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On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces
Non-local kinetic and macroscopic models for self-organised animal aggregations
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 |
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), 196-209.
doi: 10.1016/j.mcm.2007.02.016. |
[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), 121-153.
doi: 10.1081/TT-100105365. |
[3] |
A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Cont Dyn Syst B., 19 (2014), 1249-1278. |
[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), 9703-9706.
doi: 10.1073/pnas.161285298. |
[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), 111-136.
doi: 10.1080/1027336042000288633. |
[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), 1675-1692.
doi: 10.1142/S0218202507002431. |
[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), 144-175.
doi: 10.1016/j.plrev.2009.06.002. |
[8] |
M. G. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Analysis RWA, 9 (2008), 183-196.
doi: 10.1016/j.nonrwa.2006.09.012. |
[9] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[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), 1-33 (electronic). |
[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), 1757-1779.
doi: 10.1002/mma.638. |
[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), 331-339. |
[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), 1402-1406.
doi: 10.1126/science.1125142. |
[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), 327-357.
doi: 10.1142/S0218202513400101. |
[15] |
P.-L. Buono and R. Eftimie, Codimension-two bifurcations in animal aggregation models with symmetry, SIAM J. Appl. Dyn. Syst., 13 (2014), 1542-1582.
doi: 10.1137/130932272. |
[16] |
P.-L. Buono and R. Eftimie, Symmetries and pattern formation in hyperbolic versus parabolic models for self-organised aggregations, J. Math. Biol., (2014), 1-35.
doi: 10.1007/s00285-014-0842-3. |
[17] |
M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis RWA, 8 (2007), 939-958.
doi: 10.1016/j.nonrwa.2006.04.002. |
[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), 363-378.
doi: 10.3934/krm.2009.2.363. |
[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), 218-236.
doi: 10.1137/090757290. |
[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, L. Pareschi and G. Toscani), Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010, 297-336.
doi: 10.1007/978-0-8176-4946-3_12. |
[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), 113-149.
doi: 10.1007/s10915-007-9181-5. |
[22] |
J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for quasi-morse potentials, European J. Appl. Math., 25 (2014), 553-578.
doi: 10.1017/S0956792514000126. |
[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), 1533-1552.
doi: 10.1142/S0218202510004684. |
[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), 336-361.
doi: 10.1137/110851687. |
[25] |
A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions, Math Models Methods Appl. Sci., 24 (2014), 249-275.
doi: 10.1142/S0218202513400083. |
[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), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[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), 277-325.
doi: 10.1142/S0218202513400095. |
[28] |
P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Acad. Sci. Paris Ser. I, 345 (2007), 555-560.
doi: 10.1016/j.crma.2007.10.024. |
[29] |
P. Degond and S. Motsch, Large scale dynamics of the persistent turning awlker model of fish behaviour, J. Stat. Phys., 131 (2008), 989-1021.
doi: 10.1007/s10955-008-9529-8. |
[30] |
R. Eftimie, Modeling Group Formation and Activity Patterns in Self-Organizing Communities of Organisms, PhD thesis, University of Alberta, 2008. |
[31] |
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. |
[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), 6974-6979.
doi: 10.1073/pnas.0611483104. |
[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), 1537-1565.
doi: 10.1007/s11538-006-9175-8. |
[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), 37-74.
doi: 10.1007/s00285-008-0209-8. |
[35] |
R. Fetecau, Collective behavior of biological aggregations in two dimensions: A nonlocal kinetic model, Math. Model. Method. Appl. Sci., 21 (2011), 1539-1569.
doi: 10.1142/S0218202511005489. |
[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), 787-809. |
[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), 1245-1279.
doi: 10.1137/040621041. |
[38] |
T. Goudon, On Boltzmann equations and Fokker-Plank asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776.
doi: 10.1007/BF02765543. |
[39] |
S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
[40] |
C. K. Hemelrijk and H. Kunz, Density distribution and size sorting in fish schools: An individual-based model, Behav. Ecol., 16 (2004), 178-187.
