American Institute of Mathematical Sciences

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September  2015, 8(3): 413-441. doi: 10.3934/krm.2015.8.413

Non-local kinetic and macroscopic models for self-organised animal aggregations

José A. Carrillo 1, , Raluca Eftimie 2, and  Franca Hoffmann 3,

1. 

Department of Mathematics, Imperial College, London, London SW7 2AZ
2. 

Division of Mathematics, University of Dundee, Dundee, DD1 4HN, United Kingdom
3. 

University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Received  November 2014 Revised  February 2015 Published  June 2015

The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient $\epsilon$ is varied from $\epsilon=1$ (for 1D transport models) to $\epsilon=0$ (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit $\epsilon\to 0$, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams.
Keywords: macroscopic models, Self-organised aggregations, asymptotic preserving methods., non-local interactions, kinetic models.
Mathematics Subject Classification: Primary: 35L40, 35L65, 35K55, 92B99; Secondary: 82C40, 65M0.
Citation: José A. Carrillo, Raluca Eftimie, Franca Hoffmann. Non-local kinetic and macroscopic models for self-organised animal aggregations. Kinetic & Related Models, 2015, 8 (3) : 413-441. doi: 10.3934/krm.2015.8.413
References:
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N. Bellomo, E. D. Angelis and L. Preziosi, Multiscale modeling and mathematical problems related to tumor evolution and medical therapy,, Journal of Theoretical Medicine, 5 (2003), 111. doi: 10.1080/1027336042000288633. Google Scholar

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N. Bellomo, A. Bellouquid, J. Nieto and J.Soler, Multicellular biological growing systems: Hyperbolic limits towards macroscopic description,, Math Models Appl. Sci., 17 (2007), 1675. doi: 10.1142/S0218202507002431. Google Scholar

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N. Bellomo, C. Bianca and M. Delitala, Complexity analysis and mathematical tools towards the modelling of living systems,, Physics of Life Reviews, 6 (2009), 144. doi: 10.1016/j.plrev.2009.06.002. Google Scholar

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M. G. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics,, Nonlinear Analysis RWA, 9 (2008), 183. doi: 10.1016/j.nonrwa.2006.09.012. Google Scholar

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A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683. doi: 10.1088/0951-7715/22/3/009. Google Scholar

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A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron J. Diff. Eq., 44 (2006), 1. Google Scholar

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R. Breitwisch and G. Whitesides, {Directionality of singing and non-singing behaviour of mated and unmated Northern Mockingbirds, Mimus polyglottos},, Anim. Behav., 35 (1987), 331. Google Scholar

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J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kinetic and Related Models, 2 (2009), 363. doi: 10.3934/krm.2009.2.363. Google Scholar

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J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM J. Math. Anal., 42 (2010), 218. doi: 10.1137/090757290. Google Scholar

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J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming,, in Mathematical Modelling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar

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J. A. Carrillo, T. Goudon, P. Lafitte and F. Vecil, Numerical schemes of diffusion asymptotics and moment closures for kinetic equations,, J. Sci. Comput., 36 (2008), 113. doi: 10.1007/s10915-007-9181-5. Google Scholar

[22]

J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for quasi-morse potentials,, European J. Appl. Math., 25 (2014), 553. doi: 10.1017/S0956792514000126. Google Scholar

[23]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force,, Math. Models Methods Appl. Sci., 20 (2010), 1533. doi: 10.1142/S0218202510004684. Google Scholar

[24]

J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis,, Multiscale Model. Simul., 11 (2013), 336. doi: 10.1137/110851687. Google Scholar

[25]

A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions,, Math Models Methods Appl. Sci., 24 (2014), 249. doi: 10.1142/S0218202513400083. Google Scholar

[26]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2d interactiong, self-propelled particle system,, Physica D, 232 (2007), 33. doi: 10.1016/j.physd.2007.05.007. Google Scholar

[27]

P. Degond, G. Dimarco and T. Mac, Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles,, Math Models Appl. Sci., 24 (2014), 277. doi: 10.1142/S0218202513400095. Google Scholar

[28]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction,, C.R. Acad. Sci. Paris Ser. I, 345 (2007), 555. doi: 10.1016/j.crma.2007.10.024. Google Scholar

[29]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning awlker model of fish behaviour,, J. Stat. Phys., 131 (2008), 989. doi: 10.1007/s10955-008-9529-8. Google Scholar

[30]

