Hostname: page-component-669899f699-rg895 Total loading time: 0 Render date: 2025-04-25T07:00:03.536Z Has data issue: false hasContentIssue false
  • English
  • Français

A hybrid numerical model for the collective motion of fish groups

Published online by Cambridge University Press:  14 April 2025

Xuewei Mao ,
Jiadong Wang ,
Xingyuan Mao  and
Jian Deng Open the ORCID record for Jian Deng [Opens in a new window]
Xuewei Mao
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Jiadong Wang
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Xingyuan Mao
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Jian Deng*
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
*
Corresponding author: Jian Deng, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a hybrid numerical model for the collective motion of fish groups, which integrates an agent-based model with a computational fluid dynamics (CFD)-based model. In the agent-based model, the fish group is represented by self-propelled particles (SPPs), incorporating social forces with local interactive rules. The CFD-based model treats the fish body with an undulated filament that responds to the hydrodynamic forces imposed by the surrounding fluid flow. These two models are coupled using a central pattern generator controller. We test this hybrid model with groups of 30–50 individuals. The results show that the group exhibits various collective behaviours, including tight schooling, sparse schooling and milling patterns, by adjusting the coefficients in the SPP model. Due to the hydrodynamic interactions, particularly with the obstacle avoidance model, both the individuals’ and the group’s mean speed fluctuate, differentiating it from traditional SPP models that typically consider volumeless particles. More interestingly, our findings indicate that fish benefit from collective motion in terms of energetic consumption in both schooling and milling patterns. It is important to note that the swimming fish are actuated using a very simple mechanism without any optimisation strategy. An additional study investigates the effects of the Reynolds number, demonstrating the capability of the current hybrid model to account for fish groups of varying body lengths or swimming speeds. Future applications of this model are promising, offering potential insights into the energetic advantages of collective motion in large-scale fish groups.

