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From individual to collective behaviour of coupled velocity jump processes: A locust example
1. | University of Oxford, Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB |
2. | King Abdullah University of Science and Technology, Computer, Electrical and Mathematical Sciences and Engineering, Thuwal 23955-6900, Saudi Arabia |
References:
[1] |
L. Baum and M. Katz, Convergence rates in the law of large numbers, Transactions of the American Mathematical Society, 120 (1965), 108-123.
doi: 10.1090/S0002-9947-1965-0198524-1. |
[2] |
S. Bazazi, J. Buhl, J. Hale, M. Anstey, G. Sword, S. Simpson and I. Couzin, Collective motion and cannibalism in locust migratory bands, Current Biology, 18 (2008), 735-739.
doi: 10.1016/j.cub.2008.04.035. |
[3] |
J. Buhl, D. Sumpter, I. Couzin, J. Hale, E. Despland, E. Miller and S. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.
doi: 10.1126/science.1125142. |
[4] |
J. Buhl, G. Sword, F. Clissold and S. Simpson, Group structure in locust migratory bands, Behav. Ecol. Sociobiol., 65 (2011), 265-273.
doi: 10.1007/s00265-010-1041-x. |
[5] |
J. Carrillo, M. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models, 2 (2009), 363-378. |
[6] |
J. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM Journal on Mathematical Analysis, 42 (2010), 218-236.
doi: 10.1137/090757290. |
[7] |
J. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, hydrodynamic models of swarming, in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences'' (eds. G. Naldi, L. Pareschi and G. Toscani), Modelling and Simulation in Science and Technology, Birkhäuser, (2010), 297-336. |
[8] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'' Applied Mathematical Sciences, 106, Springer-Verlag, 1994. |
[9] |
E. Codling and N. Hill, Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters, Journal of Mathematical Biology, 51 (2005), 527-556.
doi: 10.1007/s00285-005-0317-7. |
[10] |
A. Czirók, A. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Physical Review Letters, 82 (1999), 209-212.
doi: 10.1103/PhysRevLett.82.209. |
[11] |
R. Erban, I. Kevrekidis, D. Adalsteinsson and T. Elston, Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation, Journal of Chemical Physics, 124 (2006), 084106. |
[12] |
R. Erban and H. Othmer, From individual to collective behaviour in bacterial chemotaxis, SIAM Journal on Applied Mathematics, 65 (2004), 361-391.
doi: 10.1137/S0036139903433232. |
[13] |
R. Erban and H. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multi-scale modeling in biology>, Multiscale Modeling and Simulation, 3 (2005), 362-394. |
[14] |
C. Escudero, C. Yates, J. Buhl, I. Couzin, R. Erban, I. Kevrekidis and P. Maini, Ergodic directional switching in mobile insect groups, Physical Review E, 82 (2010), 011926. |
[15] |
L. Evans, "Partial Differential Equations,'' American Mathematical Society, Providence, Rhode Island, 1998. |
[16] |
W. Feller, "An Introduction to Probability Theory and its Applications,'' $3^{rd}$ edition, Viley, New York, Sydney, 1967. |
[17] |
M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via the povzner-boltzmann equation, Physica D: Nonlinear Phenomena, 240 (2011), 21-31.
doi: 10.1016/j.physd.2010.08.003. |
[18] |
D. Gillespie, "Markov Processes, An Introduction for Physical Scientists,'' Academic Press, Inc., Harcourt Brace Jovanowich, 1992. |
[19] |
S. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. |
[20] |
P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after Kramers, Reviews of Modern Physics, 62 (1990), 251-341.
doi: 10.1103/RevModPhys.62.251. |
[21] |
J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, Journal of Statistical Physics, 135 (2009), 133-151.
doi: 10.1007/s10955-009-9717-1. |
[22] |
J. Haskovec and C. Schmeiser, Convergence analysis of a stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, Communications in Partial Differential Equations, 36 (2011), 940-960.
doi: 10.1080/03605302.2010.538783. |
[23] |
N. Hill and D. Häder, A biased random walk model for the trajectories of swimming micro-organisms, Journal of Theoretical Biology, 186 (1997), 503-526.
doi: 10.1006/jtbi.1997.0421. |
[24] |
M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain Journal of Mathematics, 4 (1974), 497-509. |
[25] |
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[26] |
W. Rudin, "Functional Analysis,'' McGraw-Hill Science, 1991. |
[27] |
A. Sznitman, "Topics in Propagation of Chaos,'' Lecture notes in mathematics, 1464, Springer-Verlag, 1991. |
[28] |
N. van Kampen, "Stochastic Processes in Physics and Chemistry,'' $3^{rd}$ edition, North-Holland, Amsterdam, 2007. |
[29] |
C. Yates, R. Erban, C. Escudero, I. Couzin, J. Buhl, I. Kevrekidis, P. Maini and D. Sumpter, Inherent noise can facilitate coherence in collective swarm motion, Proceedings of the National Academy of Sciences USA, 106 (2009), 5464-5469.
