Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
We study numerically a coagulation-fragmentation model derived by Niwa [
Citation: |
Figure 1.
The equilibrium distribution is approximated by the Newton scheme (Section 5.2.3). In Fig. 1a, we take mass
Figure 2.
The equilibrium distribution is approximated by simulating the time evolution of the distribution via the Euler scheme. Starting with a uniform distribution, the equilibrium, is reached at
Figure 3.
The equilibrium distribution is approximated by the recursive scheme (Section 5.1). In Fig. 3a, the mass is
Figure 4.
Comparison of the equilibria for model D' ((5.5)-(5.6)) and model D ((2.15)-(2.18)). We take truncation
Figure 5.
The equilibrium distribution is generated by the recursive scheme, for mass
Figure 6.
The large-size behaviours of the discrete and continuous equilibrium distributions are compared, for mass
Figure 7.
In Fig. 7a we plot
Figure 8.
Starting with a uniform distribution (Fig. 8a) and with an exponential distribution (Fig. 8b), the time-dependent solution of model C ((2.19)-(2.23)),
Table 1.
Starting with a uniform distribution the time-dependent solution of model C ((2.19)-(2.23)),
Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
Table 2.
Starting with an exponential distribution the time-dependent solution of model C ((2.19)-(2.23)),
Time | | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
[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. |