JFM classification

Biological Fluid Dynamics: Collective behaviour Biological Fluid Dynamics: Swimming/flying Biological Fluid Dynamics: Propulsion
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1009 , 25 April 2025 , A26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alben, S. 2021 Collective locomotion of two-dimensional lattices of flapping plates. Part 2. Lattice flows and propulsive efficiency. J. Fluid Mech. 915, A21.CrossRefGoogle Scholar
Andersen, A., Bohr, T., Schnipper, T. & Walther, J.H. 2017 Wake structure and thrust generation of a flapping foil in two-dimensional flow. J. Fluid Mech. 812, R4.CrossRefGoogle Scholar
Arranz, G., Flores, O. & García-Villalba, M. 2022 Flow interaction of three-dimensional self-propelled flexible plates in tandem. J. Fluid Mech. 931, A5.CrossRefGoogle Scholar
Ashraf, I., Bradshaw, H., Ha, T.-T., Halloy, J., Godoy-Diana, R. & Thiria, B. 2017 Simple phalanx pattern leads to energy saving in cohesive fish schooling. Proc. Natl Acad. Sci. USA 114 (36), 9599–9604.CrossRefGoogle ScholarPubMed
Ashraf, I., Godoy-Diana, R., Halloy, J., Collignon, B. & Thiria, B. 2016 Synchronization and collective swimming patterns in fish (Hemigrammus bleheri). J. R. Soc. Interface 13 (123), 20160734.CrossRefGoogle ScholarPubMed
Baggaley, A.W. 2015 Model flocks in a steady vortical flow. Phys. Rev. E 91 (5), 053019.CrossRefGoogle Scholar
Ballerini, M. et al. 2008 Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl Acad. Sci. USA 105 (4), 1232–1237.CrossRefGoogle ScholarPubMed
Bao, Y., Zhou, D., Tao, J.J., Peng, Z., Zhu, H.B., Sun, Z.L. & Tong, H.L. 2017 Dynamic interference of two anti-phase flapping foils in side-by-side arrangement in an incompressible flow. Phys. Fluids 29 (3), 033601.CrossRefGoogle Scholar
Bastien, R. & Romanczuk, P. 2020 A model of collective behavior based purely on vision. Sci. Adv. 6 (6), eaay0792.CrossRefGoogle Scholar
Boschitsch, B.M., Dewey, P.A. & Smits, A.J. 2014 Propulsive performance of unsteady tandem hydrofoils in an in-line configuration. Phys. Fluids 26 (5), 051901.CrossRefGoogle Scholar
Brock, V.E. & Riffenburgh, R.H. 1960 Fish schooling: a possible factor in reducing predation. ICES J. Mar. Sci. 25 (3), 307–317.CrossRefGoogle Scholar
Calovi, D.S., Lopez, U., Ngo, S., Sire, C., Chaté, H. & Theraulaz, G. 2014 Swarming, schooling, milling: phase diagram of a data-driven fish school model. New J. Phys. 16 (1), 015026.CrossRefGoogle Scholar
Calovi, D.S., Lopez, U., Schuhmacher, P., Chaté, H., Sire, C. & Theraulaz, G. 2015 Collective response to perturbations in a data-driven fish school model. J. R. Soc. Interface 12 (104), 20141362.CrossRefGoogle Scholar
Camperi, M., Cavagna, A., Giardina, I., Parisi, G. & Silvestri, E. 2012 Spatially balanced topological interaction grants optimal cohesion in flocking models. Interface Focus 2 (6), 715–725.CrossRefGoogle ScholarPubMed
Carling, J., Williams, T.L. & Bowtell, G. 1998 Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional Navier–Stokes equations and Newton’s laws of motion. J. Exp. Biol. 201 (23), 3143–3166.CrossRefGoogle ScholarPubMed
Cavagna, A., Cimarelli, A., Giardina, I., Parisi, G., Santagati, R., Stefanini, F. & Tavarone, R. 2010 From empirical data to inter-individual interactions: unveiling the rules of collective animal behavior. Math. Models Meth. Appl. Sci. 20 (supp01), 1491–1510.CrossRefGoogle Scholar
Chaté, H., Ginelli, F., Grégoire, G., Raynaud, F. 2008 Collective motion of self-propelled particles interacting without cohesion. Phys. Rev. E 77 (4), 046113.CrossRefGoogle ScholarPubMed
Chen, S.-Y., Fei, Y.-H.J., Chen, Y.-C., Chi, K.-J. & Yang, J.-T. 2016 The swimming patterns and energy-saving mechanism revealed from three fish in a school. Ocean Engng 122, 22–31.CrossRefGoogle Scholar
Cushing, D.H. & Jones, F.R.H. 1968 Why do fish school? Nature 218 (5145), 918–920.CrossRefGoogle ScholarPubMed