doi: 10.1073/pnas.0811195106. |
show all references
References:
[1] |
L. Baum and M. Katz, Convergence rates in the law of large numbers, Transactions of the American Mathematical Society, 120 (1965), 108-123.
doi: 10.1090/S0002-9947-1965-0198524-1. |
[2] |
S. Bazazi, J. Buhl, J. Hale, M. Anstey, G. Sword, S. Simpson and I. Couzin, Collective motion and cannibalism in locust migratory bands, Current Biology, 18 (2008), 735-739.
doi: 10.1016/j.cub.2008.04.035. |
[3] |
J. Buhl, D. Sumpter, I. Couzin, J. Hale, E. Despland, E. Miller and S. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.
doi: 10.1126/science.1125142. |
[4] |
J. Buhl, G. Sword, F. Clissold and S. Simpson, Group structure in locust migratory bands, Behav. Ecol. Sociobiol., 65 (2011), 265-273.
doi: 10.1007/s00265-010-1041-x. |
[5] |
J. Carrillo, M. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic and Related Models, 2 (2009), 363-378. |
[6] |
J. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM Journal on Mathematical Analysis, 42 (2010), 218-236.
doi: 10.1137/090757290. |
[7] |
J. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, hydrodynamic models of swarming, in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences'' (eds. G. Naldi, L. Pareschi and G. Toscani), Modelling and Simulation in Science and Technology, Birkhäuser, (2010), 297-336. |
[8] |
C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'' Applied Mathematical Sciences, 106, Springer-Verlag, 1994. |
[9] |
E. Codling and N. Hill, Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters, Journal of Mathematical Biology, 51 (2005), 527-556.
doi: 10.1007/s00285-005-0317-7. |
[10] |
A. Czirók, A. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Physical Review Letters, 82 (1999), 209-212.
doi: 10.1103/PhysRevLett.82.209. |
[11] |
R. Erban, I. Kevrekidis, D. Adalsteinsson and T. Elston, Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation, Journal of Chemical Physics, 124 (2006), 084106. |
[12] |
R. Erban and H. Othmer, From individual to collective behaviour in bacterial chemotaxis, SIAM Journal on Applied Mathematics, 65 (2004), 361-391.
doi: 10.1137/S0036139903433232. |
[13] |
R. Erban and H. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multi-scale modeling in biology>, Multiscale Modeling and Simulation, 3 (2005), 362-394. |
[14] |
C. Escudero, C. Yates, J. Buhl, I. Couzin, R. Erban, I. Kevrekidis and P. Maini, Ergodic directional switching in mobile insect groups, Physical Review E, 82 (2010), 011926. |
[15] |
L. Evans, "Partial Differential Equations,'' American Mathematical Society, Providence, Rhode Island, 1998. |
[16] |
W. Feller, "An Introduction to Probability Theory and its Applications,'' $3^{rd}$ edition, Viley, New York, Sydney, 1967. |
[17] |
M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via the povzner-boltzmann equation, Physica D: Nonlinear Phenomena, 240 (2011), 21-31.
doi: 10.1016/j.physd.2010.08.003. |
[18] |
D. Gillespie, "Markov Processes, An Introduction for Physical Scientists,'' Academic Press, Inc., Harcourt Brace Jovanowich, 1992. |
[19] |
S. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435. |
[20] |
P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after Kramers, Reviews of Modern Physics, 62 (1990), 251-341.
doi: 10.1103/RevModPhys.62.251. |
[21] |
J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, Journal of Statistical Physics, 135 (2009), 133-151.
doi: 10.1007/s10955-009-9717-1. |
[22] |
J. Haskovec and C. Schmeiser, Convergence analysis of a stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, Communications in Partial Differential Equations, 36 (2011), 940-960.
doi: 10.1080/03605302.2010.538783. |
[23] |
N. Hill and D. Häder, A biased random walk model for the trajectories of swimming micro-organisms, Journal of Theoretical Biology, 186 (1997), 503-526.
doi: 10.1006/jtbi.1997.0421. |
[24] |
M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain Journal of Mathematics, 4 (1974), 497-509. |
[25] |
H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[26] |
W. Rudin, "Functional Analysis,'' McGraw-Hill Science, 1991. |
[27] |
A. Sznitman, "Topics in Propagation of Chaos,'' Lecture notes in mathematics, 1464, Springer-Verlag, 1991. |
[28] |
N. van Kampen, "Stochastic Processes in Physics and Chemistry,'' $3^{rd}$ edition, North-Holland, Amsterdam, 2007. |
[29] |
C. Yates, R. Erban, C. Escudero, I. Couzin, J. Buhl, I. Kevrekidis, P. Maini and D. Sumpter, Inherent noise can facilitate coherence in collective swarm motion, Proceedings of the National Academy of Sciences USA, 106 (2009), 5464-5469.
doi: 10.1073/pnas.0811195106. |
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