From individual to collective behaviour of coupled velocity jump processes: A locust example

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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

Radek Erban ^1, and Jan Haskovec ^2,

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

Abstract
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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.

Keywords: Collective behaviour, density-dependent directional switching, kinetic equation., stochastic individual-based model.

Mathematics Subject Classification: 60G35, 35L40, 92B0.

Citation: Radek Erban, Jan Haskovec. From individual to collective behaviour of coupled velocity jump processes: A locust example. Kinetic and 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.
[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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