doi: 10.1093/beheco/arh149. |
[41] |
H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organised complex aerial displays of thousands of starlings: A model, Behavioral Ecology, 107 (2010), 1349-1359. |
[42] |
T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump process, SIAM J. Appl. Math., 61 (2000), 751-775.
doi: 10.1137/S0036139999358167. |
[43] |
E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion, Am. Nat., 142 (1993), 779-795.
doi: 10.1086/285572. |
[44] |
A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), 1073-1094 (electronic).
doi: 10.1137/S0036142996305558. |
[45] |
A. Klar, An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal., 36 (1999), 1507-1527 (electronic).
doi: 10.1137/S0036142997321765. |
[46] |
R. Larkin and R. Szafoni, Evidence for widely dispersed birds migrating together at night, Integrative and Comparative Biology, 48 (2008), 40-49. |
[47] |
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[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), 811-842.
doi: 10.1007/s002850050032. |
[49] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[50] |
I. Newton, The Migration Ecology of Birds, Academic Press, Elsevier, 2008. |
[51] |
H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[52] |
R. D. Passo and P. de Mottoni, Aggregative effects for a reaction-advection equation, J. Math. Biology, 20 (1984), 103-112.
doi: 10.1007/BF00275865. |
[53] |
B. Pfistner, A one dimensional model for the swarming behaviour of Myxobacteria, in Biological Motion (eds. W. Alt and G. Hoffmann), Lecture Notes on Biomathematics, 89, Springer, 1990, 556-563. |
[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), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[55] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[56] |
F. Venuti, L. Bruno and N. Bellomo, Crowd dynamics on a moving platform: Mathematical modelling and application to lively footbridges, Math. Comp. Model., 45 (2007), 252-269.
doi: 10.1016/j.mcm.2006.04.007. |
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), 196-209.
doi: 10.1016/j.mcm.2007.02.016. |
[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), 121-153.
doi: 10.1081/TT-100105365. |
[3] |
A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Cont Dyn Syst B., 19 (2014), 1249-1278. |
[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), 9703-9706.
doi: 10.1073/pnas.161285298. |
[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), 111-136.
doi: 10.1080/1027336042000288633. |
[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), 1675-1692.
doi: 10.1142/S0218202507002431. |
[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), 144-175.
doi: 10.1016/j.plrev.2009.06.002. |
[8] |
M. G. Bertotti and M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Analysis RWA, 9 (2008), 183-196.
doi: 10.1016/j.nonrwa.2006.09.012. |
[9] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[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), 1-33 (electronic). |
[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), 1757-1779.
doi: 10.1002/mma.638. |
[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), 331-339. |
[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), 1402-1406.
doi: 10.1126/science.1125142. |
[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), 327-357.
doi: 10.1142/S0218202513400101. |
[15] |
P.-L. Buono and R. Eftimie, Codimension-two bifurcations in animal aggregation models with symmetry, SIAM J. Appl. Dyn. Syst., 13 (2014), 1542-1582.
doi: 10.1137/130932272. |
[16] |
P.-L. Buono and R. Eftimie, Symmetries and pattern formation in hyperbolic versus parabolic models for self-organised aggregations, J. Math. Biol., (2014), 1-35.
doi: 10.1007/s00285-014-0842-3. |
[17] |
M. Burger, V. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Analysis RWA, 8 (2007), 939-958.
doi: 10.1016/j.nonrwa.2006.04.002. |
[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), 363-378.
doi: 10.3934/krm.2009.2.363. |
[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), 218-236.
doi: 10.1137/090757290. |
[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, L. Pareschi and G. Toscani), Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Inc., Boston, MA, 2010, 297-336.
doi: 10.1007/978-0-8176-4946-3_12. |
[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), 113-149.
doi: 10.1007/s10915-007-9181-5. |
[22] |
J. A. Carrillo, Y. Huang and S. Martin, Explicit flock solutions for quasi-morse potentials, European J. Appl. Math., 25 (2014), 553-578.
