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Stochastic dynamics of order and disorder in milling whirligig beetles

Jitesh Jhawar1,2,3,4,*, Vishwesha Guttal3,†, and Shashi Thutupalli5,6,†

  • *Contact author: jitesh.jhawar@ahduni.edu.in
  • †These authors contributed equally to this work.
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Phys. Rev. Research 7, 043308 – Published 17 December, 2025

DOI: https://doi.org/10.1103/kmbg-sqqc

Abstract

The dynamics of polar order have received much attention in the literature on collective motion. However, milling dynamics, where organisms exhibit rotational motion around the group center, is understudied. Here, we investigate the stochastic dynamics of milling swarms of whirligig beetles [Dineutus discolor (Coleoptera: Gyrinidae)] in small to medium groups (50 to 200) where noise arising due to the finite sizes of the swarms cannot be ignored. We quantify the milling order parameter of beetle swarms and show that they exhibit a “transition” from a disordered state to a milling state as a function of group size. We employ a method of data-driven discovery of dynamical equations to obtain stochastic differential equations (SDEs) that disentangle deterministic and stochastic factors driving the milling dynamics. Crucially, we find that the milling order at large group sizes can be explained largely by the deterministic terms of the SDE, while intrinsic stochasticity switches the system between clockwise and counterclockwise milling states. Furthermore, we find that an agent-based model in which each agent aligns with the average direction of its local neighbors and attracts toward neighbors on a larger spatial scale produces stochastic dynamics of milling (and corresponding SDEs) that are consistent with the data-derived SDEs. However, contrary to expectations from mean field theory, both data and explicit agent-based models show that the dynamical equations can change their qualitative features with the group size even when individual interaction rules do not change. In summary, we show evidence for the milling order transition and characterize the role of intrinsic stochastic dynamics in swarming beetles.

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References (74)

