Special Issue on Mathematics and Complexity in Life and Human SciencesNo Access

SELF-PROPELLED INTERACTING PARTICLE SYSTEMS WITH ROOSTING FORCE

JOSÉ A. CARRILLO

ICREA — Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Search for more papers by this author

AXEL KLAR

Department of Mathematics, University of Kaiserslautern, P. O. Box 3049, 67653, Kaiserslautern, Germany

Fraunhofer ITWM Kaiserslautern, 67663 Kaiserslautern, Germany

Search for more papers by this author

STEPHAN MARTIN

Department of Mathematics, University of Kaiserslautern, P. O. Box 3049, 67653 Kaiserslautern, Germany

Search for more papers by this author

, and

SUDARSHAN TIWARI

Department of Mathematics, University of Kaiserslautern, P. O. Box 3049, 67653 Kaiserslautern, Germany

Search for more papers by this author

https://doi.org/10.1142/S0218202510004684Cited by:71 (Source: Crossref)

Abstract

We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.

Keywords:

AMSC: 92D50, 35B40, 82C22, 92C15

References

M. Balleriniet al., Proc. Natl Acad. Sci. USA 105, 1232 (2008), DOI: 10.1073/pnas.0711437105. Crossref, Web of Science, Google Scholar
A. Barbaroet al., Math. Comput. Simul. 79, 3397 (2009), DOI: 10.1016/j.matcom.2008.11.018. Crossref, Web of Science, Google Scholar
A. Barbaroet al., ICES J. Mar. Sci. 66, 826 (2009), DOI: 10.1093/icesjms/fsp067. Crossref, Web of Science, Google Scholar
A. Bensoussan , J.-L. Lions and G. Papanicolaou , Asymptotic Analysis for Periodic Structures ( North-Holland , 1978 ) . Google Scholar
F. Bolley, J. A. Canizo and J. A. Carrillo, Propagation of chaos for some non-globally Lipschitz particle systems, preprint UAB . Google Scholar
E. Bonabeau , M. Dorigo and G. Theraulaz , Swarm Intelligence: From Natural to Artificial Systems ( Oxford Univ. Press , 1999 ) . Crossref, Google Scholar
L. Bonillaet al., SIAM Appl. Math. 68, 648 (2007), DOI: 10.1137/070692728. Crossref, Web of Science, Google Scholar
W. Braun and K. Hepp, Commun. Math. Phys. 56, 101 (1977), DOI: 10.1007/BF01611497. Crossref, Web of Science, Google Scholar
M. Burger, V. Capasso and D. Morale, Nonlinear Anal. 8, 939 (2007). Crossref, Web of Science, Google Scholar
S. Camazine et al. , Self-Organization in Biological Systems ( Princeton Univ. Press , 2003 ) . Google Scholar
J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, preprint UAB . Google Scholar
J. A. Cañizo, J. A. Carrillo and J. Rosado, Collective behavior of animals: Swarming and complex patterns, to appear . Google Scholar
J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Kinetic Relat. Model. 2, 363 (2009), DOI: 10.3934/krm.2009.2.363. Crossref, Web of Science, Google Scholar
J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker–Smale model, to appear in SIAM J. Math. Anal . Google Scholar
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, preprint UAB . Google Scholar
Y. L. Chuanget al., Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE Int. Conf. on Robotics and Automation (IEEE, 2007) pp. 2292–2299. Google Scholar
Y. L. Chuanget al., Physica D 232, 33 (2007), DOI: 10.1016/j.physd.2007.05.007. Crossref, Web of Science, Google Scholar
I. D. Couzinet al., Nature 433, 513 (2005), DOI: 10.1038/nature03236. Crossref, Web of Science, Google Scholar
I. D. Couzinet al., J. Theor. Biol. 218, 1 (2002), DOI: 10.1006/jtbi.2002.3065. Crossref, Web of Science, Google Scholar
