Asymptotic dynamics of attractive-repulsive swarms*

SIAM Journal on Applied Dynamical Systems

Volumn 8, Issue 3, 2009, Pages 890-908

  Leverentz, Andrew J ^a   Topaz, Chad M ^b   Bernoff, Andrew J ^a  

^a HARVEY MUDD COLLEGE   (United States)

^b MACALESTER COLLEGE   (United States)

Author keywords

Aggregation; Asymptotic dynamics; Attractive repulsive; Blow up; Burgers' equation; Integrodifferential equation; Porous medium; Swarm

Indexed keywords

AGGREGATION; ASYMPTOTIC DYNAMICS; ATTRACTIVE-REPULSIVE; BLOW-UP; BURGERS' EQUATION; POROUS MEDIUM; SWARM;

ASYMPTOTIC ANALYSIS; INTEGRODIFFERENTIAL EQUATIONS; POROUS MATERIALS;

DYNAMICS;

EID: 68649097592     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/090749037     Document Type: Article

References (29)

1. D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Math. 1224, Springer, Berlin, 1986, pp. 1-46.
2. A. J. Bernoff and C. M. Topaz, Equilibrium Configurations of Interacting Particles in One Dimension, manuscript, 2009.
3. A. L. Bertozzi and T. Laurent, Finite-time blow-up of solutions of an aggregation equation in Rn, Comm. Math. Phys., 274 (2007), pp. 717-735.
4. M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: A formal approach, Math. Methods Appl. Sci., 28 (2005), pp. 1757-1779.
5. M. Bodnar and J. J. L. Velazquez, An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), pp. 341-380.
6. 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.
7. J. A. Carrillo and G. Toscani, Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), pp. 113-142.
8. Y. L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi, and L. S. Chayes, State transitions and the continuum limit for a 2d interacting, self-propelled particle system, Phys. D, 232 (2007), pp. 33-47.
9. 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.
10. M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), pp. 851-1112.
11. M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), paper 104302.
12. R. Eftimie, G. de Vries, and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 6974-6979.
13. R. Eftimie, G. de Vries, M. A. Lewis, and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), pp. 1537-1565.
14. S. Gueron, S. A. Levin, and D. I. Rubenstein, The dynamics of herds: From individuals to aggregations, J. Theoret. Biol., 182 (1996), pp. 85-98.
15. A. J. Leverentz, An Integrodifferential Equation Modeling 1-D Swarming Behavior, Senior thesis, Department of Mathematics, Harvey Mudd College, Claremont, CA, 2008.
16. H. Levine, W. J. Rappel, and I. Cohen, Self-organization in systems of self-propelled particles, Phys. Rev. E, 63 (2001), paper 017101.
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. A. Okubo, D. Grünbaum, and L. Edelstein-Keshet, The dynamics of animal grouping, in Diffusion and Ecological Problems, Interdiscip. Appl. Math. 14, 2nd ed., A. Okubo and S. A. Levin, eds., Springer, New York, 2001, Ch. 7, pp. 197-237.
20. J. K. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 284 (1999), pp. 99-101.
21. S. R. Partan and P. Marler, Issues in the classification of multimodal communication signals, Amer. Naturalist, 166 (2005), pp. 231-245.
22. K. M. Passino, Biomimicry for Optimization, Control, and Automation, Springer, London, 2005.
23. D. Tilman and P. Kareiva, eds., Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University Press, Princeton, NJ, 1998.
24. C. M. Topaz, A. J. Bernoff, S. Logan, and W. Toolson, A model for rolling swarms of locusts, Eur. Phys. J. Special Topics, 157 (2008), pp. 93-109.
25. C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), pp. 152-174.
26. C. M. Topaz, A. L. Bertozzi, and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), pp. 1601-1623.
27. G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
28. T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), pp. 153-193.
29. Y. B. Zeldovich and G. I. Barenblatt, Asymptotic properties of self-preserving solutions of equations of unsteady motion of gas through porous media, Dokl. Akad. Nauk SSSR, 118 (1958), pp. 671-674.