  1. D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature (London) 393, 440 (1998).
  2. S. J. Altschuler, S. B. Angenent, Y. Wang, and L. F. Wu, On the spontaneous emergence of cell polarity, Nature (London) 454, 886 (2008).
  3. F. Larroya and T. Galla, Demographic noise in complex ecological communities, J. Phys.: Complexity 4, 025012 (2023).
  4. A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Adv. Biophys. 22, 1 (1986).
  5. B. Partridge, The effect of school size on the structure and dynamics of minnow schools, Anim. Behav. 28, 68 (1980).
  6. C. Becco, N. Vandewalle, J. Delcourt, and P. Poncin, Experimental evidences of a structural and dynamical transition in fish school, Physica A 367, 487 (2006).
  7. O. Feinerman, I. Pinkoviezky, A. Gelblum, E. Fonio, and N. S. Gov, The physics of cooperative transport in groups of ants, Nat. Phys. 14, 683 (2018).
  8. A. Kolpas, J. Moehlis, and I. G. Kevrekidis, Coarse-grained analysis of stochasticity-induced switching between collective motion states, Proc. Natl. Acad. Sci. USA 104, 5931 (2007).
  9. C. A. Yates, R. Erban, C. Escudero, I. D. Couzin, J. Buhl, I. G. Kevrekidis, P. K. Maini, and D. J. Sumpter, Inherent noise can facilitate coherence in collective swarm motion, Proc. Natl. Acad. Sci. USA 106, 5464 (2009).
  10. A. M. Reynolds, Intrinsic stochasticity and the emergence of collective behaviours in insect swarms, Eur. Phys. J. E 44, 1 (2021).
  11. J. Jhawar, R. G. Morris, and V. Guttal, Deriving mesoscopic models of collective behavior for finite populations, in Integrated Population Biology and Modeling, Part B, Handbook of Statistics, edited by S. R. Arni, S. Rao, and C. R. Rao (Elsevier, Amsterdam, Netherlands, 2019), Vol. 40, Chap. 13, pp. 551–594.
  12. N. Miller and R. Gerlai, From schooling to shoaling: Patterns of collective motion in zebrafish (Danio rerio), PLoS One 7, e48865 (2012).
  13. D. Sankey, L. O’Bryan, S. Garnier, G. Cowlishaw, P. Hopkins, M. Holton, I. Fürtbauer, and A. King, Consensus of travel direction is achieved by simple copying, not voting, in free-ranging goats, R. Soc. Open Sci. 8, 201128 (2021).
  14. A. Ordemann, Vortex-swarming of the zooplankton Daphnia, Biol. Phys. 2, 5 (2002).
  15. R. Mach and F. Schweitzer, Modeling vortex swarming in Daphnia, Bull. Math. Biol. 69, 539 (2007).
  16. S. Bazazi, K. S. Pfennig, N. O. Handegard, and I. D. Couzin, Vortex formation and foraging in polyphenic spadefoot toad tadpoles, Behav. Ecol. Sociobiol. 66, 879 (2012).
  17. P. Patra, K. Beyer, A. Jaiswal, A. Battista, K. Rohr, F. Frischknecht, and U. S. Schwarz, Collective migration reveals mechanical flexibility of malaria parasites, Nat. Phys. 18, 586 (2022).
  18. C. Castellano, S. Fortunato, and V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591 (2009).
  19. T. Biancalani, L. Dyson, and A. J. McKane, Noise-induced bistable states and their mean switching time in foraging colonies, Phys. Rev. Lett. 112, 038101 (2014).
  20. J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: How birds fly together, Phys. Rev. Lett. 75, 4326 (1995).
  21. A. J. McKane and T. J. Newman, Stochastic models in population biology and their deterministic analogs, Phys. Rev. E 70, 041902 (2004).
  22. S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  23. S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, Data-driven discovery of partial differential equations, Sci. Adv. 3, e1602614 (2017).
  24. A. Nabeel, V. Jadhav, D. Raj, C. Sire, G. Theraulaz, R. Escobedo, S. K. Iyer, and V. Guttal, Data-driven discovery of stochastic dynamical equations of collective motion, Phys. Biol. 20, 056003 (2023).
  25. A. Nabeel, A. Karichannavar, S. Palathingal, J. Jhawar, D. B. Brückner, D. Raj M, and V. Guttal, Discovering stochastic dynamical equations from ecological time series data, Am. Nat. 205, E100 (2025).
