Stability Analysis of Line Patterns of an Anisotropic Interaction Model
Abstract
Motivated by the formation of fingerprint patterns, we consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In addition, the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. Central to this pattern formation are straight line patterns. For a given spatially homogeneous tensor field, we show that there exists a preferred direction of straight lines, i.e., straight vertical lines can be stable for sufficiently many particles, while many other rotations of the straight lines are unstable steady states, both for a sufficiently large number of particles and in the continuum limit. For straight vertical lines we consider specific force coefficients for the stability analysis of steady states, show that stability can be achieved for exponentially decaying force coefficients for a sufficiently large number of particles, and relate these results to the Kücken--Champod model for simulating fingerprint patterns. The mathematical analysis of the steady states is completed with numerical results.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
G. Albi, D. Balagué, J. A. Carrillo, and J. von Brecht, Stability analysis of flock and mill rings for second order models in swarming, SIAM J. Appl. Math., 74 (2014), pp. 794--818.
2.
L. A. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Math., Birkhäuser, Basel, 2005.
3.
D. Balagué, J. A. Carrillo, T. Laurent, and G. Raoul, Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), pp. 1055--1088.
4.
D. Balagué, J. A. Carrillo, T. Laurent, and G. Raoul, Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), pp. 5--25.
5.
D. Balagué, J. A. Carrillo, and Y. Yao, Confinement for repulsive-attractive kernels, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 1227--1248.
6.
P. Ball, Nature's Patterns: A Tapestry in Three Parts, Oxford University Press, Oxford, 2009.
7.
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behaviour depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), pp. 1232--1237.
8.
A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 212--250.
9.
A. L. Bertozzi, J. A. Carrillo, and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), pp. 683--710.
10.
A. L. Bertozzi, T. Laurent, and F. Léger, Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005.
11.
A. L. Bertozzi, H. Sun, T. Kolokolnikov, D. Uminsky, and J. H. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, Commun. Math. Sci., 13 (2015), pp. 955--985.
12.
B. Birnir, An ODE model of the motion of pelagic fish, J. Stat. Phys., 128 (2007), pp. 535--568.
13.
A. Blanchet, V. Calvez, and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak--Keller--Segel model, SIAM J. Numer. Anal., 46 (2008), pp. 691--721.
14.
A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, 44 (2006), pp. 1--33.
15.
S. Boi, V. Capasso, and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens, Nonlinear Anal. Real World Appl., 1 (2000), pp. 163--176.
16.
M. Burger, V. Capasso, and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), pp. 939--958.
17.
M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich, and C.-B. Schönlieb, Pattern formation of a nonlocal, anisotropic interaction model, Math. Models Methods Appl. Sci., 28 (2018), pp. 409--451.
18.
S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz, and E. Bonabeau, Self-Organization in Biological Systems, Princeton University Press, Princeton, NJ, 2003.
19.
J. A. Can͂izo, J. A. Carrillo, and F. S. Patacchini, Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), pp. 1197--1217.
20.
J. A. Carrillo, M. G. Delgadino, and A. Mellet, Regularity of local minimizers of the interaction energy via obstacle problems, Comm. Math. Phys., 343 (2016), pp. 747--781.
21.
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), pp. 229--271.
22.
J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepčev, Confinement in nonlocal interaction equations, Nonlinear Anal., 75 (2012), pp. 550--558.
23.
J. A. Carrillo, L. C. F. Ferreira, and J. C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231 (2012), pp. 306--327.
24.
J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker--Smale model, SIAM J. Math. Anal., 42 (2010), pp. 218--236.
25.
J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, 2010, pp. 297--336.
26.
J. A. Carrillo and Y. Huang, Explicit equilibrium solutions for the aggregation equation with power-law potentials, Kinet. Relat. Models, 10 (2017), pp. 171--192.
27.
J. A. Carrillo, Y. Huang, and S. Martin, Explicit flock solutions for quasi-Morse potentials, European J. Appl. Math., 25 (2014), pp. 553--578.
28.
J. A. Carrillo, Y. Huang, and S. Martin, Nonlinear stability of flock solutions in second-order swarming models, Nonlinear Anal. Real World Appl., 17 (2014), pp. 332--343.
