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Two-species particle aggregation and stability of co-dimension one solutions
1. | University of California, Los Angeles, Department of Mathematics, Box 951555, Los Angeles, CA 90095-1555, United States |
2. | Dalhousie University, Department of Mathematics and Statistics, Halifax, Nova Scotia, B3H 3J5 |
3. | University of California Los Angeles, Department of Mathematics, 520 Portola Plaza Box 951555, Los Angeles, CA 90095-1555 |
References:
[1] |
E. Altschuler, T. Williams, E. Ratner, R. Tipton, R. Stong, F. Dowla and F. Wooten, Possible global minimum lattice configurations for Thomson's problem of charges on a sphere, Physical Review Letters, 78 (1997), 2681-2685.
doi: 10.1103/PhysRevLett.78.2681. |
[2] |
D. Balague, J.A. Carrillo, T. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Archive for Rational Mechanics and Analysis, 209 (2013), 1055-1088.
doi: 10.1007/s00205-013-0644-6. |
[3] |
A. Bernoff and C. Topaz, A primer of swarm equilibria, SIAM Journal on Applied Dynamical Systems, 10 (2011), 212-250.
doi: 10.1137/100804504. |
[4] |
A. L. Bertozzi, H. Sun, T. Kolokolnikov, D. Uminsky and J. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, preprint, 2013. Available from: http://www.math.ucla.edu/~bertozzi/papers/bigring-final.pdf. |
[5] |
H. Cabral and D. Schmidt, Stability of relative equilibria in the problem of $n+1$ vortices, SIAM Journal on Mathematical Analysis, 31 (1999), 231-250.
doi: 10.1137/S0036141098302124. |
[6] |
Y. Chen, T. Kolokolnikov and D. Zhirov, Collective behaviour of large number of vortices in the plane, Proceedings of the Royal Society A, 469 (2013), 20130085.
doi: 10.1098/rspa.2013.0085. |
[7] |
H. Cohn and A. Kumar, Algorithmic design of self-assembling structures, PNAS, 106 (2009), 9570-9575.
doi: 10.1073/pnas.0901636106. |
[8] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, European Journal of Applied Mathematics, 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[9] |
I. Couzin, J. Krause, N. Franks and S. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
[10] |
G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications, 20 (2013), 523-537.
doi: 10.1007/s00030-012-0164-3. |
[11] |
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, Physical Review Letters, 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[12] |
B. Düring, P. Markowich, J. F. Pietschmann and M. T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society A, 465 (2009), 3687-3708.
doi: 10.1098/rspa.2009.0239. |
[13] |
C. Escudero, F. Macià, and J.J.L. Velázquez, Two-species-coagulation approach to consensus by group level interactions, Physical Review E, 82 (2010), 016113, 6pp.
doi: 10.1103/PhysRevE.82.016113. |
[14] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[15] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.
doi: 10.1088/0951-7715/24/10/002. |
[16] |
M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808.
doi: 10.1088/0951-7715/26/10/2777. |
[17] |
J. M. Haile, Molecular Dynamics Simulation: Elementary Methods, 1st ed., John Wiley & Sons, Inc., New York, NY, USA, 1992. |
[18] |
D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Physical Review Letters, 95 (2005), 226106.
doi: 10.1103/PhysRevLett.95.226106. |
[19] |
D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[20] |
T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential, Physica D, 260 (2013), 65-76.
doi: 10.1016/j.physd.2012.10.009. |
[21] |
T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Physical Review E, 84 (2011), 015203.
doi: 10.1103/PhysRevE.84.015203. |
[22] |
T. Kostić and A. L. Bertozzi, Statistical density estimation using threshold dynamics for geometric motion, Journal of Scientific Computing, 54 (2013), 513-530.
doi: 10.1007/s10915-012-9615-6. |
[23] |
H. Levine, E. Ben-Jacob, I. Cohen and W. Rappel, Swarming patterns in microorganisms: Some new modeling results, in Decision and Control, 2006 45th IEEE Conference on, (2006), 5073-5077.
doi: 10.1109/CDC.2006.377435. |
[24] |
H. Levine, W. Rappel and I. Cohen, Self-organization in systems of self-propelled particles, Physical Review E, 63 (2000), 017101.
doi: 10.1103/PhysRevE.63.017101. |
[25] |
Y. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[26] |
Y. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, in IEEE International Conference on Robotics and Automation, (2007), 2292-2299.