  26. D. B. Brückner, A. Fink, C. Schreiber, P. J. Röttgermann, J. O. Rädler, and C. P. Broedersz, Stochastic nonlinear dynamics of confined cell migration in two-state systems, Nat. Phys. 15, 595 (2019).
  27. J. Jhawar, R. G. Morris, U. Amith-Kumar, M. D. Raj, T. Rogers, H. Rajendran, and V. Guttal, Noise-induced schooling of fish, Nat. Phys. 16, 488 (2020).
  28. I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theor. Biol. 218, 1 (2002).
  29. H. L. Devereux, C. R. Twomey, M. S. Turner, and S. Thutupalli, Whirligig beetles as corralled active Brownian particles, J. R. Soc. Interface 18, 20210114 (2021).
  30. See Supplemental Material at http://link.aps.org/supplemental/10.1103/kmbg-sqqc for movies, data files, and analysis code.
  31. MATLAB, version 9.0.0.341360 (R2016a), The MathWorks Inc., Natick, MA, 2016.
  32. V. Guttal and C. Jayaprakash, Impact of noise on bistable ecological systems, Ecol. Modell. 201, 420 (2007).
  33. C. Gardiner, Stochastic Methods (Springer, Berlin, 2009), Vol. 4.
  34. R. Friedrich, J. Peinke, M. Sahimi, and M. R. R. Tabar, Approaching complexity by stochastic methods: From biological systems to turbulence, Phys. Rep. 506, 87 (2011).
  35. L. Boninsegna, F. Nüske, and C. Clementi, Sparse learning of stochastic dynamical equations, J. Chem. Phys. 148, 241723 (2018).
  36. J. Jhawar and V. Guttal, Noise-induced effects in collective dynamics and inferring local interactions from data, Philos. Trans. R. Soc. B 375, 20190381 (2020).
  37. K. Tunstrøm, Y. Katz, C. C. Ioannou, C. Huepe, M. J. Lutz, and I. D. Couzin, Collective states, multistability and transitional behavior in schooling fish, PLoS Comput. Biol. 9, e1002915 (2013).
  38. A. Rabani, G. Ariel, and A. Be'er, Collective motion of spherical bacteria, PLoS One 8, e83760 (2013).
  39. N. T. Ouellette, Empirical questions for collective-behaviour modelling, Pramana 84, 353 (2015).
  40. P. Bera, A. Wasim, J. Mondal, and P. Ghosh, Mechanistic underpinning of cell aspect ratio-dependent emergent collective motions in swarming bacteria, Soft Matter 17, 7322 (2021).
  41. D. Martin, H. Chaté, C. Nardini, A. Solon, J. Tailleur, and F. Van Wijland, Fluctuation-induced phase separation in metric and topological models of collective motion, Phys. Rev. Lett. 126, 148001 (2021).
  42. I. D. Couzin, J. Krause, N. R. Franks, and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature (London) 433, 513 (2005).
  43. B. Yuce, E. Mastrocinque, M. S. Packianather, A. Lambiase, and D. T. Pham, The bees algorithm and its applications, in Handbook of Research on Artificial Intelligence techniques and Algorithms, edited by P. Vasant (IGI Global, 2015), pp. 122–151.
  44. D. J. Sumpter and M. Beekman, From nonlinearity to optimality: Pheromone trail foraging by ants, Anim. Behav. 66, 273 (2003).
  45. L. Dyson, C. A. Yates, J. Buhl, and A. J. McKane, Onset of collective motion in locusts is captured by a minimal model, Phys. Rev. E 92, 052708 (2015).
  46. J. E. Herbert-Read, A. Perna, R. P. Mann, T. M. Schaerf, D. J. Sumpter, and A. J. Ward, Inferring the rules of interaction of shoaling fish, Proc. Natl. Acad. Sci. USA 108, 18726 (2011).
  47. D. S. Calovi, A. Litchinko, V. Lecheval, U. Lopez, A. Pérez Escudero, H. Chaté, C. Sire, and G. Theraulaz, Disentangling and modeling interactions in fish with burst-and-coast swimming reveal distinct alignment and attraction behaviors, PLoS Comput. Biol. 14, e1005933 (2018).
  48. L. Jiang, Y. Cheng, S. Gao, Y. Zhong, C. Ma, T. Wang, and Y. Zhu, Emergence of social cluster by collective pairwise encounters in Drosophila, eLife 9, e51921 (2020).
  49. J. G. Puckett, R. Ni, and N. T. Ouellette, Time-frequency analysis reveals pairwise interactions in insect swarms, Phys. Rev. Lett. 114, 258103 (2015).
  50. Y. Katz, K. Tunstrøm, C. C. Ioannou, C. Huepe, and I. D. Couzin, Inferring the structure and dynamics of interactions in schooling fish, Proc. Natl. Acad. Sci. USA 108, 18720 (2011).
