American Institute of Mathematical Sciences

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

Two-species particle aggregation and stability of co-dimension one solutions

Alan Mackey 1, , Theodore Kolokolnikov 2, and  Andrea L. Bertozzi 3,

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.
Keywords: measure solutions, Aggregation equation, continuum limit, linear well-posedness, linear stability, asymptotics., pattern formation.
Mathematics Subject Classification: Primary: 35B35, 25B20, 35B36; Secondary: 35D30, 41A6.
Citation: Alan Mackey, Theodore Kolokolnikov, Andrea L. Bertozzi. Two-species particle aggregation and stability of co-dimension one solutions. Discrete & 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.  doi: 10.1103/PhysRevLett.78.2681.  Google Scholar

[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.  doi: 10.1007/s00205-013-0644-6.  Google Scholar

[3]

A. Bernoff and C. Topaz, A primer of swarm equilibria,, SIAM Journal on Applied Dynamical Systems, 10 (2011), 212.  doi: 10.1137/100804504.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1137/S0036141098302124.  Google Scholar

[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).  doi: 10.1098/rspa.2013.0085.  Google Scholar

[7]

H. Cohn and A. Kumar, Algorithmic design of self-assembling structures,, PNAS, 106 (2009), 9570.  doi: 10.1073/pnas.0901636106.  Google Scholar

[8]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$,, European Journal of Applied Mathematics, 22 (2011), 553.  doi: 10.1017/S0956792511000258.  Google Scholar

[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.  doi: 10.1038/nature03236.  Google Scholar

[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.  doi: 10.1007/s00030-012-0164-3.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[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.  doi: 10.1098/rspa.2009.0239.  Google Scholar

[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).  doi: 10.1103/PhysRevE.82.016113.  Google Scholar

[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.  doi: 10.1524/anly.2009.1029.  Google Scholar

[15]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model,, Nonlinearity, 24 (2011), 2681.  doi: 10.1088/0951-7715/24/10/002.  Google Scholar

[16]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species,, Nonlinearity, 26 (2013), 2777.  doi: 10.1088/0951-7715/26/10/2777.  Google Scholar

[17]

J. M. Haile, Molecular Dynamics Simulation: Elementary Methods,, 1st ed., (1992).   Google Scholar

[18]

D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility,, Physical Review Letters, 95 (2005).  doi: 10.1103/PhysRevLett.95.226106.  Google Scholar

[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.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[20]

T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential,, Physica D, 260 (2013), 65.  doi: 10.1016/j.physd.2012.10.009.  Google Scholar

[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).  doi: 10.1103/PhysRevE.84.015203.  Google Scholar

[22]

T. Kostić and A. L. Bertozzi, Statistical density estimation using threshold dynamics for geometric motion,, Journal of Scientific Computing, 54 (2013), 513.  doi: 10.1007/s10915-012-9615-6.  Google Scholar

[23]

H. Levine, E. Ben-Jacob, I. Cohen and W. Rappel, Swarming patterns in microorganisms: Some new modeling results,, in Decision and Control, (2006), 5073.  doi: 10.1109/CDC.2006.377435.  Google Scholar

[24]

H. Levine, W. Rappel and I. Cohen, Self-organization in systems of self-propelled particles,, Physical Review E, 63 (2000).  doi: 10.1103/PhysRevE.63.017101.  Google Scholar

[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.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[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.  doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[27]

T. Liu, Hydrophilic macroionic solutions: What happens when soluble ions reach the size of nanometer scale?,, Langmuir, 26 (2010), 9202.  doi: 10.1021/la902917q.  Google Scholar

[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.  doi: 10.1126/science.1201121.  Google Scholar

[29]

A. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar

[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.  doi: 10.1007/s00285-003-0209-7.  Google Scholar

[31]

R. Ramírez and T. Pöschel, Coefficient of restitution of colliding viscoelastic spheres,, Physica Review E, 60 (1999).  doi: 10.1103/physreve.60.4465.  Google Scholar

[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).  doi: 10.1142/S0218202511500230.  Google Scholar

[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.  doi: 10.1137/110819883.  Google Scholar

[34]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source,, Nonlinearity, 25 (2012), 1413.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[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.  doi: 10.1140/epjst/e2008-00633-y.  Google Scholar

[36]

C. Topaz, A. L. Bertozzi and M. E. Lewis, A nonlocal continuum model for biological aggregations,, Bulletin of Mathematical Biology, 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[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.  doi: 10.1103/PhysRevLett.75.1859.  Google Scholar

[38]

J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics,, Journal of Nonlinear Science, 22 (2012), 935.  doi: 10.1007/s00332-012-9132-7.  Google Scholar

[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).  doi: 10.1142/S0218202511400021.  Google Scholar

