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A Conceptual Model for Milling Formations in Biological Aggregates

  • Ryan Lukeman
  • Yue-Xian Li
  • Leah Edelstein-Keshet
Original Article

Abstract

Collective behavior of swarms and flocks has been studied from several perspectives, including continuous (Eulerian) and individual-based (Lagrangian) models. Here, we use the latter approach to examine a minimal model for the formation and maintenance of group structure, with specific emphasis on a simple milling pattern in which particles follow one another around a closed circular path.

We explore how rules and interactions at the level of the individuals lead to this pattern at the level of the group. In contrast to many studies based on simulation results, our model is sufficiently simple that we can obtain analytical predictions. We consider a Newtonian framework with distance-dependent pairwise interaction-force. Unlike some other studies, our mill formations do not depend on domain boundaries, nor on centrally attracting force-fields or rotor chemotaxis.

By focusing on a simple geometry and simple distant-dependent interactions, we characterize mill formations and derive existence conditions in terms of model parameters. An eigenvalue equation specifies stability regions based on properties of the interaction function. We explore this equation numerically, and validate the stability conclusions via simulation, showing distinct behavior inside, outside, and on the boundary of stability regions. Moving mill formations are then investigated, showing the effect of individual autonomous self-propulsion on group-level motion. The simplified framework allows us to clearly relate individual properties (via model parameters) to group-level structure. These relationships provide insight into the more complicated milling formations observed in nature, and suggest design properties of artificial schools where such rotational motion is desired.

Keywords

Milling Self-propelled particles Collective motion Schooling behavior Self-organization 

