- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Stability of flocking in the reciprocal two-species Vicsek model: Effects of relative population, motility, and noise

Aditya Kumar Dutta1,*, Matthieu Mangeat2,†, Heiko Rieger2,‡, Raja Paul1,§, and Swarnajit Chatterjee2,3,∥

  • 1School of Mathematical & Computational Sciences, Indian Association for the Cultivation of Science, Kolkata–700032, India
  • 2Center for Biophysics & Department for Theoretical Physics, Saarland University, 66123 Saarbrücken, Germany
  • 3Laboratoire de Physique Théorique et Modélisation, UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise, France

  • *Contact author: saisakd2137@iacs.res.in
  • †Contact author: mangeat@lusi.uni-sb.de
  • ‡Contact author: heiko.rieger@uni-saarland.de
  • §Contact author: raja.paul@iacs.res.in
  • ∥Contact author: swarnajit.chatterjee@cyu.fr
PDF

Phys. Rev. E 112, 024137 – Published 27 August, 2025

DOI: https://doi.org/10.1103/gkhv-rp16

Abstract

Natural flocks need to cope with various forms of heterogeneities, for instance, their composition, motility, interaction, or environmental factors. Here, we study the effects of such heterogeneities on the flocking dynamics of the reciprocal two-species Vicsek model [Phys. Rev. E 107, 024607 (2023)], which comprises two groups of self-propelled agents with antialigning interspecies interactions and exhibits either parallel or antiparallel flocking states. The parallel and antiparallel flocking states vanish upon reducing the size of one group, and the system transitions to a single-species flock of the majority species. At sufficiently low noise (or high density), the minority species can exhibit collective behavior, antialigning with the liquid state of the majority species. Unequal self-propulsion speeds of the two species strongly encourage antiparallel flocking over parallel flocking. However, when activity landscapes with region-dependent motilities are introduced, parallel flocking is retained if the faster region is given more space, highlighting the role of environmental constraints. Under noise heterogeneity, the colder species (subjected to lower noise) attain higher band velocity compared to the hotter one, temporarily disrupting any parallel flocking, which is subsequently restored. These findings collectively reveal how different forms of heterogeneity, both intrinsic and environmental, can qualitatively reshape flocking behavior in this class of reciprocal two-species models.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (70)