doi: 10.1109/ROBOT.2007.363661. |
[27] |
T. Liu, Hydrophilic macroionic solutions: What happens when soluble ions reach the size of nanometer scale?, Langmuir, 26 (2010), 9202-9213.
doi: 10.1021/la902917q. |
[28] |
T. Liu, M. Langston, D. Li, J. M. Pigga, C. Pichon, A. Todea and A. Müller, Self-recognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 1590-1592.
doi: 10.1126/science.1201121. |
[29] |
A. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. |
[30] |
A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, Journal of mathematical biology, 47 (2003), 353-389.
doi: 10.1007/s00285-003-0209-7. |
[31] |
R. Ramírez and T. Pöschel, Coefficient of restitution of colliding viscoelastic spheres, Physica Review E, 60 (1999), 4465.
doi: 10.1103/physreve.60.4465. |
[32] |
R.M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150023,34p.
doi: 10.1142/S0218202511500230. |
[33] |
H. Sun, D. Uminsky and A. L. Bertozzi, A generalized Birkhoff-Rott equation for two-dimensional active scalar problems, SIAM Journal on Applied Mathematics, 72 (2012), 382-404.
doi: 10.1137/110819883. |
[34] |
J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[35] |
C. Topaz, A. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts, The European Physical Journal Special Topics, 157 (2008), 93-109.
doi: 10.1140/epjst/e2008-00633-y. |
[36] |
C. Topaz, A. L. Bertozzi and M. E. Lewis, A nonlocal continuum model for biological aggregations, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[37] |
L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, O. Shochet and W. N. Reynolds, Aggregation patterns in stressed bacteria, Physical review letters, 75 (1995), 1859-1862.
doi: 10.1103/PhysRevLett.75.1859. |
[38] |
J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, Journal of Nonlinear Science, 22 (2012), 935-959.
doi: 10.1007/s00332-012-9132-7. |
[39] |
J. von Brecht, D. Uminsky, T. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31 pp.
doi: 10.1142/S0218202511400021. |
[40] |
G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European Journal of Applied Mathematics, 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
show all references
References:
[1] |
E. Altschuler, T. Williams, E. Ratner, R. Tipton, R. Stong, F. Dowla and F. Wooten, Possible global minimum lattice configurations for Thomson's problem of charges on a sphere, Physical Review Letters, 78 (1997), 2681-2685.
doi: 10.1103/PhysRevLett.78.2681. |
[2] |
D. Balague, J.A. Carrillo, T. Laurent and G. Raoul, Dimensionality of local minimizers of the interaction energy, Archive for Rational Mechanics and Analysis, 209 (2013), 1055-1088.
doi: 10.1007/s00205-013-0644-6. |
[3] |
A. Bernoff and C. Topaz, A primer of swarm equilibria, SIAM Journal on Applied Dynamical Systems, 10 (2011), 212-250.
doi: 10.1137/100804504. |
[4] |
A. L. Bertozzi, H. Sun, T. Kolokolnikov, D. Uminsky and J. von Brecht, Ring patterns and their bifurcations in a nonlocal model of biological swarms, preprint, 2013. Available from: http://www.math.ucla.edu/~bertozzi/papers/bigring-final.pdf. |
[5] |
H. Cabral and D. Schmidt, Stability of relative equilibria in the problem of $n+1$ vortices, SIAM Journal on Mathematical Analysis, 31 (1999), 231-250.
doi: 10.1137/S0036141098302124. |
[6] |
Y. Chen, T. Kolokolnikov and D. Zhirov, Collective behaviour of large number of vortices in the plane, Proceedings of the Royal Society A, 469 (2013), 20130085.
doi: 10.1098/rspa.2013.0085. |
[7] |
H. Cohn and A. Kumar, Algorithmic design of self-assembling structures, PNAS, 106 (2009), 9570-9575.
doi: 10.1073/pnas.0901636106. |
[8] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, European Journal of Applied Mathematics, 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[9] |
I. Couzin, J. Krause, N. Franks and S. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
[10] |
G. Crippa and M. Lécureux-Mercier, Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, Nonlinear Differential Equations and Applications, 20 (2013), 523-537.
doi: 10.1007/s00030-012-0164-3. |
[11] |
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, Physical Review Letters, 96 (2006), 104302.
doi: 10.1103/PhysRevLett.96.104302. |
[12] |
B. Düring, P. Markowich, J. F. Pietschmann and M. T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society A, 465 (2009), 3687-3708.