[40]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts,, European Journal of Applied Mathematics, 13 (2002), 641.  doi: 10.1017/S0956792501004843.  Google Scholar

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.  doi: 10.1103/PhysRevLett.78.2681.  Google Scholar

[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.  doi: 10.1007/s00205-013-0644-6.  Google Scholar

[3]

A. Bernoff and C. Topaz, A primer of swarm equilibria,, SIAM Journal on Applied Dynamical Systems, 10 (2011), 212.  doi: 10.1137/100804504.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1137/S0036141098302124.  Google Scholar

[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).  doi: 10.1098/rspa.2013.0085.  Google Scholar

[7]

H. Cohn and A. Kumar, Algorithmic design of self-assembling structures,, PNAS, 106 (2009), 9570.  doi: 10.1073/pnas.0901636106.  Google Scholar

[8]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$,, European Journal of Applied Mathematics, 22 (2011), 553.  doi: 10.1017/S0956792511000258.  Google Scholar

[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.  doi: 10.1038/nature03236.  Google Scholar

[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.  doi: 10.1007/s00030-012-0164-3.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.96.104302.  Google Scholar

[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.  doi: 10.1098/rspa.2009.0239.  Google Scholar

[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).  doi: 10.1103/PhysRevE.82.016113.  Google Scholar

[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.  doi: 10.1524/anly.2009.1029.  Google Scholar

[15]

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model,, Nonlinearity, 24 (2011), 2681.  doi: 10.1088/0951-7715/24/10/002.  Google Scholar

[16]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species,, Nonlinearity, 26 (2013), 2777.  doi: 10.1088/0951-7715/26/10/2777.  Google Scholar

[17]

J. M. Haile, Molecular Dynamics Simulation: Elementary Methods,, 1st ed., (1992).   Google Scholar

[18]

D. Holm and V. Putkaradze, Aggregation of finite-size particles with variable mobility,, Physical Review Letters, 95 (2005).  doi: 10.1103/PhysRevLett.95.226106.  Google Scholar

[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.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[20]

T. Kolokolnikov, Y. Huang and M. Pavlovski, Singular patterns for an aggregation model with a confining potential,, Physica D, 260 (2013), 65.  doi: 10.1016/j.physd.2012.10.009.  Google Scholar

[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).  doi: 10.1103/PhysRevE.84.015203.  Google Scholar

[22]

T. Kostić and A. L. Bertozzi, Statistical density estimation using threshold dynamics for geometric motion,, Journal of Scientific Computing, 54 (2013), 513.  doi: 10.1007/s10915-012-9615-6.  Google Scholar

[23]

H. Levine, E. Ben-Jacob, I. Cohen and W. Rappel, Swarming patterns in microorganisms: Some new modeling results,, in Decision and Control, (2006), 5073.  doi: 10.1109/CDC.2006.377435.  Google Scholar

[24]

H. Levine, W. Rappel and I. Cohen, Self-organization in systems of self-propelled particles,, Physical Review E, 63 (2000).  doi: 10.1103/PhysRevE.63.017101.  Google Scholar

[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.  doi: 10.1016/j.physd.2007.05.007.  Google Scholar

[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.  doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[27]

T. Liu, Hydrophilic macroionic solutions: What happens when soluble ions reach the size of nanometer scale?,, Langmuir, 26 (2010), 9202.  doi: 10.1021/la902917q.  Google Scholar

[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.  doi: 10.1126/science.1201121.  Google Scholar

[29]

A. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar

[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.  doi: 10.1007/s00285-003-0209-7.  Google Scholar

[31]

R. Ramírez and T. Pöschel, Coefficient of restitution of colliding viscoelastic spheres,, Physica Review E, 60 (1999).  doi: 10.1103/physreve.60.4465.  Google Scholar

[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).  doi: 10.1142/S0218202511500230.  Google Scholar

[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.  doi: 10.1137/110819883.  Google Scholar

[34]

J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source,, Nonlinearity, 25 (2012), 1413.  doi: 10.1088/0951-7715/25/5/1413.  Google Scholar

[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.  doi: 10.1140/epjst/e2008-00633-y.  Google Scholar

[36]

C. Topaz, A. L. Bertozzi and M. E. Lewis, A nonlocal continuum model for biological aggregations,, Bulletin of Mathematical Biology, 68 (2006), 1601.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[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.  doi: 10.1103/PhysRevLett.75.1859.  Google Scholar

[38]

J. von Brecht and D. Uminsky, On soccer balls and linearized inverse statistical mechanics,, Journal of Nonlinear Science, 22 (2012), 935.  doi: 10.1007/s00332-012-9132-7.  Google Scholar

[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).  doi: 10.1142/S0218202511400021.  Google Scholar

[40]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts,, European Journal of Applied Mathematics, 13 (2002), 641.  doi: 10.1017/S0956792501004843.  Google Scholar

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