2014, 11(2): 303-315. doi: 10.3934/mbe.2014.11.303

Local versus nonlocal barycentric interactions in 1D agent dynamics

1. 

Ecole Polytechnique Fédérale de Lausanne, STI-IMT-LPM, Station 17, CH-1015 Lausanne, Switzerland

2. 

Bern University of Applied Sciences, Quellgasse 21, CH-2501 Biel, Switzerland

3. 

IBM Zurich Research Laboratory, Saeumerstrasse 4, CH-8803 Rueschlikon, Switzerland

Received  September 2012 Revised  March 2013 Published  October 2013

The mean-field dynamics of a collection of stochastic agents evolving under local and nonlocal interactions in one dimension is studied via analytically solvable models. The nonlocal interactions between agents result from $(a)$ a finite extension of the agents interaction range and $(b)$ a barycentric modulation of the interaction strength. Our modeling framework is based on a discrete two-velocity Boltzmann dynamics which can be analytically discussed. Depending on the span and the modulation of the interaction range, we analytically observe a transition from a purely diffusive regime without definite pattern to a flocking evolution represented by a solitary wave traveling with constant velocity.
Citation: Max-Olivier Hongler, Roger Filliger, Olivier Gallay. Local versus nonlocal barycentric interactions in 1D agent dynamics. Mathematical Biosciences & Engineering, 2014, 11 (2) : 303-315. doi: 10.3934/mbe.2014.11.303
References:
[1]

D. Armbruster, E. Gel and J. Murakami, Bucket brigades with worker learning, Eur. J. of Oper. Res., 176 (2007), 264-274. doi: 10.1016/j.ejor.2005.06.052.

[2]

M. Balázs, M. Rácz and B. Tóth, Modeling flocks and prices: Jumping particles with an attractive interaction,, preprint, (). 

[3]

A. D. Banner, R. Fernholz and I. Karatzas, On Atlas models of equity markets, Ann. of App. Prob., 15 (2005), 2296-2330. doi: 10.1214/105051605000000449.

[4]

J. J. Bartholdi III and D. D. Eisenstein, A production line that balances itself, Oper. Res., 44 (1996), 21-34.

[5]

J. J. Bartholdi III, D. D. Eisenstein and R. D. Foley, Performance of bucket brigades when work is stochastic, Oper. Res., 49 (2001), 710-719.

[6]

S. Bazazi, P. Romanczuk, S. Thomas, L. Schimansky-Geier, J. J. Hale, G. A. Miller, G. A. Sword, S. J. Simpson and I. D. Couzin, Nutritional state and collective motion: From individuals to mass migration, Proc. of the Roy. Soc. Lond. Ser. B., 278 (2011), 356-363.

[7]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.

[8]

E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101, 4 pp. doi: 10.1103/PhysRevE.74.022101.

[9]

E. Bertin, M. Droz nad G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001.

[10]

W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale and A. M. Walczak, Statistical mechanics for natural flocks of birds, Proc. Natl. Acad. Sci., 109 (2012), 4786-4791. doi: 10.1073/pnas.1118633109.

[11]

L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and J. Soler, Exact solutions and dynamics of globally coupled oscillators, Math. Mod. Meth. App. Sci., 16 (2006), 1919-1959. doi: 10.1142/S0218202506001765.

[12]

J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142.

[13]

A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale, Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci., 107 (2010), 11865-11870. doi: 10.1073/pnas.1005766107.

[14]

H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113, 15 pp. doi: 10.1103/PhysRevE.77.046113.

[15]

S. Chatterjee and S. Pal, A phase transition behavior for Brownian motions interacting through their ranks, Prob. Th. Rel. Fiel., 147 (2010), 123-159. doi: 10.1007/s00440-009-0203-0.

[16]

Y.-L. Chou, R. Wolfe and T. Ihle, Kinetic theory for systems of self-propelled particles with metric-free interactions, Phys. Rev. E, 86 (2012), 021120, 20 pp. doi: 10.1103/PhysRevE.86.021120.

[17]

F. Comets, M. V. Menshikov, S. Volkov and A. R. Wade, Random walk with barycentric self-interaction, J. Stat Phys., 143 (2011), 855-888. doi: 10.1007/s10955-011-0218-7.

[18]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. on Autom. Contr., 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[19]

V. Dosetti, Cohesive motion in one-dimensional flocking, J. Phys. A, 45 (2012), 035003, 20 pp. doi: 10.1088/1751-8113/45/3/035003.

[20]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75. doi: 10.1007/s00285-011-0452-2.