Daghooghi, M. & Borazjani, I. 2015 The hydrodynamic advantages of synchronized swimming in a rectangular pattern. Bioinspir. Biomim. 10 (5), 056018.CrossRefGoogle Scholar
Dai, L., He, G., Zhang, X. & Zhang, X. 2018 Stable formations of self-propelled fish-like swimmers induced by hydrodynamic interactions. J. R. Soc. Interface 15 (147), 20180490.CrossRefGoogle ScholarPubMed
Das, R., Peterson, S.D. & Porfiri, M. 2023 Stability of schooling patterns of a fish pair swimming against a flow. Flow 3, E31.CrossRefGoogle Scholar
De Lellis, P. & Porfiri, M. 2022 Inferring the size of a collective of self-propelled Vicsek particles from the random motion of a single unit. Commun. Phys. 5 (1), 86.CrossRefGoogle Scholar
Deng, J. & Liu, D. 2021 Spontaneous response of a self-organized fish school to a predator. Bioinspir. Biomim. 16 (4), 046013.CrossRefGoogle ScholarPubMed
Deng, J., Mao, X. & Xie, F. 2019 Dynamics of two-dimensional flow around a circular cylinder with flexible filaments attached. Phys. Rev. E 100 (5), 053107.CrossRefGoogle ScholarPubMed
Deng, J. & Shao, X.-M. 2006 Hydrodynamics in a diamond-shaped fish school. J. Hydrodyn. 18 (S1), 428–432.CrossRefGoogle Scholar
Deng, J., Shao, X.-M. & Yu, Z.-S. 2007 Hydrodynamic studies on two traveling wavy foils in tandem arrangement. Phys. Fluids 19 (11), 113104.CrossRefGoogle Scholar
Dong, G.-J. & Lu, X.-Y. 2007 Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Phys. Fluids 19 (5), 057107.CrossRefGoogle Scholar
Escobedo, R., Lecheval, V., Papaspyros, V., Bonnet, F., Mondada, F., Sire, C. & Theraulaz, G. 2020 A data-driven method for reconstructing and modelling social interactions in moving animal groups. Phil. Trans. R. Soc. Lond. B: Biol. Sci. 375 (1807), 1807,–20190380.CrossRefGoogle ScholarPubMed
Filella, A., Nadal, F., Sire, C., Kanso, E. & Eloy, C. 2018 Model of collective fish behavior with hydrodynamic interactions. Phys. Rev. Lett. 120 (19), 198101.CrossRefGoogle ScholarPubMed
Gautrais, J., Ginelli, F., Fournier, R., Blanco, S., Soria, M., Chaté, H. & Theraulaz, G. 2012 Deciphering interactions in moving animal groups. PLoS Computational Biology 8 (9), e1002678.CrossRefGoogle ScholarPubMed
Gautrais, J., Jost, C., Soria, M., Campo, A., Motsch, S., Fournier, R., Blanco, S. & Theraulaz, G. 2009 Analyzing fish movement as a persistent turning walker. J. Math. Biol. 58 (3), 429–445.CrossRefGoogle ScholarPubMed
Gazzola, M., Tchieu, A A., Alexeev, D., de Brauer, A. & Koumoutsakos, P. 2016 Learning to school in the presence of hydrodynamic interactions. J. Fluid Mech. 789, 726–749.CrossRefGoogle Scholar
Ginelli, F. 2016 The physics of the Vicsek model. Eur. Phys. J. Special Topics 225 (11-12), 2099–2117.CrossRefGoogle Scholar
Ginelli, F. & Chaté, H. 2010 Relevance of metric-free interactions in flocking phenomena. Phys. Rev. Lett. 105 (16), 168103.CrossRefGoogle ScholarPubMed
Goldstein, D., Handler, R. & Sirovich, L. 1993 Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 105 (2), 354–366.CrossRefGoogle Scholar
Grégoire, G., Chaté, H. & Tu, Y. 2003 Moving and staying together without a leader. Physica D: Nonlinear Phenom. 181 (3-4), 157–170.CrossRefGoogle Scholar
Hemelrijk, C.K., Reid, D.A.P., Hildenbrandt, H. & Padding, J.T. 2015 The increased efficiency of fish swimming in a school. Fish Fish 16 (3), 511–521.CrossRefGoogle Scholar
Heydari, S. & Kanso, E. 2021 School cohesion, speed and efficiency are modulated by the swimmers flapping motion. J. Fluid Mech. 922, A27.CrossRefGoogle Scholar
Huang, C., Ling, F. & Kanso, E. 2024 Collective phase transitions in confined fish schools. Proc. Natl Acad. Sci. USA 121(44), e2406293121.CrossRefGoogle ScholarPubMed
Ioannou, C.C., Guttal, V. & Couzin, I.D. 2012 Predatory fish select for coordinated collective motion in virtual prey. Science 337 (6099), 1212–1215.CrossRefGoogle ScholarPubMed