R. Eftimie, Modeling Group Formation and Activity Patterns in Self-Organizing Communities of Organisms,, PhD thesis, (2008). Google Scholar

[31]

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

[32]

R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms,, Proc. Natl. Acad. Sci., 104 (2007), 6974. doi: 10.1073/pnas.0611483104. Google Scholar

[33]

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. doi: 10.1007/s11538-006-9175-8. Google Scholar

[34]

R. Eftimie, G. de Vries and M. Lewis, Weakly nonlinear analysis of a hyperbolic model for animal group formation,, J. Math. Biol., 59 (2009), 37. doi: 10.1007/s00285-008-0209-8. Google Scholar

[35]

R. Fetecau, Collective behavior of biological aggregations in two dimensions: A nonlocal kinetic model,, Math. Model. Method. Appl. Sci., 21 (2011), 1539. doi: 10.1142/S0218202511005489. Google Scholar

[36]

E. Geigant, K. Ladizhansky and A. Mogilner, An integrodifferential model for orientational distributions of {F-actin} in cells,, SIAM J. Appl. Math., 59 (1998), 787. Google Scholar

[37]

P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium, and nonequilibrium diffusion asymptotics,, Multiscale Model. Simul., 4 (2005), 1245. doi: 10.1137/040621041. Google Scholar

[38]

T. Goudon, On Boltzmann equations and Fokker-Plank asymptotics: Influence of grazing collisions,, J. Stat. Phys., 89 (1997), 751. doi: 10.1007/BF02765543. Google Scholar

[39]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,, Kinetic and Related Models, 1 (2008), 415. doi: 10.3934/krm.2008.1.415. Google Scholar

[40]

C. K. Hemelrijk and H. Kunz, Density distribution and size sorting in fish schools: An individual-based model,, Behav. Ecol., 16 (2004), 178. doi: 10.1093/beheco/arh149. Google Scholar

[41]

H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organised complex aerial displays of thousands of starlings: A model,, Behavioral Ecology, 107 (2010), 1349. Google Scholar

[42]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump process,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar

[43]

E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion,, Am. Nat., 142 (1993), 779. doi: 10.1086/285572. Google Scholar

[44]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit,, SIAM J. Numer. Anal., 35 (1998), 1073. doi: 10.1137/S0036142996305558. Google Scholar

[45]

A. Klar, An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit,, SIAM J. Numer. Anal., 36 (1999), 1507. doi: 10.1137/S0036142997321765. Google Scholar

[46]

R. Larkin and R. Szafoni, Evidence for widely dispersed birds migrating together at night,, Integrative and Comparative Biology, 48 (2008), 40. Google Scholar

[47]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534. doi: 10.1007/s002850050158. Google Scholar

[48]

A. Mogilner, L. Edelstein-Keshet and G. B. Ermentrout, Selecting a common direction. II. Peak-like solutions representing total alignment of cell clusters,, J. Math. Biol., 34 (1996), 811. doi: 10.1007/s002850050032. Google Scholar

[49]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations,, J. Math. Biol., 50 (2005), 49. doi: 10.1007/s00285-004-0279-1. Google Scholar

[50]

I. Newton, The Migration Ecology of Birds,, Academic Press, (2008). Google Scholar

[51]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[52]

R. D. Passo and P. de Mottoni, Aggregative effects for a reaction-advection equation,, J. Math. Biology, 20 (1984), 103. doi: 10.1007/BF00275865. Google Scholar

[53]

B. Pfistner, A one dimensional model for the swarming behaviour of Myxobacteria,, in Biological Motion (eds. W. Alt and G. Hoffmann), (1990), 556. Google Scholar

[54]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,, PLoS Comput. Biol., 6 (2010). doi: 10.1371/journal.pcbi.1000890. Google Scholar

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C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation,, Bull. Math. Bio., 68 (2006), 1601. doi: 10.1007/s11538-006-9088-6. Google Scholar

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show all references

References:
[1]

E. D. Angelis and B. Lods, On the kinetic theory for active particles: A model for tumor-immune system competition,, Math. Comp. Model., 47 (2008), 196. doi: 10.1016/j.mcm.2007.02.016. Google Scholar

[2]

A. Arnold, J. A. Carrillo, I. Gamba and C.-w. Shu, Low and high field scaling limits for the vlasov- and wigner-poisson-fokker-planck systems,, Transp. Theory Stat. Phys., 30 (2001), 121. doi: 10.1081/TT-100105365. Google Scholar

