dð2Þ ÞNð1Þ Dð2Þ NÞþ 1Þ;
ð8Þ
ð1Þ ð2Þ where Dð1Þ sð1Þ and Dð2Þ sð2Þ are defined as the 1 s2 2 s1 differences of preferences. In Figs. 6 and 7, we show that the simulation data can be fitted by the formula, Eq. (8) using the listed values for b. In Fig. 7, b is the same for all three curves, since P1 has same fixed values at Nð1Þ ¼ 0. Although further rigorous studies are needed for the strict validity of the above derivation, we expect that Eq. (8) would provide a simple picture for the problems like collective decision making for swarms. In order to check the usefulness of the present two-component swarm formalism to the particular problem of pest control using robots, we compare our simulation results with the recent experimental results reported in Halloy et al. (2007). They carried out experimental study on collective selection of shelters by a mixed group of four socially integrated autonomous robots and 12 cockroaches. Autonomous robots were perceived by cockroaches as congeners and used as interactive decoys. Therefore, the present model of two-component swarm with different responses to shelters maps exactly to this problem. When there exist two types of shelters, light (shelter 1) and dark (shelter 2), it is known that cockroaches (particle group 2) prefer the dark shelter over the light one. However, the robots (particle group 1) can be programmed to favor either light or dark shelter depending on experimental design. It was found that by programming the robots to favor light shelter over dark one, the whole group can be lured to the light shelter with high probability. In order to apply the two-component swarm formalism to this problem, we use Eq. (8) to determine the relations between b, sð2Þ 1 (preference of the (preference of the dark shelter by light shelter by cockroach), sð2Þ 2 ð1Þ cockroach), sð1Þ 1 (preference of the light shelter by robot), and s2 (preference of the dark shelter by robot). From Eq. (8), we obtain ð2Þ bDð2Þ ¼ bðsð2Þ 2 s1 Þ ¼ 0:062 for the cockroach-only-group choosing the dark shelter with 73% probability, and bDð1Þ ¼ bðsð1Þ 1 sð1Þ 2 Þ ¼ 0:298 for the mixed group (four robots and 12 cockroaches) choosing the light shelter to become 61% (Fig. 8(a) and Halloy et ð1Þ al., 2007). We use fixed values of sð2Þ 1 ¼ s2 ¼ 0:2 without loss of generality, since only the preference differences appear in the calculation. Here, we note that the parameters cannot be determined uniquely due to lack of experimental data. Thus, we choose a set of parameters that produce the probabilities of shelter selections close enough to the experimental values (Fig. 8(b)). We note that this process corresponds to the interpolation of the probability curve, P1 using two points. In principle, one more experimental point will allow to fix the parameter values uniquely. Even with this uncertainty, ranges for allowed values are not large. Chosen parameters are b ¼ 1:05, ð2Þ sð1Þ 1 ¼ 0:484, and s2 ¼ 0:259. Since, we have all the parameters necessary for the simulations, we can describe any detailed
ARTICLE IN PRESS S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
499
results are in good agreement with the actual experiments, thus indicating that the chosen parameters are suitable to describe the detailed dynamics of the mixed group. In Halloy et al. (2007), the experiments were simulated using a different mathematical model. In their mathematical model, the shelter choosing probability was calculated from the individual rates for entering and quitting shelters. Although both methods give satisfactory description of the experiments, we believe that the present twocomponent swarm method provides a more direct and useful tool to analyze the pest control problem, because it can be used to visualize real time spatial dynamics of the total group in addition to the final shelter selection probability. Also the parameters used in the present model is easier to control, because they are directly related to the physical properties of swarms.
4. Conclusions
Fig. 8. (a) Experimental selection probabilities for the light and the dark shelters (Halloy et al., 2007). (b) Simulation results. (Dark bars: cockroach-only-group, gray bars: mixed group (four robots and 12 cockroaches).)
In this paper, we present a particle-based-simulation study on the model of two-component swarms. When two components differ only in masses, it is shown that the heavier particles rotate outside while the lighter ones inside. This seemingly protective behavior of living flocks could emerge from a simply dynamic many-particle nature without any intelligent trait of each particle. When the self-propelling strengthes are allowed to differ for the two components, strongly propelling group lead the swarm motion in front. It is shown that there exists a linear relationship between the swarm velocity and the difference of the selfpropelling strengths Da as given in Eq. (5). When the strength difference reaches a critical value Dac , the swarm splits into two groups moving with different velocities. Thus, the picture emerging from the present analysis suggests that stronger members of a group not only lead the swarm, but also assist the weaker members to maintain the same speed through the body force, although there is no intelligent intention is involved. The case when two components of the swarm have different responses to shelters is also studied. As an application of the model, a control of cockroaches using robots is studied. It is found that the present formalism of the two-component swarm can describe the dynamics of the sheltering process of the mixed group of cockroaches and robots satisfactorily. It raises a possibility of interesting applications of the present model in the problems of pest control and fishing using robots and decoys. Although the present studies have been confined to swarms having only two kinds of components, the extension to multicomponent swarms is straightforward.