Kanso, E. & Newton, P.K. 2010 Locomotory advantages to flapping out of phase. Expl. Mech. 50 (9), 1367–1372.CrossRefGoogle Scholar
Kanso, E. & Tsang, A.C.H. 2014 Dipole models of self-propelled bodies. Fluid Dyn. Res. 46 (6), 061407.CrossRefGoogle Scholar
Khalid, M.S.U., Akhtar, I. & Dong, H. 2016 Hydrodynamics of a tandem fish school with asynchronous undulation of individuals. J. Fluid Struct. 66, 19–35.CrossRefGoogle Scholar
Ko, H., Lauder, G. & Nagpal, R. 2023 The role of hydrodynamics in collective motions of fish schools and bioinspired underwater robots. J. R. Soc. Interface 20 (207), 20230357.CrossRefGoogle ScholarPubMed
Krebs, J.R., MacRoberts, M.H. & Cullen, J.M. 1972 Flocking and feeding in the great tit parus major–an experimental study. Ibis 114 (4), 507–530.CrossRefGoogle Scholar
Kunz, H. & Hemelrijk, C.K. 2012 Simulations of the social organization of large schools of fish whose perception is obstructed. Appl. Animal Behav. Sci. 138 (3-4), 142–151.CrossRefGoogle Scholar
Lafoux, B., Moscatelli, J., Godoy-Diana, R. & Thiria, B. 2023 Illuminance-tuned collective motion in fish. Commun. Biol. 6 (1), 585.CrossRefGoogle ScholarPubMed
Landeau, L. & Terborgh, J. 1986 Oddity and the confusion effectin predation. Animal Behav. 34 (5), 1372–1380.CrossRefGoogle Scholar
Lazarus, J. 1979 The early warning function of flocking in birds: an experimental study with captive quelea. Animal Behav. 27, 855–865.CrossRefGoogle Scholar
Li, G., Kolomenskiy, D., Liu, H., Thiria, B., Godoy-Diana, R. & Gurka, R. 2019 On the energetics and stability of a minimal fish school. PLoS One 14 (8), e0215265.CrossRefGoogle ScholarPubMed
Li, L., Liu, D., Deng, J., Lutz, M.J. & Xie, G. 2021 a Fish can save energy via proprioceptive sensing. Bioinspir. Biomim. 16 (5), 056013.CrossRefGoogle ScholarPubMed
Li, L., Zheng, X., Mao, R. & Xie, G. 2021 b Energy saving of schooling robotic fish in three-dimensional formations. IEEE Robot. Autom. Lett. 6 (2), 1694–1699.CrossRefGoogle Scholar
Liao, J.C., Beal, D.N., Lauder, G.V. & Triantafyllou, M.S. 2003 Fish exploiting vortices decrease muscle activity. Science 302 (5650), 1566–1569.CrossRefGoogle ScholarPubMed
Lighthill, S.J. 1975 Mathematical biofluiddynamics. SIAM.CrossRefGoogle Scholar
Ligman, M., Lund, J. & Fürth, M. 2024 A comprehensive review of hydrodynamic studies on fish schooling. Bioinspir. Biomim. 19 (1), 011002.CrossRefGoogle Scholar
Lin, X., Wu, J., Zhang, T. & Yang, L. 2020 Self-organization of multiple self-propelling flapping foils: energy saving and increased speed. J. Fluid Mech. 884, R1.CrossRefGoogle Scholar
Lin, X., Wu, J., Zhang, T. & Yang, L. 2021 Flow-mediated organization of two freely flapping swimmers. J. Fluid Mech. 912, A37.CrossRefGoogle Scholar
Ling, H., Mclvor, G.E., Westley, J., van der Vaart, K., Vaughan, R.T., Thornton, A.& Ouellette, N.T. 2019 Behavioural plasticity and the transition to order in jackdaw flocks. Nat. Commun. 10 (1), 5174.CrossRefGoogle ScholarPubMed
Liu, D., Liang, Y., Deng, J. & Zhang, W. 2023 a Modeling three-dimensional bait ball collective motion. Phys. Rev. E 107 (1), 014606.CrossRefGoogle ScholarPubMed
Liu, D., Wang, J., Mao, X. & Deng, J. 2023 b Energetic benefits in coordinated circular swimming motion of two swimmers. Phys. Rev. E 108 (5), 054603.CrossRefGoogle ScholarPubMed
Liu, X., Liu, K. & Huang, H. 2022 Collective behavior and hydrodynamic advantage of side-by-side self-propelled flapping foils. Phys. Rev. E 105 (6), 065105.CrossRefGoogle ScholarPubMed
Lopez, U., Gautrais, J., Couzin, I.D. & Theraulaz, G. 2012 From behavioural analyses to models of collective motion in fish schools. Interface Focus 2 (6), 693–707.CrossRefGoogle ScholarPubMed
Mudaliar, R.K., Zvezdin, A.V., Bratt, G.S. & Schaerf, T.M. 2022 Systematic analysis of emergent collective motion produced by a 3D hybrid zonal model. Bull. Math. Biol. 84 (1), 1–25.Google ScholarPubMed
