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Numerical approximation of a coagulationfragmentation model for animal group size statistics
Department of Mathematics, Imperial College London South Kensington Campus London SW7 2AZ, UK 
We study numerically a coagulationfragmentation model derived by Niwa [
References:
[1] 
E. Bonabeau, L. Dagorn, Possible universality in the size distribution of fish schools, Phys. Rev. E, 51 (1995), 52205223. doi: 10.1103/PhysRevE.51.R5220. 
[2] 
E. Bonabeau, L. Dagorn, P. Freon, Scaling in animal groupsize distributions, Proc. Natl. Acad. Sci. USA, 96 (1999), 44724477. doi: 10.1073/pnas.96.8.4472. 
[3] 
J. P. Bourgade, F. Filbet, Convergence of a finite volume scheme for coagulationfragmentation equations, Comm. Math. Sciences, 77 (2008), 851882. doi: 10.4310/CMS.2008.v6.n2.a1. 
[4]  H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, SpringerVerlag, New York, 2011. 
[5] 
P. Degond, J. G. Liu, R. L. Pego, Coagulationfragmentation model for animal groupsize statistics, J. Nonlinear Sci., 27 (2017), 379424. doi: 10.1007/s0033201693363. 
[6] 
F. Filbet, P. Laurencot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 20042028. doi: 10.1137/S1064827503429132. 
[7] 
L. ForestierCoste, S. Mancini, A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM J. Sci. Comput., 34 (2012), B840B860. doi: 10.1137/110847998. 
[8] 
S. Gueron, The steadystate distributions of coagulationfragmentation processes, J. Math. Biol., 37 (1998), 127. doi: 10.1007/s002850050117. 
[9] 
S. Gueron, S. A. Levin, The dynamics of group formations, Math. Biosc., 128 (1995), 243264. doi: 10.1016/00255564(94)00074A. 
[10] 
J. Kumar, G. Kaur, E. Tsotsas, An accurate and efficient discrete formulation of aggregation population balance equation, Kinet. Relat. Models, 9 (2016), 373391. doi: 10.3934/krm.2016.9.373. 
[11] 
R. Kumar, J. Kumar, G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Math. Models Methods Appl. Sci., 23 (2013), 12351273. doi: 10.1142/S0218202513500085. 
[12] 
Q. Ma, A. Johansson, D. J. T. Sumpter, A first principles derivation of animal group size distributions, J. Theoret. Biol., 283 (2011), 3543. doi: 10.1016/j.jtbi.2011.04.031. 
[13] 
A. W. Mahoney, D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chem. Eng. Sci., 57 (2002), 11071119. doi: 10.1016/S00092509(01)004274. 
[14] 
M. Nicmanis, M. J. Hounslow, A finite element analysis of the steady state population balance equation for particulate systems: Aggregation and growth, Comput. Chem. Eng., 20 (1996), 261266. doi: 10.1016/00981354(96)000543. 
[15] 
H. Niwa, Mathematical model for the size distributions of fish schools, Comp. Math. Appl., 32 (1996), 7988. doi: 10.1016/S08981221(96)00199X. 
[16] 
H. Niwa, School size statistics of fish, J. Theoret. Biol., 195 (1998), 351361. doi: 10.1006/jtbi.1998.0801. 
[17] 
H. Niwa, PowerLaw versus exponential distributions of animal group sizes, J. Theoret. Biol., 224 (2003), 451457. doi: 10.1016/S00225193(03)001929. 
[18] 
H. Niwa, Spaceirrelevant scaling law for fish school sizes, J. Theoret. Biol., 228 (2004), 347357. doi: 10.1016/j.jtbi.2004.01.011. 
[19] 
A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, rocks, and herds, Adv. Biophys., 22 (1986), 194. 
[20] 
S. Rigopoulos, A. G. Jones, Finiteelement scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 11271139. doi: 10.1002/aic.690490507. 
