### Mathematical Biosciences and Engineering

2020, Issue 6: 7692-7707. doi: 10.3934/mbe.2020391
Research article Special Issues

# Asymptotic flocking for the three-zone model

• Received: 06 July 2020 Accepted: 15 October 2020 Published: 05 November 2020
• We prove the asymptotic flocking behavior of a general model of swarming dynamics. The model describing interacting particles encompasses three types of behavior: repulsion, alignment and attraction. We refer to this dynamics as the three-zone model. Our result expands the analysis of the so-called Cucker-Smale model where only alignment rule is taken into account. Whereas in the Cucker-Smale model, the alignment should be strong enough at long distance to ensure flocking behavior, here we only require that the attraction is described by a confinement potential. The key for the proof is to use that the dynamics is dissipative thanks to the alignment term which plays the role of a friction term. Several numerical examples illustrate the result and we also extend the proof for the kinetic equation associated with the three-zone dynamics.

Citation: Fei Cao, Sebastien Motsch, Alexander Reamy, Ryan Theisen. Asymptotic flocking for the three-zone model[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7692-7707. doi: 10.3934/mbe.2020391

### Related Papers:

• We prove the asymptotic flocking behavior of a general model of swarming dynamics. The model describing interacting particles encompasses three types of behavior: repulsion, alignment and attraction. We refer to this dynamics as the three-zone model. Our result expands the analysis of the so-called Cucker-Smale model where only alignment rule is taken into account. Whereas in the Cucker-Smale model, the alignment should be strong enough at long distance to ensure flocking behavior, here we only require that the attraction is described by a confinement potential. The key for the proof is to use that the dynamics is dissipative thanks to the alignment term which plays the role of a friction term. Several numerical examples illustrate the result and we also extend the proof for the kinetic equation associated with the three-zone dynamics.

 [1] I. Aoki, A Simulation Study on the Schooling Mechanism in Fish, Bulletin of the Japanese Society of Scientific Fisheries (Japan), 1982. [2] I. D. Couzin, J. Krause, R. James, G. D Ruxton, N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theor. Biol., 218 (2002), 1-11. doi: 10.1006/jtbi.2002.3065 [3] A. Huth, C. Wissel, The simulation of the movement of fish schools, J. Theor. Biol., 156 (1992), 365-385. doi: 10.1016/S0022-5193(05)80681-2 [4] Y. X. Li, R. Lukeman, L. Edelstein-Keshet, Minimal mechanisms for school formation in self-propelled particles, Phys. D: Nonlinear Phenomena, 237 (2008), 699-720. [5] C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, in ACM SIGGRAPH Computer Graphics, (1987), 25-34. [6] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Automatic Control, 52 (2007), 852. [7] F. Cucker, S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x [8] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226 [9] M. Agueh, R. Illner, A. Richardson, Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type, Kinetic Related Models, 4 (2011), 1-16. doi: 10.3934/krm.2011.4.1 [10] S. Motsch, E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9 [11] S. Motsch, E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866 [12] J. A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236. doi: 10.1137/090757290 [13] S. Y. Ha, J. G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2 [14] S. Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic Related Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415 [15] J. A. Carrillo, Y. P. Choi, S. Pérez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, preprint, arXiv: 1605.00232. [16] J. A. Carrillo, Y. Huang, S. Martin, Explicit flock solutions for Quasi-Morse potentials, Eur. J. Appl. Math., (2014), 1-26. [17] J. Von Brecht, D. Uminsky, T. Kolokolnikov, A. Bertozzi, Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012). [18] D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal., 254 (2008), 727-759. doi: 10.1016/j.jfa.2007.11.002 [19] C. Villani, Hypocoercivity, Memoirs of the American Mathematical Society, 2009. [20] P. E. Jabin, S. Motsch, Clustering and asymptotic behavior in opinion formation, J. Differ. Equations, 257 (2014), 4165-4187. doi: 10.1016/j.jde.2014.08.005 [21] T. Karper, A. Mellet, K. Trivisa, On strong local alignment in the kinetic Cucker-Smale model, in Hyperbolic Conservation Laws and Related Analysis with Applications, Springer, (2014), 227-242. [22] J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D: Nonlinear Phenomena, 261 (2013), 42-51. doi: 10.1016/j.physd.2013.06.006 [23] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, et al., Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proceed. Natl. Acad. Sci., 105 (2008), 1232. [24] A. Blanchet, P. Degond, Topological interactions in a Boltzmann-type framework, J. Stat. Phys., 163 (2016), 41-60. doi: 10.1007/s10955-016-1471-6 [25] J. A. Carrillo, Y. Huang, Explicit Equilibrium solutions for the aggregation equation with power-law potentials, preprint, arXiv: 1602.06615. [26] R. Fetecau, Y. Huang, T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681. doi: 10.1088/0951-7715/24/10/002 [27] T. Kolokolnikov, H. Sun, D. Uminsky, A. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E, 84 (2011), 015203. doi: 10.1103/PhysRevE.84.015203 [28] Y. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi, L. S. Chayes, State transitions and the continuum limit for a 2d interacting, self-propelled particle system, Phys. D: Nonlinear Phenomena, 232 (2007), 33-47. [29] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi, L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302. [30] D. Ruelle, Statistical Mechanics: Rigorous Results. World Scientific, 1969. [31] D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul, Dimensionality of local minimizers of the Interaction energy, Arch. Rational Mech. Anal., 209 (2013), 1055-1088. doi: 10.1007/s00205-013-0644-6 [32] L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Inventiones Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9 [33] F. Filbet, On deterministic approximation of the Boltzmann equation in a bounded domain, Multiscale Modell. Simul., 10 (2012), 792-817. doi: 10.1137/11082419X [34] F. Bolley, J. A. Canizo, J. A. Carrillo, Stochastic mean-field limit: non-Lipschitz forces and swarming, Math. Models Methods. Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702 [35] J. Carrillo, Y. P. Choi, M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, Springer, (2014), 1-46. [36] P. Degond, G. Dimarco, T. B. N. Mac, N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615-1638. doi: 10.4310/CMS.2015.v13.n6.a12 [37] P. Degond, J. G. Liu, S. Motsch, V. Panferov, Hydrodynamic models of self-organized dynamics: derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114. [38] P. Degond, S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005 [39] P. E. Jabin, A review of the mean field limits for Vlasov equations, Kinetic Related Models, 7 (2014), 661-711. doi: 10.3934/krm.2014.7.661 [40] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, 1991.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Article outline

Figures(5)

• On This Site