P. Degond and S. Mas-Gallic, Transp. Th. Statist. Phys. 16, 589 (1987), DOI: 10.1080/00411458708204307. Crossref, Google Scholar
P. Degond and S. Motsch, Math. Models Methods Appl. Sci. 18, 1193 (2008), DOI: 10.1142/S0218202508003005. Link, Web of Science, Google Scholar
R. Dobrushin, Funct. Anal. Appl. 13, 115 (1979), DOI: 10.1007/BF01077243. Crossref, Web of Science, Google Scholar
M. R. D'Orsognaet al., Phys. Rev. Lett. 96, 104302 (2006). Crossref, Web of Science, Google Scholar
T. Götzet al., SIAM Appl. Math. 67, 1704 (2007). Crossref, Web of Science, Google Scholar
G. Grégoire and H. Chaté, Phy. Rev. Lett. 92, 025702 (2004). Crossref, Web of Science, Google Scholar
S.-Y. Ha and J.-G. Liu, Comm. Math. Sci. 7, 297 (2009). Crossref, Web of Science, Google Scholar
S.-Y. Ha and E. Tadmor, Kinetic Relat. Models 1, 415 (2008). Crossref, Web of Science, Google Scholar
C. K. Hemelrijk and H. Kunz, Behav. Ecol. 16, 178 (2005), DOI: 10.1093/beheco/arh149. Crossref, Web of Science, Google Scholar
M. Hertyet al., Kinetic Relat. Models 2, 480 (2009). Google Scholar
H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organised complex aerial displays of thousands of starlings: A model , arXiv:0908.2677 . Google Scholar
A. Huth and C. Wissel, J. Theor. Biol. 152, 365 (1992). Google Scholar
A. Klar, P. Reuterswärd and M. Seaïd, J. Sci. Comput. 38, 349 (2009), DOI: 10.1007/s10915-008-9244-2. Crossref, Web of Science, Google Scholar
H. Kunz and C. K. Hemelrijk, Artificial Life 3, 237 (2003). Google Scholar
H. Levine, W. J. Rappel and I. Cohen, Phys. Rev. E 63, 017101 (2000), DOI: 10.1103/PhysRevE.63.017101. Crossref, Web of Science, Google Scholar
Y. X. Li, R. Lukeman and L. Edelstein-Keshet, Physica D 237, 699 (2008), DOI: 10.1016/j.physd.2007.10.009. Crossref, Web of Science, Google Scholar
Y. X. Li, R. Lukeman and L. Edelstein-Keshet, Bull. Math. Biol. 71, 352 (2008). Web of Science, Google Scholar
A. Mogilneret al., J. Math. Biol. 47, 353 (2003), DOI: 10.1007/s00285-003-0209-7. Crossref, Web of Science, Google Scholar
H. Neunzert, Trans. Fluid Dynam. 18, 663 (1977). Google Scholar
G. Papanicolaou , D. Stroock and S. Varadhan , Statistical Mechanics and Dynamical Systems , Duke University Mathematics Series III , ed. D. Ruelle ( Durham , 1978 ) . Google Scholar
J. Parrish and L. Edelstein-Keshet, Science 294, 99 (1999). Web of Science, Google Scholar
H. Risken , The Fokker–Planck Equation , Springer Series in Synergetics 18 ( Springer-Verlag , 1989 ) . Crossref, Google Scholar
A.-S. Sznitman, École d'Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math. 1464 (Springer, 1991) pp. 165–251. Crossref, Google Scholar
H. Spohn , Large Scale Dynamics of Interacting Particles ( Springer , 1991 ) . Crossref, Google Scholar
S. Tiwari and J. Kuhnert, J. Comp. Appl. Math. 203, 376 (2007), DOI: 10.1016/j.cam.2006.04.048. Crossref, Web of Science, Google Scholar
J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995), DOI: 10.1103/PhysRevLett.75.4326. Crossref, Web of Science, Google Scholar
C. M. Topaz and A. L. Bertozzi, SIAM J. Appl. Math. 65, 152 (2004), DOI: 10.1137/S0036139903437424. Crossref, Web of Science, Google Scholar
C. M. Topaz, A. L. Bertozzi and M. A. Lewis, Bull. Math. Biol. 68, 1601 (2006), DOI: 10.1007/s11538-006-9088-6. Crossref, Web of Science, Google Scholar
T. Vicseket al., Phys. Rev. Lett. 75, 1226 (1995), DOI: 10.1103/PhysRevLett.75.1226. Crossref, Web of Science, Google Scholar