[21]

R. Eftimie, G. de Vries, M. A. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565. doi: 10.1007/s11538-006-9175-8.

[22]

E. R. Fernholz and I. Karatzas, Stochastic portfolio theory: A survey, in Handbook of Numerical Analysis. Mathematical Modeling and Numerical Methods in Finance (ed. A. Bensoussan), Elsevier, Amsterdam, 2009, 89-168.

[23]

T. D. Frank, Nonlinear Fokker Planck Equations. Fundamentals and Applications, Springer Series in Synergetics, Springer-Verlag, Berlin, 2005.

[24]

E. Gambetta and B. Perthame, Scaling limits for the Ruijgrok-Wu model of the Boltzmann equation, Math. Meth. in App. Sci., 24 (2001), 949-967. doi: 10.1002/mma.251.

[25]

G. Grégoire, H. Chaté and Y. Tu, Moving and staying together without a leader, Physica D, 181 (2003), 157-170. doi: 10.1016/S0167-2789(03)00102-7.

[26]

G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92 (2004), 025702.

[27]

E. Gutkin and M. Kac, Propagation of chaos and the Burgers equation, SIAM J. Appl. Math., 43 (1983), 971-980. doi: 10.1137/0143063.

[28]

F. Hashemi, M.-O. Hongler and O. Gallay, Spatio-temporal patterns for a generalized innovation diffusion model, Theor. Econ. Lett., 2 (2012), 1-9. doi: 10.4236/tel.2012.21001.

[29]

D. Helbing, Traffic Flow: Encyclopedia of Nonlinear Science, Routledge, New York, 2005.

[30]

P. C. Hemmer, On a generalization of Smoluchowski's diffusion equation, Physica, 27 (1961), 79-82. doi: 10.1016/0031-8914(61)90022-2.

[31]

M.-O. Hongler and R. Filliger, Mesoscopic derivation of a fundamental diagram of one-lane traffic, Phys. Lett. A, 301 (2002), 408-412. doi: 10.1016/S0375-9601(02)01082-4.

[32]

M.-O. Hongler and L. Streit, A probabilistic connection between the Burger and a discrete Boltzmann equation, Europhys. Lett., 12 (1990), 193-197. doi: 10.1209/0295-5075/12/3/001.

[33]

M.-O. Hongler, O. Gallay, M. Hülsmann, P. Cordes and R. Colmorn, Centralized versus decentralized control - A solvable stylized model in transportation, Phys. A, 389 (2010), 4162-4171. doi: 10.1016/j.physa.2010.05.047.

[34]

T. Ihle, Kinetic theory of flocking: Derivation of hydrodynamic equations, Phys. Rev E, 83 (2011), 030901, 4 pp. doi: 10.1103/PhysRevE.83.030901.

[35]

M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mount. J. Math., 4 (1974), 497-509. doi: 10.1216/RMJ-1974-4-3-497.

[36]

P. D. Lorch, G. A. Sword, D. T. Gwynne and G. L. Anderson, Radiotelemetry reveals differences in individual movement patterns between outbreak and non-outbreak Mormon cricket populations, Ecol. Entom., 30 (2005), 548-555. doi: 10.1111/j.0307-6946.2005.00725.x.

[37]

F. Lutscher and A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems, J. Nonlin. Sci., 12 (2002), 619-640. doi: 10.1007/s00332-002-0510-4.

[38]

R. J. LeVeque, Lectures in Mathematics ETHZ, Birkhäuser, Basel, 1999.

[39]

H. P. McKean, Jr., Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., 1967, 41-57.

[40]

K. Nishihara, A note on the stability of travelling wave solutions of Burgers' equation, Jap. J. Appl. Math., 2 (1985), 27-35. doi: 10.1007/BF03167037.

[41]

S. Pal and J. Pitman, One-dimensional Brownian particle systems with rank-dependent drifts, Ann. of App. Prob., 18 (2008), 2179-2207. doi: 10.1214/08-AAP516.

[42]

A. Peshkov, S. Ngo, E. Bertin, H. Chaté and F. Ginelli, Continuous theory of active matter systems with metric-free interactions, Phys. Rev. Lett., 109 (2012), 098101, 6 pp. doi: 10.1103/PhysRevLett.109.098101.

[43]

S. Ramaswamy, The mechanics and statistics of active matter, Ann. Rev. of Cond. Matt. Phys., 1 (2010), 323-345. doi: 10.1146/annurev-conmatphys-070909-104101.

[44]

P. Romanczuk, I. D. Couzin, L. Schimansky-Geier, Collective motion due to individual escape and pursuit response, Phys. Rev. Lett., 102 (2009), 010602, 4 pp. doi: 10.1103/PhysRevLett.102.010602.

