
-
Previous Article
Transport of measures on networks
- NHM Home
- This Issue
-
Next Article
Stability estimates for scalar conservation laws with moving flux constraints
Numerical approximation of a coagulation-fragmentation model for animal group size statistics
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK |
We study numerically a coagulation-fragmentation model derived by Niwa [
References:
[1] |
E. Bonabeau and L. Dagorn,
Possible universality in the size distribution of fish schools, Phys. Rev. E, 51 (1995), 5220-5223.
doi: 10.1103/PhysRevE.51.R5220. |
[2] |
E. Bonabeau, L. Dagorn and P. Freon,
Scaling in animal group-size distributions, Proc. Natl. Acad. Sci. USA, 96 (1999), 4472-4477.
doi: 10.1073/pnas.96.8.4472. |
[3] |
J. P. Bourgade and F. Filbet,
Convergence of a finite volume scheme for coagulation-fragmentation equations, Comm. Math. Sciences, 77 (2008), 851-882.
doi: 10.4310/CMS.2008.v6.n2.a1. |
[4] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
![]() |
[5] |
P. Degond, J. G. Liu and R. L. Pego,
Coagulation-fragmentation model for animal group-size statistics, J. Nonlinear Sci., 27 (2017), 379-424.
doi: 10.1007/s00332-016-9336-3. |
[6] |
F. Filbet and P. Laurencot,
Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028.
doi: 10.1137/S1064827503429132. |
[7] |
L. Forestier-Coste and S. Mancini,
A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM J. Sci. Comput., 34 (2012), B840-B860.
doi: 10.1137/110847998. |
[8] |
S. Gueron,
The steady-state distributions of coagulation-fragmentation processes, J. Math. Biol., 37 (1998), 1-27.
doi: 10.1007/s002850050117. |
[9] |
S. Gueron and S. A. Levin,
The dynamics of group formations, Math. Biosc., 128 (1995), 243-264.
doi: 10.1016/0025-5564(94)00074-A. |
[10] |
J. Kumar, G. Kaur and E. Tsotsas,
An accurate and efficient discrete formulation of aggregation population balance equation, Kinet. Relat. Models, 9 (2016), 373-391.
doi: 10.3934/krm.2016.9.373. |
[11] |
R. Kumar, J. Kumar and 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), 1235-1273.
doi: 10.1142/S0218202513500085. |
[12] |
Q. Ma, A. Johansson and D. J. T. Sumpter,
A first principles derivation of animal group size distributions, J. Theoret. Biol., 283 (2011), 35-43.
doi: 10.1016/j.jtbi.2011.04.031. |
[13] |
A. W. Mahoney and D. Ramkrishna,
Efficient solution of population balance equations with discontinuities by finite elements, Chem. Eng. Sci., 57 (2002), 1107-1119.
doi: 10.1016/S0009-2509(01)00427-4. |
[14] |
M. Nicmanis and 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), 261-266.
doi: 10.1016/0098-1354(96)00054-3. |
[15] |
H. Niwa,
Mathematical model for the size distributions of fish schools, Comp. Math. Appl., 32 (1996), 79-88.
doi: 10.1016/S0898-1221(96)00199-X. |
[16] |
H. Niwa,
School size statistics of fish, J. Theoret. Biol., 195 (1998), 351-361.
doi: 10.1006/jtbi.1998.0801. |
[17] |
H. Niwa,
Power-Law versus exponential distributions of animal group sizes, J. Theoret. Biol., 224 (2003), 451-457.
doi: 10.1016/S0022-5193(03)00192-9. |
[18] |
H. Niwa,
Space-irrelevant scaling law for fish school sizes, J. Theoret. Biol., 228 (2004), 347-357.