[21] 
D. Verkoeijen, G. A. Pouw, G. M. H. Meesters, B. Scarlett, Population balances for particulate processesa volume approach, Chem. Eng. Sci., 57 (2002), 22872303. doi: 10.1016/S00092509(02)001185. 
show all references
References:
[1] 
E. Bonabeau, L. Dagorn, Possible universality in the size distribution of fish schools, Phys. Rev. E, 51 (1995), 52205223. doi: 10.1103/PhysRevE.51.R5220. 
[2] 
E. Bonabeau, L. Dagorn, P. Freon, Scaling in animal groupsize distributions, Proc. Natl. Acad. Sci. USA, 96 (1999), 44724477. doi: 10.1073/pnas.96.8.4472. 
[3] 
J. P. Bourgade, F. Filbet, Convergence of a finite volume scheme for coagulationfragmentation equations, Comm. Math. Sciences, 77 (2008), 851882. doi: 10.4310/CMS.2008.v6.n2.a1. 
[4]  H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, SpringerVerlag, New York, 2011. 
[5] 
P. Degond, J. G. Liu, R. L. Pego, Coagulationfragmentation model for animal groupsize statistics, J. Nonlinear Sci., 27 (2017), 379424. doi: 10.1007/s0033201693363. 
[6] 
F. Filbet, P. Laurencot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 20042028. doi: 10.1137/S1064827503429132. 
[7] 
L. ForestierCoste, S. Mancini, A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM J. Sci. Comput., 34 (2012), B840B860. doi: 10.1137/110847998. 
[8] 
S. Gueron, The steadystate distributions of coagulationfragmentation processes, J. Math. Biol., 37 (1998), 127. doi: 10.1007/s002850050117. 
[9] 
S. Gueron, S. A. Levin, The dynamics of group formations, Math. Biosc., 128 (1995), 243264. doi: 10.1016/00255564(94)00074A. 
[10] 
J. Kumar, G. Kaur, E. Tsotsas, An accurate and efficient discrete formulation of aggregation population balance equation, Kinet. Relat. Models, 9 (2016), 373391. doi: 10.3934/krm.2016.9.373. 
[11] 
R. Kumar, J. Kumar, G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms, Math. Models Methods Appl. Sci., 23 (2013), 12351273. doi: 10.1142/S0218202513500085. 
[12] 
Q. Ma, A. Johansson, D. J. T. Sumpter, A first principles derivation of animal group size distributions, J. Theoret. Biol., 283 (2011), 3543. doi: 10.1016/j.jtbi.2011.04.031. 
[13] 
A. W. Mahoney, D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chem. Eng. Sci., 57 (2002), 11071119. doi: 10.1016/S00092509(01)004274. 
[14] 
M. Nicmanis, M. J. Hounslow, A finite element analysis of the steady state population balance equation for particulate systems: Aggregation and growth, Comput. Chem. Eng., 20 (1996), 261266. doi: 10.1016/00981354(96)000543. 
[15] 
H. Niwa, Mathematical model for the size distributions of fish schools, Comp. Math. Appl., 32 (1996), 7988. doi: 10.1016/S08981221(96)00199X. 
[16] 
H. Niwa, School size statistics of fish, J. Theoret. Biol., 195 (1998), 351361. doi: 10.1006/jtbi.1998.0801. 
[17] 
H. Niwa, PowerLaw versus exponential distributions of animal group sizes, J. Theoret. Biol., 224 (2003), 451457. doi: 10.1016/S00225193(03)001929. 
[18] 
H. Niwa, Spaceirrelevant scaling law for fish school sizes, J. Theoret. Biol., 228 (2004), 347357. doi: 10.1016/j.jtbi.2004.01.011. 
[19] 
A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, rocks, and herds, Adv. Biophys., 22 (1986), 194. 
[20] 
S. Rigopoulos, A. G. Jones, Finiteelement scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 11271139. doi: 10.1002/aic.690490507. 
[21] 
D. Verkoeijen, G. A. Pouw, G. M. H. Meesters, B. Scarlett, Population balances for particulate processesa volume approach, Chem. Eng. Sci., 57 (2002), 22872303. doi: 10.1016/S00092509(02)001185. 
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