[45]

T. W. Ruijgrok and T. T. Wu, A completely solvable model of the nonlinear Boltzmann equation, Physica A, 113 (1982), 401-416. doi: 10.1016/0378-4371(82)90147-9.

[46]

D. Strömbom, Collective motion from local attraction, J. Theor Biol., 283 (2011), 145-151. doi: 10.1016/j.jtbi.2011.05.019.

[47]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E (3), 58 (1998), 4828-4858. doi: 10.1103/PhysRevE.58.4828.

[48]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

show all references

References:
[1]

D. Armbruster, E. Gel and J. Murakami, Bucket brigades with worker learning, Eur. J. of Oper. Res., 176 (2007), 264-274. doi: 10.1016/j.ejor.2005.06.052.

[2]

M. Balázs, M. Rácz and B. Tóth, Modeling flocks and prices: Jumping particles with an attractive interaction,, preprint, (). 

[3]

A. D. Banner, R. Fernholz and I. Karatzas, On Atlas models of equity markets, Ann. of App. Prob., 15 (2005), 2296-2330. doi: 10.1214/105051605000000449.

[4]

J. J. Bartholdi III and D. D. Eisenstein, A production line that balances itself, Oper. Res., 44 (1996), 21-34.

[5]

J. J. Bartholdi III, D. D. Eisenstein and R. D. Foley, Performance of bucket brigades when work is stochastic, Oper. Res., 49 (2001), 710-719.

[6]

S. Bazazi, P. Romanczuk, S. Thomas, L. Schimansky-Geier, J. J. Hale, G. A. Miller, G. A. Sword, S. J. Simpson and I. D. Couzin, Nutritional state and collective motion: From individuals to mass migration, Proc. of the Roy. Soc. Lond. Ser. B., 278 (2011), 356-363.

[7]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.

[8]

E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101, 4 pp. doi: 10.1103/PhysRevE.74.022101.

[9]

E. Bertin, M. Droz nad G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001. doi: 10.1088/1751-8113/42/44/445001.

[10]

W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale and A. M. Walczak, Statistical mechanics for natural flocks of birds, Proc. Natl. Acad. Sci., 109 (2012), 4786-4791. doi: 10.1073/pnas.1118633109.

[11]

L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and J. Soler, Exact solutions and dynamics of globally coupled oscillators, Math. Mod. Meth. App. Sci., 16 (2006), 1919-1959. doi: 10.1142/S0218202506001765.

[12]

J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406. doi: 10.1126/science.1125142.

[13]

A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini and M. Viale, Scale-free correlations in starling flocks, Proc. Natl. Acad. Sci., 107 (2010), 11865-11870. doi: 10.1073/pnas.1005766107.

[14]

H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113, 15 pp. doi: 10.1103/PhysRevE.77.046113.

[15]

S. Chatterjee and S. Pal, A phase transition behavior for Brownian motions interacting through their ranks, Prob. Th. Rel. Fiel., 147 (2010), 123-159. doi: 10.1007/s00440-009-0203-0.

[16]

Y.-L. Chou, R. Wolfe and T. Ihle, Kinetic theory for systems of self-propelled particles with metric-free interactions, Phys. Rev. E, 86 (2012), 021120, 20 pp. doi: 10.1103/PhysRevE.86.021120.

[17]

F. Comets, M. V. Menshikov, S. Volkov and A. R. Wade, Random walk with barycentric self-interaction, J. Stat Phys., 143 (2011), 855-888. doi: 10.1007/s10955-011-0218-7.

[18]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. on Autom. Contr., 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[19]

V. Dosetti, Cohesive motion in one-dimensional flocking, J. Phys. A, 45 (2012), 035003, 20 pp. doi: 10.1088/1751-8113/45/3/035003.

[20]

R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: A brief review, J. Math. Biol., 65 (2012), 35-75. doi: 10.1007/s00285-011-0452-2.

[21]

R. Eftimie, G. de Vries, M. A. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol., 69 (2007), 1537-1565. doi: 10.1007/s11538-006-9175-8.

[22]

E. R. Fernholz and I. Karatzas, Stochastic portfolio theory: A survey, in Handbook of Numerical Analysis. Mathematical Modeling and Numerical Methods in Finance (ed. A. Bensoussan), Elsevier, Amsterdam, 2009, 89-168.

[23]

T. D. Frank, Nonlinear Fokker Planck Equations. Fundamentals and Applications, Springer Series in Synergetics, Springer-Verlag, Berlin, 2005.