References

  1. Ben-Jacob, E., et al., 1997. Chemomodulation of cellular movement, collective formation of vortices by swarming bacteria, and colonial development. Physica A 238, 181–197. CrossRefGoogle Scholar
  2. Camazine, S., et al., 2001. Self-Organization in Biological Systems, Princeton University Press, Princeton. Google Scholar
  3. Chuang, Y.L., et al., 2007. State transitions and the continuum limit for a 2d interacting, self-propelled particle system. Physica D 232, 33–47. zbMATHCrossRefMathSciNetGoogle Scholar
  4. Couzin, I.D., et al., 2002. Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11. CrossRefMathSciNetGoogle Scholar
  5. Csahók, Z., Czirók, A., 2008. Hydrodynamics of bacterial motion. Physica A 243, 304–318. CrossRefGoogle Scholar
  6. Czirók, A., et al., 1996. Formation of complex bacterial colonies via self-generated vortices. Phys. Rev. E 54(2), 1792–1801. CrossRefGoogle Scholar
  7. Davis, P.J., 1979. Circulant Matrices, Wiley, New York. zbMATHGoogle Scholar
  8. D’Orsogna, M.R., et al., 2006. Self-propelled particles with soft-core interactions: Patterns, stability, and collapse. Phys. Rev. Lett. 96, 104302. CrossRefGoogle Scholar
  9. Grossman, D., Aranson, I.S., Ben-Jacob, E., 2008. Emergence of agent swarm migration and vortex formation through inelastic collisions. New J. Phys. 10, 023036. CrossRefGoogle Scholar
  10. Harvey-Clark, C.J., et al., 1999. Putative mating behavior in basking sharks off the Nova Scotia coast. Copeia 3, 780–782. CrossRefGoogle Scholar
  11. Hemmingsson, J., 1995. Modellization of self-propelling particles with a coupled map lattice model. J. Phys. A 28, 4245–4250. CrossRefGoogle Scholar
  12. Huth, A., Wissel, C., 1992. The simulation of movement of fish schools. J. Theor. Biol. 156, 365–385. CrossRefGoogle Scholar
  13. Kulinskii, V.L., et al., 2005. Hydrodynamic model for a system of self-propelling particles with conservative kinematic constraints. Europhys. Lett. 71(2), 207–213. CrossRefMathSciNetGoogle Scholar
  14. Levine, H., Rappel, W., Cohen, I., 2001. Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101. CrossRefGoogle Scholar
  15. Li, Y.-X., Lukeman, R., Edelstein-Keshet, L., 2008. Minimal mechanisms for school formation in self-propelled particles. Physica D 237(5), 699–720. zbMATHCrossRefMathSciNetGoogle Scholar
  16. Mach, R., Schweitzer, F., 2007. Modeling vortex swarming in daphnia. Bull. Math. Biol. 69(2), 539–562. zbMATHCrossRefMathSciNetGoogle Scholar
  17. Marshall, J.A., Broucke, M.E., Francis, B.A., 2004. Formations of vehicles in cyclic pursuit. IEEE Trans. Automat. Contr. 49(11), 1963–1974. CrossRefMathSciNetGoogle Scholar
  18. Mogilner, A., et al., 2003. Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47, 353–389. zbMATHCrossRefMathSciNetGoogle Scholar
  19. Niwa, H.-S., 1994. Self-organizing dynamic model of fish schooling. J. Theor. Biol. 171, 123–136. CrossRefGoogle Scholar
  20. Niwa, H.-S., 1996. Newtonian dynamical approach to fish schooling. J. Theor. Biol. 181, 47–63. CrossRefGoogle Scholar
  21. Niwa, H.-S., 1998. Migration of fish schools in heterothermal environments. J. Theor. Biol. 193, 215–231. CrossRefGoogle Scholar
  22. Okubo, A., 1980. Diffusion and Ecological Problems: Mathematical Models, Springer, New York. zbMATHGoogle Scholar
  23. Okubo, A., Grunbaum, D., Edelstein-Keshet, L., 2001. The dynamics of animal grouping. In: Okubo, A., Levin, S. (Eds.), Diffusion and Ecological Problems: Modern Perspectives. Springer, New York Google Scholar
  24. Parr, A.E., 1927. A contribution to the theoretical analysis of the schooling behaviour of fishes. Occ. Pap. Bingham Oceanogr. Collect. 1, 1–32. Google Scholar
  25. Parrish, J., Edelstein-Keshet, L., 1999. Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science 284, 99–101. CrossRefGoogle Scholar
  26. Parrish, J., Viscido, S., Grunbaum, D., 2002. Self-organized fish schools: an examination of emergent properties. Biol. Bull. 202, 296–305. CrossRefGoogle Scholar
  27. Ratushnaya, V.I., et al., 2007. Stability properties of the collective stationary motion of self-propelling particles with conservative kinematic constraints. J. Phys. A 40, 2573–2581. zbMATHMathSciNetGoogle Scholar
  28. Sakai, S., 1973. A model for group structure and its behavior. Biophys. Jpn. 13, 82–90. Google Scholar
  29. Schneirla, T.C., 1944. A unique case of circular milling in ants, considered in relation to trail following and the general problem of orientation. Am. Mus. Novit. 1253, 1–25. Google Scholar
  30. Silvester, J.R., 2000. Determinants of block matrices. Math. Gaz. 84, 460–467. CrossRefGoogle Scholar
  31. Suzuki, R., Sakai, S., 1973. Movement of a group of animals. Biophys. Jpn. 13, 281–282. CrossRefGoogle Scholar
  32. Topaz, C., Bertozzi, A., 2004. Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65(1), 152–174. zbMATHCrossRefMathSciNetGoogle Scholar
  33. Vicsek, T., et al., 1995. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229. CrossRefGoogle Scholar
  34. Weihs, D., 1974. Energetic advantages of burst swimming of fish. J. Theor. Biol. 48, 215–229. CrossRefGoogle Scholar
  35. Wilson, S.G., 2004. Basking sharks schooling in the southern Gulf of Maine. Fish. Oceanogr. 13, 283–286. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2008

Authors and Affiliations

  • Ryan Lukeman
    • 1
  • Yue-Xian Li
    • 1
  • Leah Edelstein-Keshet
    • 1
  1. 1.Department of Mathematics, and Institute of Applied MathematicsUniversity of British ColumbiaVancouverCanada

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