  1. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  2. A. Bottinelli, D. T. J. Sumpter, and J. L. Silverberg, Emergent structural mechanisms for high-density collective motion inspired by human crowds, Phys. Rev. Lett. 117, 228301 (2016).
  3. L. Gómez-Nava, R. Bon, and F. Peruani, Intermittent collective motion in sheep results from alternating the role of leader and follower, Nat. Phys. 18, 1494 (2022).
  4. M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA 105, 1232 (2008).
  5. 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).
  6. F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Andersen, A. Deutsch, and M. Bär, Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria, Phys. Rev. Lett. 108, 098102 (2012).
  7. F. Giavazzi, M. Paoluzzi, M. Macchi, D. Bi, G. Scita, M. L. Manning, R. Cerbino, and M. C. Marchetti, Flocking transitions in confluent tissues, Soft Matter 14, 3471 (2018).
  8. V. Schaller, C. Weber, C. Semmrich, E. Frey, and A. Bausch, Polar patterns of driven filaments, Nature (London) 467, 73 (2010).
  9. A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature (London) 503, 95 (2013).
  10. A. Kaiser, A. Snezhko, and I. S. Aranson, Flocking ferromagnetic colloids, Sci. Adv. 3, e1601469 (2017).
  11. J. Deseigne, O. Dauchot, and H. Chaté, Collective motion of vibrated polar disks, Phys. Rev. Lett. 105, 098001 (2010).
  12. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
  13. 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).
  14. J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E 58, 4828 (1998).
  15. A. P. Solon, H. Chaté, and J. Tailleur, From phase to microphase separation in flocking models: The essential role of nonequilibrium fluctuations, Phys. Rev. Lett. 114, 068101 (2015).
  16. A. M. Menzel, Collective motion of binary self-propelled particle mixtures, Phys. Rev. E 85, 021912 (2012).
  17. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  18. K. L. Kreienkamp and S.H.L. Klapp, Clustering and flocking of repulsive chiral active particles with non-reciprocal couplings, New J. Phys. 24, 123009 (2022).
  19. S. Chatterjee, M. Mangeat, C-U Woo, H. Rieger, and J. D. Noh, Flocking of two unfriendly species: The two-species Vicsek model, Phys. Rev. E 107, 024607 (2023).
  20. D. Martin, D. Seara, Y. Avni, M. Fruchart, and V. Vitelli, The transition to collective motion in nonreciprocal active matter: Coarse graining agent-based models into fluctuating hydrodynamics, arXiv:2307.08251.
  21. M. Mangeat, S. Chatterjee, J. D. Noh, and H. Rieger, Emergent complex phases in a discrete flocking model with reciprocal and non-reciprocal interactions, Commun. Phys. 8, 186 (2025).
  22. J. Grauer, H. Löwen, A. Be'er, and B. Liebchen, Swarm hunting and cluster ejections in chemically communicating active mixtures, Sci. Rep. 10, 5594 (2020).
  23. G. Tucci, R. Golestanian, and S. Saha, Nonreciprocal collective dynamics in a mixture of phoretic Janus colloids, New J. Phys. 26, 073006 (2024).
  24. G. Tucci, G. Pisegna, R. Golestanian, and S. Saha, Hydrodynamic stresses in a multi-species suspension of active Janus colloids, Phys. Rev. Res. 7, 033003 (2025).
  25. E. Ben-Jacob, A. Finkelshtein, G. Ariel, and C. Ingham, Multispecies swarms of social microorganisms as moving ecosystems, Trends Microbiol. 24, 257 (2016).
  26. J. Herbert-Read, S. Krause, L. Morrell, T. Schaerf, J. Krause, and A. Ward, The role of individuality in collective group movement, Proc. R. Soc. B 280, 20122564 (2013).
  27. B. Ilkanaiv, D. B. Kearns, G. Ariel, and A. Be'er, Effect of cell aspect ratio on swarming bacteria, Phys. Rev. Lett. 118, 158002 (2017).
  28. S. Peled, S. D. Ryan, S. Heidenreich, M. Bär, G. Ariel, and A. Be'er, Heterogeneous bacterial swarms with mixed lengths, Phys. Rev. E 103, 032413 (2021).
  29. A. Jose, G. Ariel, and A. Be'er, Physical characteristics of mixed-species swarming colonies, Phys. Rev. E 105, 064404 (2022).
  30. G. Natan, V. Worlitzer, G. Ariel, and A. Be'er, Mixed-species bacterial swarms show an interplay of mixing and segregation across scales, Sci. Rep. 12, 16500 (2022).
  31. W. Zuo and Y. Wu, Dynamic motility selection drives population segregation in a bacterial swarm, Proc. Natl. Acad. Sci. USA 117, 4693 (2020).
  32. T. Kolb and D. Klotsa, Active binary mixtures of fast and slow hard spheres, Soft Matter 16, 1967 (2020).
  33. S. Pattanayak, J. P. Singh, M. Kumar, and S. Mishra, Speed inhomogeneity accelerates information transfer in polar flock, Phys. Rev. E 101, 052602 (2020).
  34. M. Forget, S. Adiba, L. G. Brunnet, and S. De Monte, Heterogeneous individual motility biases group composition in a model of aggregating cells, Front. Ecol. Evol. 10, 1052309 (2022).
  35. S. Maity and A. Morin, Spontaneous demixing of binary colloidal flocks, Phys. Rev. Lett. 131, 178304 (2023).
  36. S. Pigolotti and R. Benzi, Selective advantage of diffusing faster, Phys. Rev. Lett. 112, 188102 (2014).
  37. G. Book, C. Ingham, and G. Ariel, Modeling cooperating micro-organisms in antibiotic environment, PLoS One 12, e0190037 (2017).
  38. V. Khodygo, M. T. Swain, and A. Mughal, Homogeneous and heterogeneous populations of active rods in two-dimensional channels, Phys. Rev. E 99, 022602 (2019).
  39. E. Lardet, L. Chen, and T. Bertrand, Flocking beyond one species: Novel phase coexistence in a generalized two-species Vicsek model, arXiv:2503.17617.
