Abstract
This paper proposes an agent-based model which reproduces different structures of animal groups. The shape and structure of the group is the effect of simple interaction rules among individuals: each animal deploys itself depending on the position of a limited number of close group mates. The proposed model is shown to produce clustered formations, as well as lines and V-like formations. The key factors which trigger the onset of different patterns are argued to be the relative strength of attraction and repulsion forces and, most important, the anisotropy in their application.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aoki I (1980) An analysis of the schooling behavior of fish: internal organization and communication process. Bull Ocean Res Inst Univ Tokyo 12: 1–65
Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Lecomte V, Orlandi A, Parisi G, Procaccini A, Viale M, Zdravkovic V (2008a) Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc Natl Acad Sci 105: 1232–1237
Ballerini M, Cabibbo N, Candelier R, Cavagna A, Cisbani E, Giardina I, Orlandi A, Parisi G, Procaccini A, Viale M, Zdravkovic V (2008b) Empirical investigation of starling flocks: a benchmark study in collective animal behaviour. Anim Behav 76: 201–215
Bullo F, Cortés J, Martínez S (2009) Distributed control of robotic networks: a mathematical approach to motion coordination algorithms. In: Princeton series in applied mathematics. Princeton University Press, Princeton
Chao D, Levin SA (1999) A simulation of herding behavior: the emergence of large-scale phenomena from local interactions. In: Ruan S, Wolkowicz GSK, Wu J (eds) Differential equations with applications to biology, vol 21 of Fields Institute Communications, AMS Providence, RI, pp 81–95
Chuang Y, D’Orsogna MR, Marthaler D, Bertozzi AL, Chayes LS (2007) State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Physica D 232: 33–47
Couzin ID, Krause J, James R, Ruxton GD, Franks NR (2002) Collective memory and spatial sorting in animal groups. J Theor Biol 218: 1–11
Couzin ID, Krause J, Franks NR, Levin SA (2005) Effective leadership and decision-making in animal groups on the move. Nature 433: 513–516
Cucker F, Smale S (2007) Emergent behavior in flocks. IEEE Trans Automat Contr 52: 852–862
Cutts C, Speakman J (1994) Energy savings in formation flight of pink-footed geese. J Exp Biol 189: 251–261
Filippov AF (1988) Differential equations with discontinuous righthand sides, vol 18 of Mathematics and its applications. Kluwer, Dordrecht
Frasca P, Mason P, Piccoli B (2009) Detection of Gaussian signals via hexagonal sensor networks. Int J Math Model Numer Optim 1: 39–55
Giardina I (2008) Collective behavior in animal groups: theoretical models and empirical studies. HFSP J 2: 205–219
Gould LL, Heppner FH (1974) The vee formation of Canada geese. Auk 91: 494–506
Grégoire G, Chaté H, Tu Y (2003) Moving and staying together without a leader. Physica D 181: 157–170
Gueron S, Levin SA, Rubenstein DI (1996) The dynamics of herds: from individuals to aggregations. J Theor Biol 182: 85–98
Hainsworth FR (1987) Precision and dynamics of positioning by Canada geese flying in formation. J Exp Biol 128: 445–462
Hamilton WD (1971) Geometry for the selfish herd. J Theor Biol 31: 295–311
Hemelrijk CK, Hildenbrandt H (2008) Self-organized shape and frontal density of fish schools. Ethology 114: 245–254
Heppner FH (1974) Avian flight formations. Bird-Banding 45: 160–169
Huth A, Wissel C (1992) The simulation of the movement of fish schools. J Theor Biol 156: 365–385
Inada Y, Kawachi K (2002) Order and flexibility in the motion of fish schools. J Theor Biol 214: 371–387
Krause J, Ruxton GD (2002) Living in groups. In: Oxford series in ecology and evolution. Oxford University Press, New York
Kunz H, Hemelrijk CK (2003) Artificial fish schools: collective effects of school size, body size, and body form. Artif Life 9: 237–253
Li YX, Lukeman R, Edelstein-Keshet L (2008) Minimal mechanisms for school formations in self- propelled particles. Physica D 237: 699–720
Liberzon D (2003) Switching in systems and control. Birkhäuser, Boston
Lukeman R, Edelstein-Keshet L (2009) Personal communication
Lukeman R, Li YX, Edelstein-Keshet L (2009) A conceptual model for milling formations in biological aggregates. Bull Math Biol 71: 352–382
Mogilner A, Edelstein-Keshet L, Bent L, Spiros A (2003) Mutual interactions, potentials, and individual distance in a social aggregation. J Math Biol 47: 353–389
Nathan A, Barbosa VC (2008) V-like formations in flocks of artificial birds. Artif Life 14: 179–188
Parrish JK, Viscido SV, Grunbaum D (2002) Self-organized fish schools: an examination of emergent properties. Biol Bull 202: 296–305
Seiler P, Pant A, Hedrick JK (2003) A systems interpretation for observations of bird V-formations. J Theor Biol 221: 279–287
Seiler P, Pant A, Hedrick JK (2004) Disturbance propagation in vehicle strings. IEEE Trans Automat Contr 49: 1835–1842
Shi H, Wang L, Chu T (2006) Virtual leader approach to coordinated control of multiple mobile agents with asymmetric interactions. Physica D 213: 51–65
Speakman JR, Banks D (1998) The function of flight formations in Greylag Geese Anser anser; energy saving or orientation? Ibis 140: 280–287
Sumpter DJT (2006) The principles of collective animal behaviour. Phil Trans R Soc B 361: 5–22
Swaroop D, Hedrick JK (1996) String stability of interconnected systems. IEEE Trans Automat Contr 41: 349–357
Tanner HG, Jadbabaie A, Pappas GJ (2007) Flocking in fixed and switching networks. IEEE Trans Automat Contr 52: 863–868
Tien JH, Levin SA, Rubenstein DI (2004) Dynamics of fish shoals: identifying key decision rules. Evol Ecol Res 6: 555–565
Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O (1995) Novel type of phase transition in a system of self-driven particles. Phys Rev Lett 75: 1226–1229
Warburton K, Lazarus J (1991) Tendency-distance models of social cohesion in animal groups. J Theor Biol 150: 473–488
Weimerskirch H, Martin J, Clerquin Y, Alexandre P, Jiraskova S (2001) Energy saving in flight formation. Nature 413: 697–698
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cristiani, E., Frasca, P. & Piccoli, B. Effects of anisotropic interactions on the structure of animal groups. J. Math. Biol. 62, 569–588 (2011). https://doi.org/10.1007/s00285-010-0347-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-010-0347-7