[3]

A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents,, Discrete Cont Dyn Syst B., 19 (2014), 1249. Google Scholar

[4]

M. Beekman, D. J. T. Sumpter and F. L. W. Ratnieks, Phase transitions between disordered and ordered foraging in pharaoh's ants,, Proc. Natl. Acad. Sci., 98 (2001), 9703. doi: 10.1073/pnas.161285298. Google Scholar

[5]

N. Bellomo, E. D. Angelis and L. Preziosi, Multiscale modeling and mathematical problems related to tumor evolution and medical therapy,, Journal of Theoretical Medicine, 5 (2003), 111. doi: 10.1080/1027336042000288633. Google Scholar

[6]

N. Bellomo, A. Bellouquid, J. Nieto and J.Soler, Multicellular biological growing systems: Hyperbolic limits towards macroscopic description,, Math Models Appl. Sci., 17 (2007), 1675. doi: 10.1142/S0218202507002431. Google Scholar

[7]

N. Bellomo, C. Bianca and M. Delitala, Complexity analysis and mathematical tools towards the modelling of living systems,, Physics of Life Reviews, 6 (2009), 144. doi: 10.1016/j.plrev.2009.06.002. Google Scholar

[8]

M. G. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics,, Nonlinear Analysis RWA, 9 (2008), 183. doi: 10.1016/j.nonrwa.2006.09.012. Google Scholar

[9]

A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels,, Nonlinearity, 22 (2009), 683. doi: 10.1088/0951-7715/22/3/009. Google Scholar

[10]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions,, Electron J. Diff. Eq., 44 (2006), 1. Google Scholar

[11]

M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: A formal approach,, Math. Meth. Appl. Sci., 28 (2005), 1757. doi: 10.1002/mma.638. Google Scholar

[12]

R. Breitwisch and G. Whitesides, {Directionality of singing and non-singing behaviour of mated and unmated Northern Mockingbirds, Mimus polyglottos},, Anim. Behav., 35 (1987), 331. Google Scholar

[13]

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. doi: 10.1126/science.1125142. Google Scholar

[14]

P.-L. Buono and R. Eftimie, Analysis of Hopf/Hopf bifurcations in nonlocal hyperbolic models for self-organised aggregations,, Math. Models Methods Appl. Sci., 24 (2014), 327. doi: 10.1142/S0218202513400101. Google Scholar

[15]

P.-L. Buono and R. Eftimie, Codimension-two bifurcations in animal aggregation models with symmetry,, SIAM J. Appl. Dyn. Syst., 13 (2014), 1542. doi: 10.1137/130932272. Google Scholar

[16]

P.-L. Buono and R. Eftimie, Symmetries and pattern formation in hyperbolic versus parabolic models for self-organised aggregations,, J. Math. Biol., (2014), 1. doi: 10.1007/s00285-014-0842-3. Google Scholar

[17]

M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions,, Nonlinear Analysis RWA, 8 (2007), 939. doi: 10.1016/j.nonrwa.2006.04.002. Google Scholar

[18]

J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory,, Kinetic and Related Models, 2 (2009), 363. doi: 10.3934/krm.2009.2.363. Google Scholar

[19]

J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model,, SIAM J. Math. Anal., 42 (2010), 218. doi: 10.1137/090757290. Google Scholar

[20]

J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming,, in Mathematical Modelling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar

[21]

J. A. Carrillo, T. Goudon, P. Lafitte and F. Vecil, Numerical schemes of diffusion asymptotics and moment closures for kinetic equations,, J. Sci. Comput., 36 (2008), 113. doi: 10.1007/s10915-007-9181-5. Google Scholar

[22]

J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for quasi-morse potentials,, European J. Appl. Math., 25 (2014), 553. doi: 10.1017/S0956792514000126. Google Scholar

[23]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force,, Math. Models Methods Appl. Sci., 20 (2010), 1533. doi: 10.1142/S0218202510004684. Google Scholar

[24]

J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis,, Multiscale Model. Simul., 11 (2013), 336. doi: 10.1137/110851687. Google Scholar

[25]

A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions,, Math Models Methods Appl. Sci., 24 (2014), 249. doi: 10.1142/S0218202513400083. Google Scholar

[26]

Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2d interactiong, self-propelled particle system,, Physica D, 232 (2007), 33. doi: 10.1016/j.physd.2007.05.007. Google Scholar

[27]