Acknowledgment This work was partially supported by the Korea Science and Engineering Foundation (R01-2006-000-10083-0). References Fig. 9. Fraction of population under the light shelter for the cockroach-only-group (a) and for the mixed group (b) as a function of time. They are averaged only with the light shelter selection processes. Dots are the experimental data from Halloy et al. (2007). The line represents averaged values from 200 simulations and time is rescaled to the experiment time scale.
dynamics of the cockroach–robot mixed group. We have simulated the dynamic shelter choosing processes of the cockroach-only-group and the mixed group (four robots and 12 cockroaches). The results are compared with the experimental results of Halloy et al. (2007) (Fig. 9). We note that the simulated
Aldana, M., Dossetti, V., Huepe, C., Kenkre, V.M., Larralde, H., 2007. Phase transitions in systems of self-propelled agents and related network models. Phys. Rev. Lett. 98, 095702(4). Ame , J.-M., Halloy, J., Rivault, C., Detrain, C., Deneubourg, J.-L., 2006. Collegial decision making based on social amplification leads to optimal group formation. Proc. Natl. Acad. Sci. USA 103, 5835–5840. Chen, H.-Y., Leung, K., 2006. Rotating states of self-propelling particles in two dimensions. Phys. Rev. E 73, 056107(9). Couzin, I.D., Krause, J., James, R., Ruxton, G.D., Franks, N.R., 2002. Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11. Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A., 2005. Leadership and decisionmaking in animal groups on the move. Nature 433, 513–516. Couzin, I.D., 2007. Collective minds. Nature 445, 715.
ARTICLE IN PRESS 500
S.K. You et al. / Journal of Theoretical Biology 261 (2009) 494–500
Couzin, I.D., 2009. Collective cognition in animal groups. Trends Cogntive Sci. 13, 36–43. D’Orsogna, M.R., Chuang, Y.L., Bertozzi, A.L., Chayes, L.S., 2006. Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302(4). Erdmann, U., Ebeling, W., Mikhailov, A.S., 2005. Noise-induced transition from translational to rotational motion of swarms. Phys. Rev. E 71, 051904(7). Gre goire, G., Chate, H., 2004. Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702(4). Halloy, J., Sempo, G., Caprari, G., Rivault, C., Asadpour, M., et al., 2007. Social integration of robots into groups of cockroaches to control self-organized choices. Science 318, 1155–1158. Krause, J., Ruxton, G.D., 2002. Living in Groups. Oxford University Press, USA. Levine, H., Rappel, W.-J., Cohen, T., 2000. Self-organization in systems of selfpropelled particles. Phys. Rev. E 63, 017101(4). Levine, H., Ben-Jacob, E., Cohen, I., Rappel, W.-J., 2006. Swarming patterns in microorganisms. In: 45th IEEE Conference on Decision and Control, 2006, pp. 5073–5077. Li, W., Wang, X., 2007. Adaptive velocity strategy for swarm aggregation. Phys. Rev. E 75, 021917(7). Mach, R., Schweitzer, F., 2007. Modeling vortex swarming in Daphnia. Bull. Math. Biol. 69, 539–562 and references therein.
Mailleux, A.-C., Furey, R., Saffre, F., Krafft, B., Deneubourg, J.-L., 2008. How nonnestmates affect the cohesion of swarming groups in social spiders. Insect. Soc. 55, 355–359. McInnes, C.R., 2007. Vortex formation in swarms of interacting particles. Phys. Rev. E 75, 032904(3). Nagey, M., Daruka, I., Vicsek, T., 2007. New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion. Physica A 373, 445–454. Schmickl, T., Thenius, R., Moeslinger, C., Radspieler, G., Kernbach, S., et al., 2009. Get in touch: cooperative decision making based on robot-to-robot collisions. Auton. Agent Multi-Agent Syst. 18, 133–155. Schweitzer, F., 2003. Brownian Agents and Active Particles. Springer, Berlin. Sumpter, D.J.T., 2006. The principles of collective animal behavior. Philos. Trans. R. Soc. B 29, 5–22. Toner, J., Tu, Y., 1995. Long-range order in a two-dimensional dynamical XY model: How birds fly together. Phys. Rev. Lett. 75, 4326–4329. Toner, J., Tu, Y., Ramaswamy, S., 2005. Hydrodynamics and phases of flocks. Ann. Phys. 318, 170–244. ´ A., Ben-Jacob, E., Cohen, I., Shochet, O., 1995. Novel type of phase Vicsek, T., Czirok, transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229.