Swarm dynamics and equilibria for a nonlocal aggregation model

Nonlinearity

Volumn 24, Issue 10, 2011, Pages 2681-2716

  Fetecau, R C ^a   Huang, Y ^a   Kolokolnikov, T ^b  

^a SIMON FRASER UNIVERSITY   (Canada)

^b DALHOUSIE UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 80053259548     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/24/10/002     Document Type: Article

References (32)

1. Mogilner A and Edelstein-Keshet L 1999 A non-local model for a swarm J. Math. Biol. 38 534-70
2. Topaz C M, Bertozzi A L and Lewis M A 2006 A nonlocal continuum model for biological aggregation Bull. Math. Bio. 68 1601-23 (Pubitemid 44392962)
3. Holm D D and Putkaradze V 2005 Aggregation of finite-size particles with variable mobility Phys. Rev. Lett. 95 226106
4. Holm D D and Putkaradze V 2006 Formation of clumps and patches in selfaggregation of finite-size particles Physica D 220 183-96 (Pubitemid 44216019)
5. Benedetto D, Caglioti E and Pulvirenti M 1997 A kinetic equation for granular media RAIRO Mod́l Math. Anal. Nuḿr. 31 615-41 (Pubitemid 127374202)
6. Toscani G 2000 One-dimensional kinetic models of granular flows M2AN Math. Model Numer. Anal. 34 1277-91 (Pubitemid 33220541)
7. Haile J 1992 Molecular Dynamics Simulation: Elementary Methods (New York: Wiley)
8. Bernoff A J and Topaz C M 2011 A primer of swarm equilibria SIAM J. Appl. Dyn. Syst. 10 212-50
9. Bodnar M and Velazquez J J L 2006 An integro-differential equation arising as a limit of individual cell-based models J. Diff. Eqns 222 341-80 (Pubitemid 43246216)
10. Burger M and Di Francesco M 2008 Large time behavior of nonlocal aggregation models with nonlinear diffusion Netw. Heterog. Media 3 749-85
11. Laurent T 2007 Local and global existence for an aggregation equation Commun. Partial Diff. Eqns 32 1941-64
12. n Commun. Math. Phys. 274 717-35
13. p theory for the multidimensional aggregation equation Commun. Pure Appl. Math. 64 45-83
14. Leverentz A J, Topaz C M and Bernoff A J 2009 Asymptotic dynamics of attractive-repulsive swarms SIAM J. Appl. Dyn. Syst. 8 880-908
15. Fellner K and Raoul G 2010 Stable stationary states of non-local interaction equations Math. Models and Methods Appl. Sci. (M3AS) 20 2267-91
16. Bertozzi A L, Carrillo J A and Laurent T 2009 Blow-up in multidimensional aggregation equations with mildly singular interaction kernels Nonlinearity 22 683-710
17. Huang Y and Bertozzi A L 2010 Self-similar blowup solutions to an aggregation equation in SIAM J. Appl. Math. 70 2582-603
18. Topaz C M and Bertozzi A L 2004 Swarming patterns in a two-dimensional kinematic model for biological groups SIAM J. Appl. Math. 65 152-74
19. Bodnar M and Velasquez J J L 2005 Derivation of macroscopic equations for individual cell-based models: a formal approach Math. Methods Appl. Sci. 28 1757-79 (Pubitemid 41360038)
20. D'Orsogna M R, Chuang Y-L, Bertozzi A L and Chayes L S 2006 Self-propelled particles with soft-core interactions: patterns, stability and collapse Phys. Rev. Lett. 96 104302
21. Du Q and Zhang P 2003 Existence of weak solutions to some vortex density models SIAM J. Math. Anal. 34 1279-99 (electronic) (Pubitemid 37416689)
22. Bertozzi A L, Garnett T and Laurent J B 2011 Characterization of radially symmetric finite time blowup in multidimensional aggregation equations SIAM J. Math. Anal. submitted
23. Velzquez J J L 2004 Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions SIAM J. Appl. Math. 64 1198-223
24. Majda A J and Bertozzi A L 2002 Vorticity and incompressible flow (Cambridge Texts in Applied Mathematics vol 27) (Cambridge: Cambridge University Press)
25. Beale J T, Kato T and Majda A 1984 Remarks on the breakdown of smooth solutions for the 3-D Euler equations Commun. Math. Phys. 94 61-6
26. Du Y 2006 Order structure and topological methods in nonlinear partial differential equations (Series in Partial Differential Equations and Applications vols 1 and 2) (Hackensack, NJ: World Scientific)
27. Eastman S and Estep D 2007 A power method for nonlinear operators Appl. Anal. 86 1303-14
28. Kolokolnikov T, Sun H, Uminsky D and Bertozzi A L 2011 Stability of ring patterns arising from two-dimensional particle interactions Phys. Rev. E 84 015203(R)
29. Horn R A and Johnson C R 1990 Matrix Analysis (Cambridge: Cambridge University Press) corrected reprint of the 1985 original
30. Strogatz S H 2000 From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators Physica D 143 1-20
31. Horstmann D 2003 From 1970 until present: the Keller-Segel model in chemotaxis and its consequences: I Jahresber. Deutsch. Math.-Verein. 105 103-65
32. Lukeman Y L R and Edelstein-Keshet L 2010 Inferring individual rules from collective behavior Proc. Natl Acad. Sci. USA 107 12576-580