Pan, Y., Zhang, W., Kelly, J. & Dong, H. 2024 Unraveling hydrodynamic interactions in fish schools: a three-dimensional computational study of in-line and side-by-side configurations. Phys. Fluids 36 (8), 081909.CrossRefGoogle Scholar
Peng, Z.-R., Huang, H. & Lu, X.-Y. 2018 Hydrodynamic schooling of multiple self-propelled flapping plates. J. Fluid Mech. 853, 587–600.CrossRefGoogle Scholar
Pitcher, T.J., Magurran, A.E. & Winfield, I.J. 1982 Fish in larger shoals find food faster. Behav. Ecol. Sociobiol. 10 (2), 149–151.CrossRefGoogle Scholar
Romanczuk, P., Couzin, I.D. & Schimansky-Geier, L. 2009 Collective motion due to individual escape and pursuit response. Phys. Rev. Lett. 102 (1), 010602.CrossRefGoogle ScholarPubMed
Roy, S. & Lemus, J. 2021 How does the fusion of sensory information from audition and vision impact collective behavior? Frontiers Appl. Maths Stat. 7, 758711.CrossRefGoogle Scholar
Sánchez-Rodríguez, J.A., Raufaste, C. & Argentina, M. 2023 Scaling the tail beat frequency and swimming speed in underwater undulatory swimming. Nat. Commun. 14 (1), 5569.CrossRefGoogle ScholarPubMed
Shirgaonkar, A.A., MacIver, M.A. & Patankar, N.A. 2009 A new mathematical formulation and fast algorithm for fully resolved simulation of self-propulsion. J. Comput. Phys. 228 (7), 2366–2390.CrossRefGoogle Scholar
Strömbom, D. 2011 Collective motion from local attraction. J. Theor. Biol. 283 (1), 145–151.CrossRefGoogle ScholarPubMed
Strömbom, D., Siljestam, M., Park, J. & Sumpter, D.J.T. 2015 The shape and dynamics of local attraction. Eur. Phys. J. Special Topics 224 (17), 3311–3323.CrossRefGoogle Scholar
Tchieu, A.A., Kanso, E. & Newton, P.K. 2012 The finite-dipole dynamical system. Proc. R. Soc. Lond. A: Math. Phys. Engng Sci., 468, 2146, 3006–3026,Google Scholar
Timm, M.L., Pandhare, R.S. & Masoud, H. 2024 Multi-body hydrodynamic interactions in fish-like swimming. Appl. Mech. Rev. 76 (3), 030801.CrossRefGoogle Scholar
Verma, S., Novati, G. & Koumoutsakos, P. 2018 Efficient collective swimming by harnessing vortices through deep reinforcement learning. Proc. Natl Acad. Sci. USA 115 (23), 5849–5854.CrossRefGoogle ScholarPubMed
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. 1995 Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 (6), 1226–1229.CrossRefGoogle Scholar
Vicsek, T. & Zafeiris, A. 2012 Collective motion. Phys. Rep. 517 (3-4), 71–140.CrossRefGoogle Scholar
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241 (5387), 290–291.CrossRefGoogle Scholar
Weihs, D. & Farhi, E. 2017 Passive forces aiding coordinated groupings of swimming animals. Theor. Appl. Mech. Lett. 7 (5), 276–279.CrossRefGoogle Scholar
Xu, G.D., Duan, W.Y. & Xu, W.H. 2017 The propulsion of two flapping foils with tandem configuration and vortex interactions. Phys. Fluids 29 (9), 097102.CrossRefGoogle Scholar
Zhu, X., He, G. & Zhang, X. 2014 Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration. Phys. Rev. Lett. 113 (23), 238105.CrossRefGoogle ScholarPubMed
Zumaya, M.D.;n, Larralde, Hán, Aldana, M. 2018 Delay in the dispersal of flocks moving in unbounded space using long-range interactions. Sci. Rep. UK 8 (1), 15872.CrossRefGoogle ScholarPubMed
Supplementary material: File

Mao et al. supplementary material movie 1

S0 Dynamic evolution of mode S0.
Download Mao et al. supplementary material movie 1(File)
File 25 MB
Supplementary material: File

Mao et al. supplementary material movie 2

S1 Dynamic evolution of mode S1.
Download Mao et al. supplementary material movie 2(File)
File 18.2 MB
Supplementary material: File

Mao et al. supplementary material movie 3

S2 Dynamic evolution of mode S2.
Download Mao et al. supplementary material movie 3(File)
File 19.4 MB
Supplementary material: File

Mao et al. supplementary material movie 4

M1 Dynamic evolution of mode M1.
Download Mao et al. supplementary material movie 4(File)
File 49.6 MB