29.
J. A. Carrillo, F. James, F. Lagoutière, and N. Vauchelet, The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differential Equations, 260 (2016), pp. 304--338.
30.
J. A. Carrillo, R. J. McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19 (2003), pp. 971--1018.
31.
J. A. Carrillo, R. J. McCann, and C. Villani, Contractions in the $2$-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), pp. 217--263.
32.
A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini, and R. Tavarone, From empirical data to inter-individual interactions: Unveiling the rules of collective animal behavior, Math. Models Methods Appl. Sci., 20 (2010), pp. 1491--1510.
33.
P. Degond and S. Motsch, Large scale dynamics of the persistent turning Walker model of fish behavior, J. Stat. Phys., 131 (2008), pp. 989--1021.
34.
A. M. Delprato, A. Samadani, A. Kudrolli, and L. S. Tsimring, Swarming ring patterns in bacterial colonies exposed to ultraviolet radiation, Phys. Rev. Lett., 87 (2001), 158102.
35.
M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.
36.
B. Düring, C. Gottschlich, S. Huckemann, L. M. Kreusser, and C.-B. Schönlieb, An anisotropic interaction model for simulating fingerprints, J. Math. Biol., 78 (2019), pp. 2171--2206.
37.
L. Edelstein-Keshet, J. Watmough, and D. Grunbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts, J. Math. Biol., 36 (1998), pp. 515--549.
38.
K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), pp. 2267--2291.
39.
K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), pp. 1436--1450.
40.
R. C. Fetecau, Y. Huang, and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), pp. 2681--2716.
41.
M. Kücken and C. Champod, Merkel cells and the individuality of friction ridge skin, J. Theoret. Biol., 317 (2013), pp. 229--237.
42.
T. Kolokolnikov, J. A. Carrillo, A. Bertozzi, R. Fetecau, and M. Lewis, Emergent behaviour in multi-particle systems with non-local interactions [editorial], Phys. D, 260 (2013), pp. 1--4.
43.
T. Kolokolnikov, H. Sun, D. Uminsky, and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E (3), 84 (2011), 015203.
44.
H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., 172 (2004), pp. 407--428.
45.
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), pp. 534--570.
46.
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.
47.
A. Okubo and S. A. Levin, Diffusion and Ecological Problems, in Interdisciplinary Applied Mathematics: Mathematical Biology, Springer, New York, 2001, p. 197--237.
48.
J. K. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 284 (1999), pp. 99--101.
49.
I. Prigogine and I. Stenger, Order Out of Chaos, Bantam Books, New York, 1984.
50.
G. Raoul, Non-local interaction equations: Stationary states and stability analysis, Differential Integral Equations, 25 (2012), pp. 417--440.
51.
R. Simione, Properties of Minimizers of Nonlocal Interaction Energy, PhD thesis. Carnegie Mellon University, Pittsburgh, PA, 2014.
52.
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.
53.
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.
54.
C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., American Mathematical Society, Providence, RI, 2003.
55.
J. H. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, J. Nonlinear Sci., 22 (2012), pp. 935--959.
56.
J. H. von Brecht, D. Uminsky, T. Kolokolnikov, and A. L. Bertozzi, Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012), 1140002.
Information & Authors
Information
Published In
SIAM Journal on Applied Dynamical Systems
Pages: 1798 - 1845
DOI: 10.1137/18M1181638
ISSN (online): 1536-0040
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 18 June 2018
Accepted: 8 August 2019
Published online: 15 October 2019
Keywords
MSC codes
Authors
Funding Information
Alan Turing Institute https://doi.org/10.13039/100012338
Funding Information
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/P031587/1, EP/L016516/1, EP/M00483X/1, EP/N014588/1
Funding Information
Leverhulme Trust https://doi.org/10.13039/501100000275 : RPG-2015-69, EP/M00483X/1
Funding Information
Studienstiftung des Deutschen Volkes https://doi.org/10.13039/501100004350
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Pattern Formation for a Local/Nonlocal Interaction Functional Arising in Colloidal SystemsSIAM Journal on Mathematical Analysis, Vol. 52, No. 3 | 21 May 2020