P. Degond, G. Dimarco and T. Mac, Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles,, Math Models Appl. Sci., 24 (2014), 277. doi: 10.1142/S0218202513400095. Google Scholar

[28]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction,, C.R. Acad. Sci. Paris Ser. I, 345 (2007), 555. doi: 10.1016/j.crma.2007.10.024. Google Scholar

[29]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning awlker model of fish behaviour,, J. Stat. Phys., 131 (2008), 989. doi: 10.1007/s10955-008-9529-8. Google Scholar

[30]

R. Eftimie, Modeling Group Formation and Activity Patterns in Self-Organizing Communities of Organisms,, PhD thesis, (2008). Google Scholar

[31]

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

[32]

R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms,, Proc. Natl. Acad. Sci., 104 (2007), 6974. doi: 10.1073/pnas.0611483104. Google Scholar

[33]

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. doi: 10.1007/s11538-006-9175-8. Google Scholar

[34]

R. Eftimie, G. de Vries and M. Lewis, Weakly nonlinear analysis of a hyperbolic model for animal group formation,, J. Math. Biol., 59 (2009), 37. doi: 10.1007/s00285-008-0209-8. Google Scholar

[35]

R. Fetecau, Collective behavior of biological aggregations in two dimensions: A nonlocal kinetic model,, Math. Model. Method. Appl. Sci., 21 (2011), 1539. doi: 10.1142/S0218202511005489. Google Scholar

[36]

E. Geigant, K. Ladizhansky and A. Mogilner, An integrodifferential model for orientational distributions of {F-actin} in cells,, SIAM J. Appl. Math., 59 (1998), 787. Google Scholar

[37]

P. Godillon-Lafitte and T. Goudon, A coupled model for radiative transfer: Doppler effects, equilibrium, and nonequilibrium diffusion asymptotics,, Multiscale Model. Simul., 4 (2005), 1245. doi: 10.1137/040621041. Google Scholar

[38]

T. Goudon, On Boltzmann equations and Fokker-Plank asymptotics: Influence of grazing collisions,, J. Stat. Phys., 89 (1997), 751. doi: 10.1007/BF02765543. Google Scholar

[39]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking,, Kinetic and Related Models, 1 (2008), 415. doi: 10.3934/krm.2008.1.415. Google Scholar

[40]

C. K. Hemelrijk and H. Kunz, Density distribution and size sorting in fish schools: An individual-based model,, Behav. Ecol., 16 (2004), 178. doi: 10.1093/beheco/arh149. Google Scholar

[41]

H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organised complex aerial displays of thousands of starlings: A model,, Behavioral Ecology, 107 (2010), 1349. Google Scholar

[42]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump process,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar

[43]

E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion,, Am. Nat., 142 (1993), 779. doi: 10.1086/285572. Google Scholar

[44]

A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit,, SIAM J. Numer. Anal., 35 (1998), 1073. doi: 10.1137/S0036142996305558. Google Scholar

[45]

A. Klar, An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit,, SIAM J. Numer. Anal., 36 (1999), 1507. doi: 10.1137/S0036142997321765. Google Scholar

[46]

R. Larkin and R. Szafoni, Evidence for widely dispersed birds migrating together at night,, Integrative and Comparative Biology, 48 (2008), 40. Google Scholar

[47]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm,, J. Math. Biol., 38 (1999), 534. doi: 10.1007/s002850050158. Google Scholar

[48]

A. Mogilner, L. Edelstein-Keshet and G. B. Ermentrout, Selecting a common direction. II. Peak-like solutions representing total alignment of cell clusters,, J. Math. Biol., 34 (1996), 811. doi: 10.1007/s002850050032. Google Scholar

[49]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations,, J. Math. Biol., 50 (2005), 49. doi: 10.1007/s00285-004-0279-1. Google Scholar

[50]

I. Newton, The Migration Ecology of Birds,, Academic Press, (2008). Google Scholar

[51]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar

[52]

R. D. Passo and P. de Mottoni, Aggregative effects for a reaction-advection equation,, J. Math. Biology, 20 (1984), 103. doi: 10.1007/BF00275865. Google Scholar

[53]

B. Pfistner, A one dimensional model for the swarming behaviour of Myxobacteria,, in Biological Motion (eds. W. Alt and G. Hoffmann), (1990), 556. Google Scholar

[54]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,, PLoS Comput. Biol., 6 (2010). doi: 10.1371/journal.pcbi.1000890. Google Scholar

[55]

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