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), 1-94. |
[20] |
S. Rigopoulos and A. G. Jones,
Finite-element scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 1127-1139.
doi: 10.1002/aic.690490507. |
[21] |
D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett,
Population balances for particulate processes-a volume approach, Chem. Eng. Sci., 57 (2002), 2287-2303.
doi: 10.1016/S0009-2509(02)00118-5. |
show all references
References:
[1] |
E. Bonabeau and L. Dagorn,
Possible universality in the size distribution of fish schools, Phys. Rev. E, 51 (1995), 5220-5223.
doi: 10.1103/PhysRevE.51.R5220. |
[2] |
E. Bonabeau, L. Dagorn and P. Freon,
Scaling in animal group-size distributions, Proc. Natl. Acad. Sci. USA, 96 (1999), 4472-4477.
doi: 10.1073/pnas.96.8.4472. |
[3] |
J. P. Bourgade and F. Filbet,
Convergence of a finite volume scheme for coagulation-fragmentation equations, Comm. Math. Sciences, 77 (2008), 851-882.
doi: 10.4310/CMS.2008.v6.n2.a1. |
[4] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
![]() |
[5] |
P. Degond, J. G. Liu and R. L. Pego,
Coagulation-fragmentation model for animal group-size statistics, J. Nonlinear Sci., 27 (2017), 379-424.
doi: 10.1007/s00332-016-9336-3. |
[6] |
F. Filbet and P. Laurencot,
Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028.
doi: 10.1137/S1064827503429132. |
[7] |
L. Forestier-Coste and S. Mancini,
A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM J. Sci. Comput., 34 (2012), B840-B860.
doi: 10.1137/110847998. |
[8] |
S. Gueron,
The steady-state distributions of coagulation-fragmentation processes, J. Math. Biol., 37 (1998), 1-27.
doi: 10.1007/s002850050117. |
[9] |
S. Gueron and S. A. Levin,
The dynamics of group formations, Math. Biosc., 128 (1995), 243-264.
doi: 10.1016/0025-5564(94)00074-A. |
[10] |
J. Kumar, G. Kaur and E. Tsotsas,
An accurate and efficient discrete formulation of aggregation population balance equation, Kinet. Relat. Models, 9 (2016), 373-391.
doi: 10.3934/krm.2016.9.373. |
[11] |
R. Kumar, J. Kumar and 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), 1235-1273.
doi: 10.1142/S0218202513500085. |
[12] |
Q. Ma, A. Johansson and D. J. T. Sumpter,
A first principles derivation of animal group size distributions, J. Theoret. Biol., 283 (2011), 35-43.
doi: 10.1016/j.jtbi.2011.04.031. |
[13] |
A. W. Mahoney and D. Ramkrishna,
Efficient solution of population balance equations with discontinuities by finite elements, Chem. Eng. Sci., 57 (2002), 1107-1119.
doi: 10.1016/S0009-2509(01)00427-4. |
[14] |
M. Nicmanis and 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), 261-266.
doi: 10.1016/0098-1354(96)00054-3. |
[15] |
H. Niwa,
Mathematical model for the size distributions of fish schools, Comp. Math. Appl., 32 (1996), 79-88.
doi: 10.1016/S0898-1221(96)00199-X. |
[16] |
H. Niwa,
School size statistics of fish, J. Theoret. Biol., 195 (1998), 351-361.
doi: 10.1006/jtbi.1998.0801. |
[17] |
H. Niwa,
Power-Law versus exponential distributions of animal group sizes, J. Theoret. Biol., 224 (2003), 451-457.
doi: 10.1016/S0022-5193(03)00192-9. |
[18] |
H. Niwa,
Space-irrelevant scaling law for fish school sizes, J. Theoret. Biol., 228 (2004), 347-357.
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), 1-94. |
[20] |
S. Rigopoulos and A. G. Jones,
Finite-element scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 1127-1139.
doi: 10.1002/aic.690490507. |
[21] |
D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett,
Population balances for particulate processes-a volume approach, Chem. Eng. Sci., 57 (2002), 2287-2303.
doi: 10.1016/S0009-2509(02)00118-5. |








Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
[1] |
Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541 |
[2] |
Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014 |
[3] |
Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685 |
[4] |
Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589 |
[5] |
Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393 |
[6] |
Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445 |
[7] |
Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563 |
[8] |
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23 |
[9] |
Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537 |
[10] |
Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333 |
[11] |
Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126 |
[12] |
Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177 |
[13] |
Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1 |
[14] |
Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173 |
[15] |
Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493 |
[16] |
Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995 |
[17] |
Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1 |
[18] |
Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761 |
[19] |
Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449 |
[20] |
Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070 |
2016 Impact Factor: 1.2
Tools
Metrics
Other articles
by authors
[Back to Top]