  51. R. Lukeman, Y.-X. Li, and L. Edelstein-Keshet, Inferring individual rules from collective behavior, Proc. Natl. Acad. Sci. USA 107, 12576 (2010).
  52. D. S. Calovi, U. Lopez, S. Ngo, C. Sire, H. Chaté, and G. Theraulaz, Swarming, schooling, milling: Phase diagram of a data-driven fish school model, New J. Phys. 16, 015026 (2014).
  53. L. Jiang, L. Giuggioli, A. Perna, R. Escobedo, V. Lecheval, C. Sire, Z. Han, and G. Theraulaz, Identifying influential neighbors in animal flocking., PLoS Comput. Biol. 13, e1005822 (2017).
  54. W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology (Springer, Berlin, 1984).
  55. E. Dogan and E. J. Allen, Derivation of stochastic partial differential equations for reaction-diffusion processes, Stochastic Anal. Appl. 29, 424 (2011).
  56. J. Buhl, D. J. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller, and S. J. Simpson, From disorder to order in marching locusts, Science 312, 1402 (2006).
  57. A. Attanasi et al., Information transfer and behavioural inertia in starling flocks, Nat. Phys. 10, 691 (2014).
  58. F. Ginelli, The physics of the Vicsek model, Eur. Phys. J.: Spec. Top. 225, 2099 (2016).
  59. W. L. Romey and A. R. Lamb, Flash expansion threshold in whirligig swarms, PLoS One 10, e0136467 (2015).
  60. W. Romey, A. Walston, and P. Watt, Do 3-D predators attack the margins of 2-D selfish herds? Behav. Ecol. 19, 74 (2008).
  61. J. Jhawar, Beetles data analysis: Code, data, and movies for “Stochastic dynamics of order and disorder in milling whirligig beetles”, Zenodo (2025), https://doi.org/10.5281/zenodo.17645611
  62. H. Musoff and P. Zarchan, Fundamentals of Kalman Filtering: A Practical Approach, edited by P. Zarchan, H. Musoff, and F. K. Lu (American Institute of Aeronautics and Astronautics, Inc., Virginia, 2013).
  63. C. Vamoş and M. Crǎciun, Separation of components from a scale mixture of Gaussian white noises, Phys. Rev. E 81, 051125 (2010).
  64. I. Gómez et al., Detecting events in time series using sliding noise tests: Application to atmospheric observations, J. Atmos. Sci. 71, 1133 (2014).
  65. Z. Tang, Y. Liu, and W. Zhang, Multi-level local mean decomposition for noise reduction in engineering signals, Eng. Rep. 5, e12677 (2023).
  66. F. J. Massey Jr, The Kolmogorov-Smirnov test for goodness of fit, J. Am. Stat. Assoc. 46, 68 (1951).
  67. M. Nagy, Z. Ákos, D. Biro, and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature (London) 464, 890 (2010).
  68. S. M. Buchanan, J. S. Kain, and B. L. De Bivort, Neuronal control of locomotor handedness in Drosophila, Proc. Natl. Acad. Sci. USA 112, 6700 (2015).
  69. T. W. Dunn, Y. Mu, S. Narayan, O. Randlett, E. A. Naumann, C.-T. Yang, A. F. Schier, J. Freeman, F. Engert, and M. B. Ahrens, Brain-wide mapping of neural activity controlling zebrafish exploratory locomotion, eLife 5, e12741 (2016).
  70. T. R. Stickland, N. F. Britton, and N. R. Franks, Models of information flow in ant foraging: The benefits of both attractive and repulsive signals, in Information Processing in Social Insects, edited by C. Detrain, J. L. Deneubourg, and J. M. Pasteels (Birkhäuser, Basel, 1999), pp. 83–100.
  71. A. Reina and J. A. Marshall, Negative feedback may suppress variation to improve collective foraging performance, PLoS Comput. Biol. 18, e1010090 (2022).
  72. D. T. Gillespie, The chemical Langevin equation, J. Chem. Phys. 113, 297 (2000).
  73. V. Guttal and C. Jayaprakash, Changing skewness: An early warning signal of regime shifts in ecosystems, Ecol. Lett. 11, 450 (2008).
  74. B. Lafoux, P. Bernard, B. Thiria, and R. Godoy-Diana, Confinement-driven state transition and bistability in schooling fish, Phys. Rev. E 110, 034613 (2024).

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Phys. Rev. Research 7, 043308– Published 17 December, 2025
  • Received 8 April 2024
  • Accepted 11 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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