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Explicit flock solutions for Quasi-Morse potentials

Published online by Cambridge University Press:  15 April 2014

J. A. CARRILLO ,
Y. HUANG  and
S. MARTIN
J. A. CARRILLO
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected], [email protected], [email protected]
Y. HUANG
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected], [email protected], [email protected]
S. MARTIN
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected], [email protected], [email protected]
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Abstract

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We consider interacting particle systems and their mean-field limits, which are frequently used to model collective aggregation and are known to demonstrate a rich variety of pattern formations. The interaction is based on a pairwise potential combining short-range repulsion and long-range attraction. We study particular solutions, which are referred to as flocks in the second-order models, for the specific choice of the Quasi-Morse interaction potential. Our main result is a rigorous analysis of continuous, compactly supported flock profiles for the biologically relevant parameter regime. Existence and uniqueness are proven for three space dimensions, while existence is shown for the two-dimensional case. Furthermore, we numerically investigate additional Morse-like interactions to complete the understanding of this class of potentials.

Keywords

Integral equationsSwarming patternsNon-local problemsBessel functions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 5 , October 2014 , pp. 553 - 578
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence http://creativecommons.org/licenses/by/3.0/
Copyright
Copyright © Cambridge University Press 2014

References

[1]Albi, G., Balagué, D., Carrillo, J. A. & VonBrecht, J. (to appear) Stability analysis of flock and mill rings for 2nd order models in swarming. SIAM J. Appl. Math..Google Scholar
[2]Balagué, D., Carrillo, J. A., Laurent, T. & Raoul, G. (2013) Dimensionality of local minimizers of the interaction energy. Arch. Rat. Mech. Anal. 209 (3), 1055–1088.Google Scholar
[3]Balagué, D., Carrillo, J. A., Laurent, T. & Raoul, G. (2013) Nonlocal interactions by repulsive–attractive potentials: Radial ins/stability. Phys. D 260, 5–25.Google Scholar
[4]Bernoff, A. J. & Topaz, C. M. (2011) A primer of swarm equilibria. SIAM J. Appl. Dyn. Syst. 10 (1), 212–250.Google Scholar
[5]Bernoff, A. J. & Topaz, C. M. (2013) Nonlocal aggregation models: A primer of swarm equilibria. SIAM Rev. 55 (4), 709–747.Google Scholar
[6]Bertozzi, A. L., Carrillo, J. A. & Laurent, T. (2009) Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22 (3), 683–710.Google Scholar
[7]Bertozzi, A. L., von Brecht, J. H., Sun, H., Kolokolnikov, T. & Uminsky, D. (to appear) Ring patterns and their bifurcations in a nonlocal model of biological swarms. Comm. Math. Sci.Google Scholar
[8]Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G. & Bonabeau, E. (2003) Self-Organization in Biological Systems. Princeton Studies in Complexity, Princeton University Press, Princeton, NJ (reprint of the 2001 original).Google Scholar
[9]Carrillo, J. A., D'Orsogna, M. R. & Panferov, V. (2009) Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2 (2), 363–378.Google Scholar
[10]Carrillo, J. A., Huang, Y. & Martin, S. (2014) Nonlinear stability of flock solutions in second-order swarming models. Nonlinear Anal. Real World Appl. 17, 332–343.Google Scholar
[11]Carrillo, J. A., Klar, A., Martin, S. & Tiwari, S. (2010) Self-propelled interacting particle systems with roosting force. Math. Mod. Meth. Appl. Sci. 20, 1533–1552.Google Scholar
[12]Carrillo, J. A., Martin, S. & Panferov, V. (2013) A new interaction potential for swarming models. Phys. D 260, 112–126.Google Scholar
[13]Chuang, Y.-L., D'Orsogna, M. R., Marthaler, D., Bertozzi, A. L. & Chayes, L. S. (2007) State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Phys. D 232 (1), 33–47.Google Scholar
[14]Couzin, I. D. & Krause, J. (2003) Self-organization and collective behavior of vertebrates. Adv. Study Behav. 32, 1–67.Google Scholar
[15]D'Orsogna, M. R., Chuang, Y.-L., Bertozzi, A. L. & Chayes, L. S. (2006) Self-propelled particles with soft-core interactions: Patterns, stability, and collapse. Phys. Rev. Lett. 96 (10), 104302.Google Scholar
[16]Fetecau, R. C. & Huang, Y. (2013) Equilibria of biological aggregations with nonlocal repulsive–attractive interactions. Phys. D 260, 49–64.Google Scholar
[17]Fetecau, R. C., Huang, Y. & Kolokolnikov, T. (2011) Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24 (10), 2681–2716.Google Scholar
[18]Hildenbrandt, H., Carere, C. & Hemelrijk, C. K. (2010) Self-organised complex aerial displays of thousands of starlings: A model. Behav. Ecology 107 (21), 1349–1359.Google Scholar
[19]Holm, D. D. & Putkaradze, V. (2005) Aggregation of finite-size particles with variable mobility. Phys. Rev. Lett. 95, 226106.Google Scholar
[20]Holm, D. D. & Putkaradze, V. (2006) Formation of clumps and patches in self-aggregation of finite-size particles. Phys. D 220 (2), 183–196.Google Scholar
[21]Huth, A. & Wissel, C. (1992) The simulation of the movement of fish schools. J. Theor. Biol. 156, 365–385.CrossRefGoogle Scholar
[22]Kolokolnikov, T., Huang, Y. & Pavlovski, M. (2013) Singular patterns for an aggregation model with a confining potential. Phys. D 260, 65–76.Google Scholar
[23]Kolokonikov, T., Sun, H., Uminsky, D. & Bertozzi, A. (2011) Stability of ring patterns arising from 2d particle interactions. Phys. Rev. E 84 (1), 015203.Google Scholar
[24]Levine, H., Rappel, W.-J. & Cohen, I. (2000, December) Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 017101.Google Scholar
[25]Lieb, E. H. & Loss, M. (2001) Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI.Google Scholar
[26]Lukeman, R., Li, Y. X. & Edelstein-Keshet, L. (2010) Inferring individual rules from collective behavior. Proc. Natl. Acad. Sci. USA 107 (28), 12576–12580.CrossRefGoogle ScholarPubMed
[27]Magnus, W., Oberhettinger, F. & Soni, R. P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, NY.Google Scholar
[28]Mogilner, A. & Edelstein-Keshet, L. (1999) A non-local model for a swarm. J. Math. Biol. 38, 534–570.Google Scholar
[29]Topaz, C. M., Bernoff, A. J., Logan, S. & Toolson, W. (2008) A model for rolling swarms of locusts. Eur. Phys. J. Spec. Top. 157 (1), 93–109.Google Scholar
[30]von Brecht, J. H. & Uminsky, D. (2012) On soccer balls and linearized inverse statistical mechanics. J. Nonlinear Sci. 22 (6), 935–959.Google Scholar
[31]von Brecht, J. H., Uminsky, D., Kolokolnikov, T. & Bertozzi, A. L. (2012) Predicting pattern formation in particle interactions. Math. Models Methods Appl. Sci. 22, 1140002, 31.Google Scholar