×

Explicit flock solutions for quasi-Morse potentials. (English) Zbl 1304.82049

Summary: 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.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
92D25 Population dynamics (general)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Math. Models Methods Appl. Sci. 22 pp 31– (2012)
[2] DOI: 10.1016/j.physd.2006.07.010 · Zbl 1125.82021 · doi:10.1016/j.physd.2006.07.010
[3] DOI: 10.1103/PhysRevLett.95.226106 · doi:10.1103/PhysRevLett.95.226106
[4] Behav. Ecology 107 pp 1349– (2010)
[5] DOI: 10.1088/0951-7715/24/10/002 · Zbl 1288.92031 · doi:10.1088/0951-7715/24/10/002
[6] DOI: 10.1016/j.physd.2012.11.004 · Zbl 1286.35017 · doi:10.1016/j.physd.2012.11.004
[7] DOI: 10.1103/PhysRevLett.96.104302 · doi:10.1103/PhysRevLett.96.104302
[8] DOI: 10.3934/krm.2009.2.363 · Zbl 1195.92069 · doi:10.3934/krm.2009.2.363
[9] DOI: 10.1016/S0065-3454(03)01001-5 · doi:10.1016/S0065-3454(03)01001-5
[10] Self-Organization in Biological Systems (2003) · Zbl 1130.92009
[11] DOI: 10.1016/j.physd.2007.05.007 · Zbl 1375.82103 · doi:10.1016/j.physd.2007.05.007
[12] DOI: 10.1016/j.physd.2013.02.004 · doi:10.1016/j.physd.2013.02.004
[13] DOI: 10.1088/0951-7715/22/3/009 · Zbl 1194.35053 · doi:10.1088/0951-7715/22/3/009
[14] DOI: 10.1142/S0218202510004684 · Zbl 1197.92052 · doi:10.1142/S0218202510004684
[15] DOI: 10.1137/130925669 · Zbl 1282.35003 · doi:10.1137/130925669
[16] DOI: 10.1016/j.nonrwa.2013.12.008 · Zbl 1302.34082 · doi:10.1016/j.nonrwa.2013.12.008
[17] DOI: 10.1137/100804504 · Zbl 1255.35012 · doi:10.1137/100804504
[18] DOI: 10.1016/j.physd.2012.10.002 · Zbl 1286.35038 · doi:10.1016/j.physd.2012.10.002
[19] DOI: 10.1007/s00205-013-0644-6 · Zbl 1311.49053 · doi:10.1007/s00205-013-0644-6
[20] DOI: 10.1140/epjst/e2008-00633-y · doi:10.1140/epjst/e2008-00633-y
[21] DOI: 10.1007/s002850050158 · Zbl 0940.92032 · doi:10.1007/s002850050158
[22] Formulas and Theorems for the Special Functions of Mathematical Physics (1966)
[23] DOI: 10.1073/pnas.1001763107 · doi:10.1073/pnas.1001763107
[24] Analysis (2001)
[25] DOI: 10.1103/PhysRevE.63.017101 · doi:10.1103/PhysRevE.63.017101
[26] DOI: 10.1103/PhysRevE.84.015203 · doi:10.1103/PhysRevE.84.015203
[27] DOI: 10.1016/j.physd.2012.10.009 · Zbl 1286.35036 · doi:10.1016/j.physd.2012.10.009
[28] DOI: 10.1016/S0022-5193(05)80681-2 · doi:10.1016/S0022-5193(05)80681-2
[29] DOI: 10.1007/s00332-012-9132-7 · Zbl 1302.82032 · doi:10.1007/s00332-012-9132-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.