Swarming patterns in a two-dimensional kinematic model for biological groups

SIAM Journal on Applied Mathematics

Volumn 65, Issue 1, 2005, Pages 152-174

  Topaz, Chad M ^a   Bertozzi, Andrea L ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Aggregation; Integrodifferential equation; Pattern; Swarming; Vortex

Indexed keywords

BIOLOGICAL AGGREGATIONS; PATTERN; SWARMING; VORTEX;

AGGLOMERATION; BIODIVERSITY; BIOLOGY; COMPUTER SIMULATION; DIFFERENTIAL EQUATIONS; FLUID DYNAMICS; MATHEMATICAL MODELS; POROUS MATERIALS;

KINEMATICS;

EID: 16244371027     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139903437424     Document Type: Article

References (33)

1. M. ALDANA AND C. HUEPE, Phase transitions in self-driven many-particle systems and related non-equilibrium models: A network approach, J. Statist. Phys., 112 (2003), pp. 135-153.
2. W. ALT AND G. HOFFMAN, EDS., Biological Motion: Proceedings of a Workshop Held in Königswinter, Germany, March 16-19, 1989, Lecture Notes in Biomath. 89, Springer-Verlag, Berlin, 1990.
3. E. BEN-JACOB, I. COHBN, A. CZIRÓK, T. VICSEK, AND D.L. GUTNICK, Chemomodulation of cellular movement, collective formation of vortices by swarming bacteria, and colonial development, Phys. A, 238 (1997), pp. 181-197.
4. A.L. BERTOZZI AND P. CONSTANTIN, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), pp. 19-28.
5. E. BONABEU, M. DORIGO, AND G. THERAULAZ, Swarm Intelligence: From Natural to Artificial Systems, Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press, New York, 1999.
6. J.Y. CHEMIN, Persistency of geometric structures in bidimensional incompressible fluids, Ann. Sci. Ećole Norm. Sup., 26 (1994), pp. 517-542.
7. M.C. CROSS AND P.C. HOHENBERG, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), pp. 851-1112.
8. I.D. COUZIN, J. KRAUSE, R. JAMES, G.D. RUXTON, AND N.R. FRANKS, Collective memory and spatial sorting in animal groups, J. Theoret. Biol., 218 (2002), pp. 1-11.
9. D. G. DRITSCHEL, Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modeling of vortex dynamics in two-dimensional, inviscid, incompressible flows, Comp. Phys. Rep., 10 (1989), pp. 77-146.
10. L. EDELSTEIN-KESHET, Mathematical models of swarming and social aggregation, in Proceedings of the 2001 International Symposium on Nonlinear Theory and Its Applications, Miyagi, Japan, 2001, pp. 1-7.
11. L. EDELSTEIN-KESHET, J. WATMOUGH, AND D. GRUNBAUM, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts, J. Math. Biol., 36 (1998), pp. 515-549.
12. U. ERDMANN AND W. EBELING, Collective motion of Brownian particles with hydrodynamic interactions, Fluct. Noise Lett., 3 (2003), pp. L145-L154.
13. U. ERDMANN, W. EBELING, AND V.S. ANISHCHENKO, Excitation of rotational modes in two-dimensional systems of driven Brownian particles, Phys. Rev. E, 65 (2002), paper 061106.
14. G. FLIERL, D. GRÜNBAUM, S. LEVIN, AND D. OLSON, From individuals to aggregations: The interplay between behavior and physics, J. Theoret. Biol., 196 (1999), pp. 397-454.
15. H. LEVINE, W.J. RAPPEL, AND I. COHEN, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2001), paper 017101.
16. A.J. MAJDA AND A.L. BERTOZZI, Vorticity and Incompressible Flow, Texts Appl. Math., Cambridge University Press, Cambridge, UK, 2002.
17. A. MOGILNER AND L. EDELSTEIN-KESHET, A non-local model for a swarm, J. Math. Biol., 38 (1999), pp. 534-570.
18. A. MOGILNER, L. EDELSTEIN-KESHET, L. BENT, AND A. SPIROS, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., 47 (2003), pp. 353-389.
19. J.D. MURRAY, Mathematical Biology I: An Introduction, 3rd ed., Interdiscip. Appl. Math. 17, Springer, New York, 2002.
20. A. OKUBO, Diffusion and Ecological Problems, Springer, New York, 1980.
21. A. OKUBO, D. GRUNBAUM, AND L. EDELSTEIN-KESHET, The dynamics of animal grouping, in Diffusion and Ecological Problems, 2nd ed., A. Okubo and S. Levin, eds., Interdiscip. Appl. Math. 14., Springer, New York, 1999, pp. 197-237.
22. A. ORDEMANN, F. MOSS, AND G. BALAZSI, Motions of Daphnia in a light field: Random walks with a Zooplankton, Nova Acta Leopoldina, 88 (2003), pp. 87-103.
23. J.K. PARRISH AND L. EDELSTEIN-KESHET, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 284 (1999), pp. 99-101.
24. J.K. PARRISH AND W.M. HAMNER, EDS., Animal Groups in Three Dimensions, Cambridge University Press, Cambridge, UK, 1997.
25. W.J. RAPPEL, A. NICOL, A. SARKISSIAN, AND H. LEVINE, Self-organized vortex state in two-dimensional Dictyostelium dynamics, Phys. Rev. Lett., 83 (1999), pp. 1247-1250.
26. T.C. SCHNEIRLA, Army Ants: A Study in Social Organization, W.H. Freeman, San Francisco, 1971.
27. F. SCHWEITZER, W. EBELING, AND B. TILCH, Statistical mechanics of canonical-dissipative systems and applications to swarm dynamics, Phys. Rev. E, 64 (2001), paper 021110.
28. J.C. STRIKWERDA, Finite Difference Schemes and Partial Differential Equations, Chapman & Hall, New York, 1989.
29. J. TONER AND Y. TU, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), pp. 4828-4858.
30. T. VICSEK, A. CZIRÓK, E. BEN-JACOB, I. COHEN, AND O. SHOCHET, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), pp. 1226-1229.
31. T. VICSEK, A. CZIRÓK, I. J. FARKAS, AND D. HELBING, Application of statistical mechanics to collective motion in biology, Phys. A, 274 (1999), pp. 182-189.
32. V. YUDOVICH, Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), pp. 1032-1066.
33. N.J. ZABUSKY, M.H. HUGHES, AND K.V. ROBERTS, Contour dynamics for the Euler equations in two dimensions, J. Comput. Phys., 30 (1979), pp. 96-106.