doi: 10.1098/rspa.2009.0239. |
[13] |
C. Escudero, F. Macià, and J.J.L. Velázquez, Two-species-coagulation approach to consensus by group level interactions, Physical Review E, 82 (2010), 016113, 6pp.
doi: 10.1103/PhysRevE.82.016113. |
[14] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[15] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.
doi: 10.1088/0951-7715/24/10/002. |
[16] |
M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808.
doi: 10.1088/0951-7715/26/10/2777. |
[17] |
J. M. Haile, Molecular Dynamics Simulation: Elementary Methods, 1st ed., John Wiley & Sons, Inc., New York, NY, USA, 1992. |
[18] |
D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility, Physical Review Letters, 95 (2005), 226106.
doi: 10.1103/PhysRevLett.95.226106. |
[19] |
D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[20] |
T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential, Physica D, 260 (2013), 65-76.
doi: 10.1016/j.physd.2012.10.009. |
[21] |
T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Physical Review E, 84 (2011), 015203.
doi: 10.1103/PhysRevE.84.015203. |
[22] |
T. Kostić and A. L. Bertozzi, Statistical density estimation using threshold dynamics for geometric motion, Journal of Scientific Computing, 54 (2013), 513-530.
doi: 10.1007/s10915-012-9615-6. |
[23] |
H. Levine, E. Ben-Jacob, I. Cohen and W. Rappel, Swarming patterns in microorganisms: Some new modeling results, in Decision and Control, 2006 45th IEEE Conference on, (2006), 5073-5077.
doi: 10.1109/CDC.2006.377435. |
[24] |
H. Levine, W. Rappel and I. Cohen, Self-organization in systems of self-propelled particles, Physical Review E, 63 (2000), 017101.
doi: 10.1103/PhysRevE.63.017101. |
[25] |
Y. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[26] |
Y. Chuang, Y. R. Huang, M. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, in IEEE International Conference on Robotics and Automation, (2007), 2292-2299.
doi: 10.1109/ROBOT.2007.363661. |
[27] |
T. Liu, Hydrophilic macroionic solutions: What happens when soluble ions reach the size of nanometer scale?, Langmuir, 26 (2010), 9202-9213.
doi: 10.1021/la902917q. |
[28] |
T. Liu, M. Langston, D. Li, J. M. Pigga, C. Pichon, A. Todea and A. Müller, Self-recognition among different polyprotic macroions during assembly processes in dilute solution, Science, 331 (2011), 1590-1592.
doi: 10.1126/science.1201121. |
[29] |
A. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. |
[30] |
A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, Journal of mathematical biology, 47 (2003), 353-389.
doi: 10.1007/s00285-003-0209-7. |
[31] |
R. Ramírez and T. Pöschel, Coefficient of restitution of colliding viscoelastic spheres, Physica Review E, 60 (1999), 4465.
doi: 10.1103/physreve.60.4465. |
[32] |
R.M. Colombo, M. Garavello and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150023,34p.
doi: 10.1142/S0218202511500230. |
[33] |
H. Sun, D. Uminsky and A. L. Bertozzi, A generalized Birkhoff-Rott equation for two-dimensional active scalar problems, SIAM Journal on Applied Mathematics, 72 (2012), 382-404.
doi: 10.1137/110819883. |
[34] |
J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[35] |
C. Topaz, A. Bernoff, S. Logan and W. Toolson, A model for rolling swarms of locusts, The European Physical Journal Special Topics, 157 (2008), 93-109.
doi: 10.1140/epjst/e2008-00633-y. |
[36] |
C. Topaz, A. L. Bertozzi and M. E. Lewis, A nonlocal continuum model for biological aggregations, Bulletin of Mathematical Biology, 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[37] |
L. Tsimring, H. Levine, I. Aranson, E. Ben-Jacob, I. Cohen, O. Shochet and W. N. Reynolds, Aggregation patterns in stressed bacteria, Physical review letters, 75 (1995), 1859-1862.
doi: 10.1103/PhysRevLett.75.1859. |
[38] |
J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics, Journal of Nonlinear Science, 22 (2012), 935-959.
doi: 10.1007/s00332-012-9132-7. |
[39] |
J. von Brecht, D. Uminsky, T. Kolokolnikov and A. L. Bertozzi, Predicting pattern formation in particle interactions, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1140002, 31 pp.
doi: 10.1142/S0218202511400021. |
[40] |
G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European Journal of Applied Mathematics, 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
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