[24]

E. Gambetta and B. Perthame, Scaling limits for the Ruijgrok-Wu model of the Boltzmann equation, Math. Meth. in App. Sci., 24 (2001), 949-967. doi: 10.1002/mma.251.

[25]

G. Grégoire, H. Chaté and Y. Tu, Moving and staying together without a leader, Physica D, 181 (2003), 157-170. doi: 10.1016/S0167-2789(03)00102-7.

[26]

G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett., 92 (2004), 025702.

[27]

E. Gutkin and M. Kac, Propagation of chaos and the Burgers equation, SIAM J. Appl. Math., 43 (1983), 971-980. doi: 10.1137/0143063.

[28]

F. Hashemi, M.-O. Hongler and O. Gallay, Spatio-temporal patterns for a generalized innovation diffusion model, Theor. Econ. Lett., 2 (2012), 1-9. doi: 10.4236/tel.2012.21001.

[29]

D. Helbing, Traffic Flow: Encyclopedia of Nonlinear Science, Routledge, New York, 2005.

[30]

P. C. Hemmer, On a generalization of Smoluchowski's diffusion equation, Physica, 27 (1961), 79-82. doi: 10.1016/0031-8914(61)90022-2.

[31]

M.-O. Hongler and R. Filliger, Mesoscopic derivation of a fundamental diagram of one-lane traffic, Phys. Lett. A, 301 (2002), 408-412. doi: 10.1016/S0375-9601(02)01082-4.

[32]

M.-O. Hongler and L. Streit, A probabilistic connection between the Burger and a discrete Boltzmann equation, Europhys. Lett., 12 (1990), 193-197. doi: 10.1209/0295-5075/12/3/001.

[33]

M.-O. Hongler, O. Gallay, M. Hülsmann, P. Cordes and R. Colmorn, Centralized versus decentralized control - A solvable stylized model in transportation, Phys. A, 389 (2010), 4162-4171. doi: 10.1016/j.physa.2010.05.047.

[34]

T. Ihle, Kinetic theory of flocking: Derivation of hydrodynamic equations, Phys. Rev E, 83 (2011), 030901, 4 pp. doi: 10.1103/PhysRevE.83.030901.

[35]

M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mount. J. Math., 4 (1974), 497-509. doi: 10.1216/RMJ-1974-4-3-497.

[36]

P. D. Lorch, G. A. Sword, D. T. Gwynne and G. L. Anderson, Radiotelemetry reveals differences in individual movement patterns between outbreak and non-outbreak Mormon cricket populations, Ecol. Entom., 30 (2005), 548-555. doi: 10.1111/j.0307-6946.2005.00725.x.

[37]

F. Lutscher and A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems, J. Nonlin. Sci., 12 (2002), 619-640. doi: 10.1007/s00332-002-0510-4.

[38]

R. J. LeVeque, Lectures in Mathematics ETHZ, Birkhäuser, Basel, 1999.

[39]

H. P. McKean, Jr., Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., 1967, 41-57.

[40]

K. Nishihara, A note on the stability of travelling wave solutions of Burgers' equation, Jap. J. Appl. Math., 2 (1985), 27-35. doi: 10.1007/BF03167037.

[41]

S. Pal and J. Pitman, One-dimensional Brownian particle systems with rank-dependent drifts, Ann. of App. Prob., 18 (2008), 2179-2207. doi: 10.1214/08-AAP516.

[42]

A. Peshkov, S. Ngo, E. Bertin, H. Chaté and F. Ginelli, Continuous theory of active matter systems with metric-free interactions, Phys. Rev. Lett., 109 (2012), 098101, 6 pp. doi: 10.1103/PhysRevLett.109.098101.

[43]

S. Ramaswamy, The mechanics and statistics of active matter, Ann. Rev. of Cond. Matt. Phys., 1 (2010), 323-345. doi: 10.1146/annurev-conmatphys-070909-104101.

[44]

P. Romanczuk, I. D. Couzin, L. Schimansky-Geier, Collective motion due to individual escape and pursuit response, Phys. Rev. Lett., 102 (2009), 010602, 4 pp. doi: 10.1103/PhysRevLett.102.010602.

[45]

T. W. Ruijgrok and T. T. Wu, A completely solvable model of the nonlinear Boltzmann equation, Physica A, 113 (1982), 401-416. doi: 10.1016/0378-4371(82)90147-9.

[46]

D. Strömbom, Collective motion from local attraction, J. Theor Biol., 283 (2011), 145-151. doi: 10.1016/j.jtbi.2011.05.019.

[47]

J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E (3), 58 (1998), 4828-4858. doi: 10.1103/PhysRevE.58.4828.

[48]

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.

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