doi: 10.1017/S0956792514000126. |
[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), 1533-1552.
doi: 10.1142/S0218202510004684. |
[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), 336-361.
doi: 10.1137/110851687. |
[25] |
A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev, Pedestrian flow models with slowdown interactions, Math Models Methods Appl. Sci., 24 (2014), 249-275.
doi: 10.1142/S0218202513400083. |
[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), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[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), 277-325.
doi: 10.1142/S0218202513400095. |
[28] |
P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Acad. Sci. Paris Ser. I, 345 (2007), 555-560.
doi: 10.1016/j.crma.2007.10.024. |
[29] |
P. Degond and S. Motsch, Large scale dynamics of the persistent turning awlker model of fish behaviour, J. Stat. Phys., 131 (2008), 989-1021.
doi: 10.1007/s10955-008-9529-8. |
[30] |
R. Eftimie, Modeling Group Formation and Activity Patterns in Self-Organizing Communities of Organisms, PhD thesis, University of Alberta, 2008. |
[31] |
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. |
[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), 6974-6979.
doi: 10.1073/pnas.0611483104. |
[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), 1537-1565.
doi: 10.1007/s11538-006-9175-8. |
[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), 37-74.
doi: 10.1007/s00285-008-0209-8. |
[35] |
R. Fetecau, Collective behavior of biological aggregations in two dimensions: A nonlocal kinetic model, Math. Model. Method. Appl. Sci., 21 (2011), 1539-1569.
doi: 10.1142/S0218202511005489. |
[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), 787-809. |
[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), 1245-1279.
doi: 10.1137/040621041. |
[38] |
T. Goudon, On Boltzmann equations and Fokker-Plank asymptotics: Influence of grazing collisions, J. Stat. Phys., 89 (1997), 751-776.
doi: 10.1007/BF02765543. |
[39] |
S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
[40] |
C. K. Hemelrijk and H. Kunz, Density distribution and size sorting in fish schools: An individual-based model, Behav. Ecol., 16 (2004), 178-187.
doi: 10.1093/beheco/arh149. |
[41] |
H. Hildenbrandt, C. Carere and C. K. Hemelrijk, Self-organised complex aerial displays of thousands of starlings: A model, Behavioral Ecology, 107 (2010), 1349-1359. |
[42] |
T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump process, SIAM J. Appl. Math., 61 (2000), 751-775.
doi: 10.1137/S0036139999358167. |
[43] |
E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion, Am. Nat., 142 (1993), 779-795.
doi: 10.1086/285572. |
[44] |
A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35 (1998), 1073-1094 (electronic).
doi: 10.1137/S0036142996305558. |
[45] |
A. Klar, An asymptotic preserving numerical scheme for kinetic equations in the low Mach number limit, SIAM J. Numer. Anal., 36 (1999), 1507-1527 (electronic).
doi: 10.1137/S0036142997321765. |
[46] |
R. Larkin and R. Szafoni, Evidence for widely dispersed birds migrating together at night, Integrative and Comparative Biology, 48 (2008), 40-49. |
[47] |
A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[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), 811-842.
doi: 10.1007/s002850050032. |
[49] |
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66.
doi: 10.1007/s00285-004-0279-1. |
[50] |
I. Newton, The Migration Ecology of Birds, Academic Press, Elsevier, 2008. |
[51] |
H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.
doi: 10.1137/S0036139900382772. |
[52] |
R. D. Passo and P. de Mottoni, Aggregative effects for a reaction-advection equation, J. Math. Biology, 20 (1984), 103-112.
doi: 10.1007/BF00275865. |
[53] |
B. Pfistner, A one dimensional model for the swarming behaviour of Myxobacteria, in Biological Motion (eds. W. Alt and G. Hoffmann), Lecture Notes on Biomathematics, 89, Springer, 1990, 556-563. |
[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), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[55] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Bio., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
[56] |
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