July  2014, 19(5): 1411-1436. doi: 10.3934/dcdsb.2014.19.1411

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

Received  May 2013 Revised  January 2014 Published  April 2014

Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in $\mathbb{R}^2$, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Some intriguing steady state patterns are shown through numerical examples.
Citation: Alan Mackey, Theodore Kolokolnikov, Andrea L. Bertozzi. Two-species particle aggregation and stability of co-dimension one solutions. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1411-1436. doi: 10.3934/dcdsb.2014.19.1411
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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