  40. B. Khelfa, R. Korbmacher, A. Schadschneider, and A. Tordeux, Heterogeneity-induced lane and band formation in self-driven particle systems, Sci. Rep. 12, 4768 (2022).
  41. G. Ariel, O. Rimer, and E. Ben-Jacob, Order–disorder phase transition in heterogeneous populations of self-propelled particles, J. Stat. Phys. 158, 579 (2015).
  42. G. Netzer, Y. Yarom, and G. Ariel, Heterogeneous populations in a network model of collective motion, Physica A 530, 121550 (2019).
  43. E. Ilker and J. F. Joanny, Phase separation and nucleation in mixtures of particles with different temperatures, Phys. Rev. Res. 2, 023200 (2020).
  44. A. Wilde and C. W. Mullineaux, Light-controlled motility in prokaryotes and the problem of directional light perception, FEMS Microbiol. Rev. 41, 900 (2017).
  45. G. Frangipane, D. Dell'Arciprete, S. Petracchini, C. Maggi, F. Saglimbeni, S. Bianchi, G. Vizsnyiczai, M. L. Bernardini, and R. Di Leonardo, Dynamic density shaping of photokinetic E. coli, eLife 7, e36608 (2018).
  46. N. A. Söker, S. Auschra, V. Holubec, K. Kroy, and F. Cichos, How activity landscapes polarize microswimmers without alignment forces, Phys. Rev. Lett. 126, 228001 (2021).
  47. S. Auschra, V. Holubec, N. A. Söker, F. Cichos, and K. Kroy, Polarization-density patterns of active particles in motility gradients, Phys. Rev. E 103, 062601 (2021).
  48. A. Wysocki, A. K. Dasanna, and H. Rieger, Interacting particles in an activity landscape, New J. Phys. 24, 093013 (2022).
  49. J. Huang and Z. G. Shao, Collective motion of binary chiral particle mixtures with environmental complex noise, Phys. Rev. E 110, 034135 (2024).
  50. C. J. O. Reichhardt and C. Reichhardt, Avalanche dynamics for active matter in heterogeneous media, New J. Phys. 20, 025002 (2018).
  51. R. Saavedra and F. Peruani, Self-trapping of active particles with nonreciprocal interactions in disordered media, Phys. Rev. E 110, 064602 (2024).
  52. P. Rahmani, F. Peruani, and P. Romanczuk, Topological flocking models in spatially heterogeneous environments, Commun. Phys. 4, 206 (2021).
  53. L. Chen and J. Toner, Universality for moving stripes: A hydrodynamic theory of polar active smectics, Phys. Rev. Lett. 111, 088701 (2013).
  54. T. C. Adhyapak, S. Ramaswamy, and J. Toner, Live soap: Stability, order, and fluctuations in apolar active smectics, Phys. Rev. Lett. 110, 118102 (2013).
  55. A. K. Dutta, M. Mangeat, H. Rieger, R. Paul, and S. Chatterjee, Supplementary movies for stability of flocking in the reciprocal two-species Vicsek model: Effects of relative population, motility, and noise, Zenodo (2025), https://doi.org/10.5281/zenodo.15241518.
  56. See Supplemental Material at http://link.aps.org/supplemental/10.1103/gkhv-rp16 for additional movies.
  57. N. Kumar, H. Soni, S. Ramaswamy, and A. K. Sood, Flocking at a distance in active granular matter, Nat. Commun. 5, 4688 (2014).
  58. N. Koumakis, A. Gnoli, C. Maggi, A. Puglisi, and R. Di Leonardo, Mechanism of self-propulsion in 3D-printed active granular particles, New J. Phys. 18, 113046 (2016).
  59. J. Chen, X. Lei, Y. Xiang, M. Duan, X. Peng, and H. P. Zhang, Emergent chirality and hyperuniformity in an active mixture with nonreciprocal interactions, Phys. Rev. Lett. 132, 118301 (2024).
  60. P. K. Bera and A. K. Sood, Motile dissenters disrupt the flocking of active granular matter, Phys. Rev. E 101, 052615 (2020).
  61. T. Chen, X. Yang, B. Zhang, J. Li, J. Pan, and Y. Wang, Scale-inspired programmable robotic structures with concurrent shape morphing and stiffness variation, Sci. Robot. 9, eadl0307 (2024).
  62. R. F. Storms, C. Carere, F. Zoratto, and C. K. Hemelrijk, Complex patterns of collective escape in starling flocks under predation, Behav. Ecol. Sociobiol. 73, 10 (2019).
  63. M. Papadopoulou, H. Hildenbrandt, D. W. E. Sankey, S. J. Portugal, and C. K. Hemelrijk, Emergence of splits and collective turns in pigeon flocks under predation, R. Soc. Open Sci. 9, 211898 (2022).
  64. J. E. Herbert-Read, E. Rosén, A. Szorkovszky, C. C. Ioannou, B. Rogell, A. Perna, I. W. Ramnarine, A. Kotrschal, N. Kolm, J. Krause, and D. J. T. Sumpter, How predation shapes the social interaction rules of shoaling fish, Proc. R. Soc. B 284, 20171126 (2017).
  65. B. Momeni, K. A. Brileya, M. W. Fields, and W. Shou, Strong inter-population cooperation leads to partner intermixing in microbial communities, eLife 2, e00230 (2013).
  66. D. R. Farine, L. M. Aplin, C. J. Garroway, R. P. Mann, and B. C. Sheldon, Collective decision making and social interaction rules in mixed-species flocks of songbirds, Anim. Behav. 95, 173 (2014).
  67. D. Papageorgiou, B. Nyaguthii, and D. R. Farine, Compromise or choose: Shared movement decisions in wild vulturine guineafowl, Commun. Biol. 7, 95 (2024).
  68. F. Ginelli, F. Peruani, M-H. Pillot, H. Chaté, G. Theraulaz, and R. Bon, Intermittent collective dynamics emerge from conflicting imperatives in sheep herds, Proc. Natl. Acad. Sci. USA 112, 12729 (2015).
  69. R. Martinez, F. Alarcon, D. R. Rodriguez, J. L. Aragones, and C. Valeriani, Collective behavior of Vicsek particles without and with obstacles, Eur. Phys. J. E 41, 91 (2018).
  70. M. Karmakar, S. Chatterjee, M. Mangeat, H. Rieger, and R. Paul, Jamming and flocking in the restricted active potts model, Phys. Rev. E 108, 014604 (2023).

Outline

Information

Phys. Rev. E 112, 024137– Published 27 August, 2025
  • Received 18 April 2025
  • Accepted 11 August 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation