# American Institute of Mathematical Sciences

December  2012, 5(4): 817-842. doi: 10.3934/krm.2012.5.817

## 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

Received  September 2011 Revised  August 2012 Published  November 2012

A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on the behaviour of locusts. It exhibits nontrivial dynamics with a pitchfork bifurcation and recovers the observed group directional switching. Estimates of the expected switching times, in terms of the number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations (with nonlocal and nonlinear right hand side) is derived and analyzed. The existence of its solutions is proven and the system's long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model.
Citation: Radek Erban, Jan Haskovec. From individual to collective behaviour of coupled velocity jump processes: A locust example. Kinetic & Related Models, 2012, 5 (4) : 817-842. doi: 10.3934/krm.2012.5.817
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] J. 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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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [15] L. Evans, "Partial Differential Equations,'' American Mathematical Society, Providence, Rhode Island, 1998. Google Scholar [16] W. Feller, "An Introduction to Probability Theory and its Applications,'' $3^{rd}$ edition, Viley, New York, Sydney, 1967.  Google Scholar [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.  Google Scholar [18] D. Gillespie, "Markov Processes, An Introduction for Physical Scientists,'' Academic Press, Inc., Harcourt Brace Jovanowich, 1992.  Google Scholar [19] S. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain Journal of Mathematics, 4 (1974), 497-509.  Google Scholar [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.  Google Scholar [26] W. Rudin, "Functional Analysis,'' McGraw-Hill Science, 1991.  Google Scholar [27] A. Sznitman, "Topics in Propagation of Chaos,'' Lecture notes in mathematics, 1464, Springer-Verlag, 1991.  Google Scholar [28] N. van Kampen, "Stochastic Processes in Physics and Chemistry,'' $3^{rd}$ edition, North-Holland, Amsterdam, 2007. Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,'' Applied Mathematical Sciences, 106, Springer-Verlag, 1994.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [15] L. Evans, "Partial Differential Equations,'' American Mathematical Society, Providence, Rhode Island, 1998. Google Scholar [16] W. Feller, "An Introduction to Probability Theory and its Applications,'' $3^{rd}$ edition, Viley, New York, Sydney, 1967.  Google Scholar [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.  Google Scholar [18] D. Gillespie, "Markov Processes, An Introduction for Physical Scientists,'' Academic Press, Inc., Harcourt Brace Jovanowich, 1992.  Google Scholar [19] S. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain Journal of Mathematics, 4 (1974), 497-509.  Google Scholar [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.  Google Scholar [26] W. Rudin, "Functional Analysis,'' McGraw-Hill Science, 1991.  Google Scholar [27] A. Sznitman, "Topics in Propagation of Chaos,'' Lecture notes in mathematics, 1464, Springer-Verlag, 1991.  Google Scholar [28] N. van Kampen, "Stochastic Processes in Physics and Chemistry,'' $3^{rd}$ edition, North-Holland, Amsterdam, 2007. Google Scholar [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.  Google Scholar
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