Flocking with informed agents
MathematicS In Action, Tome 1 (2008) no. 1, pp. 1-25.

Two similar Laplacian-based models for swarms with informed agents are proposed and analyzed analytically and numerically. In these models, each individual adjusts its velocity to match that of its neighbors and some individuals are given a preferred heading direction towards which they accelerate if there is no local velocity consensus. The convergence to a collective group swarming state with constant velocity is analytically proven for a range of parameters and initial conditions. Using numerical computations, the ability of a small group of informed individuals to accurately guide a swarm of uninformed agents is investigated. The results obtained in one of our two models are analogous to those found for more realistic and complex algorithms for describing biological swarms, namely, that the fraction of informed individuals required to guide the whole group is small, and that it becomes smaller for swarms with more individuals. This observation in our simple system provides insight into the possibly robust dynamics that contribute to biologically effective collective leadership and decision-making processes. In contrast with the more sophisticated models mentioned above, we can describe conditions under which convergence to consensus is ensured.

Publié le :
DOI : 10.5802/msia.1
Classification : 93C15
Mots clés : Particle systems, flocking
Felipe Cucker 1 ; Cristián Huepe 2

1 Department of Mathematics City University of Hong Kong HONG KONG
2 614 N Paulina Street, Chicago, IL 60622-6062 U.S.A.
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Felipe Cucker; Cristián Huepe. Flocking with informed agents. MathematicS In Action, Tome 1 (2008) no. 1, pp. 1-25. doi : 10.5802/msia.1. https://msia.centre-mersenne.org/articles/10.5802/msia.1/

[1] C.M. Breder. Equations descriptive of fish schools and other animal aggregations. Ecology, 35:361–370, 1954. | DOI

[2] J. Cortes, S. Martinez, and F. Bullo. Spatially-distributed coverage optimization and control with limited-range interactions. ESAIM Control Optim. Calc. Var., 11:691–719, 2005. | DOI | MR | Zbl

[3] I.D. Couzin, J. Krause, N.R. Franks, and S.A. Levin. Effective leadership and decision making in animal groups on the move. Nature, 433:513–516, 2005. | DOI

[4] I.D. Couzin, J. Krause, R. James, G.D. Ruxton, and N.R. Franks. Collective memory and spatial sorting in animal groups. Journal of Theoretical Biology, 218:1–11, 2002. | DOI | MR

[5] F. Cucker and S. Smale. Best choices for regularization parameters in learning theory. Found. Comput. Math., 2:413–428, 2002. | DOI | MR | Zbl

[6] F. Cucker and S. Smale. Emergent behavior in flocks. IEEE Trans. on Autom. Control, 52:852–862, 2007. | DOI | MR | Zbl

[7] F. Cucker and S. Smale. On the mathematics of emergence. Japan J. Math., 2:197–227, 2007. | DOI | MR | Zbl

[8] A. Czirok, H.E. Stanley, and T. Vicsek. Spontaneous ordered motion of self-propelled particles. J. Phys. A: Math. Gen., 30:1375–1385, 1997. | DOI

[9] A. Czirok and T. Vicsek. Collective behavior of interacting self-propelled particles. Physica A, 281:17–29, 2000. | DOI

[10] U. Erdmann, W. Ebeling, and A. S. Mikhailov. Noise-induced transition from translational to rotational motion of swarms. Phys. Rev. E, 71:051904, 2005. | DOI

[11] J.A. Fax and R.M. Murray. Information flow and cooperative control of vehicle formation. IEEE Trans. Aut. Contr., 49:1465–1476, 2004. | DOI | MR | Zbl

[12] G. Flierl, D. Grünbaum, S. Levin, and D. Olson. From individuals to aggregations: the interplay between behavior and physics. J. Theor. Biol., 196:397–454, 1999. | DOI

[13] V. Gazi and K.M. Passino. Stability analysis of swarms. IEEE Trans. Aut. Contr., 48:692–697, 2003. | DOI | MR | Zbl

[14] D. Grunbaum and A. Okubo. Modeling social animal aggregations, volume 100 of Lecture Notes in Biomathematics, pages 296–325. Springer-Verlag, 1994. | DOI | Zbl

[15] S.-Y. Ha and J.-G. Liu. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Preprint, 2008. | DOI | MR

[16] S.-Y. Ha and E. Tadmor. From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic and Related Models, 1:415–435, 2008. | DOI | MR | Zbl

[17] A. Jadbabaie, J. Lin, and A.S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. on Autom. Control, 48:988–1001, 2003. | DOI | MR | Zbl

[18] E.W. Justh and P.S. Krishnaprasad. Equilibria and steering laws for planar formations. Syst. and Contr. Lett., 52:25–38, 2004. | DOI | MR | Zbl

[19] A. Kolpas, J. Moehlis, and I.G. Kevrekidis. Coarse-grained analysis of stochasticity-induced switching between collective motion states. Proc. Nat. Acad. Sc., 104:5931–5935, 2007. | DOI

[20] P. Ogren, E. Fiorelli, and N. E. Leonard. Cooperative control of mobile sensor networks: adaptive gradient climbing in a distributed environment. IEEE Trans. Aut. Contr., 49:1292–1302, 2004. | DOI | MR | Zbl

[21] A. Okubo. Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds. Adv. Biophys, 22:1–94, 1986. | DOI

[22] L. Perea, P. Elosegui, and G. Gómez. Extension of the Cucker-Smale control law to space flight formations. Preprint, 2008. | DOI

[23] J. Shen. Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math, 68:694–719, 2007. | DOI | MR | Zbl

[24] I. Suzuki and M. Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal on Computing, 28:1347–1363, 1999. | DOI | MR | Zbl

[25] C.M. Topaz and A.L. Bertozzi. Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math., 65:152–174, 2004. | DOI | MR | Zbl

[26] T. Vicsek, A. Czirók, E. Ben-Jacob, and O. Shochet. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Letters, 75:1226–1229, 1995. | DOI | MR

[27] K. Warburton and J. Lazarus. Tendency-distance models of social cohesion in animal groups. J. Theoret. Biol., 150:473–488, 1991. | DOI

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