Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Noise-induced schooling of fish


We report on the dynamics of collective alignment in groups of the cichlid fish Etroplus suratensis. Focusing on small- to intermediate-sized groups (10 N 100), we demonstrate that schooling (highly polarized and coherent motion) is noise induced, arising from the intrinsic stochasticity associated with finite numbers of interacting fish. The fewer the fish, the greater the (multiplicative) noise and therefore the greater the likelihood of alignment. Such rare empirical evidence tightly constrains the possible underlying interactions that govern fish alignment, suggesting that E. suratensis either spontaneously change their direction or copy the direction of another fish, without any local averaging (the otherwise canonical mechanism of collective alignment). Our study therefore highlights the importance of stochasticity in behavioural inference. Furthermore, rather than simply obscuring otherwise deterministic dynamics, noise can be fundamental to the characterization of emergent collective behaviours.

Your institute does not have access to this article

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Capturing the stochastic dynamics of ordering in small- to medium-sized groups of fish.
Fig. 2: Steady-state statistics demonstrate high degree of schooling.
Fig. 3: Empirical fitting of the first and second jump moments reveals that schooling is noise induced.
Fig. 4: Pairwise versus ternary interactions: theory and stochastic simulations.

Data availability

All data necessary to reproduce Figs. 2–4 are available via

Code availability

All codes, including readme files necessary to reproduce Figs. 2–4, are available via


  1. Ballerini, M. et al. Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl Acad. Sci. USA 105, 1232–1237 (2008).

    ADS  Article  Google Scholar 

  2. Bialek, W. et al. Statistical mechanics for natural flocks of birds. Proc. Natl Acad. Sci. USA 109, 4786–4791 (2012).

    ADS  Article  Google Scholar 

  3. Pearce, J. G. D., Miller, A. M., Rowlands, G. & Turner, M. S. Role of projection in the control of bird flocks. Proc. Natl Acad. Sci. USA 111, 10422–10426 (2014).

    ADS  Article  Google Scholar 

  4. Becco, C., Vandewalle, N., Delcourt, J. & Poncin, P. Experimental evidences of a structural and dynamical transition in fish school. Physica A 367, 487–493 (2006).

    ADS  Article  Google Scholar 

  5. Herbert-Read, J. E. et al. Inferring the rules of interaction of shoaling fish. Proc. Natl Acad. Sci. USA 108, 18726–18731 (2011).

    ADS  Article  Google Scholar 

  6. Gautrais, J. et al. Deciphering interactions in moving animal groups. PLoS Comput. Biol. 8, e1002678 (2012).

    MathSciNet  Article  Google Scholar 

  7. Ward, A. J. et al. Local interactions and global properties of wild, free-ranging stickleback shoals. R. Soc. Open Sci. 4, 170043 (2017).

    ADS  Article  Google Scholar 

  8. Jiang, L. et al. Identifying influential neighbors in animal flocking. PLoS Comput. Biol. 13, 1–32 (2017).

    ADS  Article  Google Scholar 

  9. Buhl, J. et al. From disorder to order in marching locusts. Science 312, 1402–1406 (2006).

    ADS  Article  Google Scholar 

  10. Yates, C. A. et al. Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl Acad. Sci. USA 106, 5464–5469 (2009).

    ADS  Article  Google Scholar 

  11. Shemesh, Y. et al. High-order social interactions in groups of mice. eLife 2, e00759 (2013).

    Article  Google Scholar 

  12. Rands, S. A., Hayley, M. & Terry, N. L. Red deer synchronise their activity with close neighbours. PeerJ 2, e344 (2014).

    Article  Google Scholar 

  13. Dyson, L., Yates, C. A., Buhl, J. & McKane, A. J. Onset of collective motion in locusts is captured by a minimal model. Phys. Rev. E 92, 052708 (2015).

    ADS  Article  Google Scholar 

  14. Van Kampen, N. G. Stochastic Processes in Physics and Chemistry (Elsevier, 1992).

  15. Horsthemke, W. & Lefever, R. Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology (Springer, 1984).

  16. Ridolfi, L., D’Odorico, P. & Laio, F. Noise-Induced Phenomena in the Environmental Sciences (Cambridge University Press, 2011).

  17. Biancalani, T., Dyson, L. & McKane, A. J. Noise-induced bistable states and their mean switching time in foraging colonies. Phys. Rev. Lett. 112, 038101 (2014).

    ADS  Article  Google Scholar 

  18. Boettiger, C. From noise to knowledge: how randomness generates novel phenomena and reveals information. Ecol. Lett. 21, 1255–1267 (2018).

    Article  Google Scholar 

  19. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  20. Toner, J. & Tu, Y. Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995).

    ADS  Article  Google Scholar 

  21. Baglietto, G. & Vazquez, F. Flocking dynamics with voter-like interactions. J. Stat. Mech. 2018, 033403 (2018).

    Article  Google Scholar 

  22. Rosenthal, S. B., Twomey, C. R., Hartnett, A. T., Wu, H. S. & Couzin, I. D. Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion. Proc. Natl Acad. Sci. USA 112, 4690–4695 (2015).

    ADS  Article  Google Scholar 

  23. Jhawar, J., Morris, R. G. & Guttal, V. in Handbook of Statistics: Integrated Population Biology and Modeling, Part B Vol. 40 (eds. Rao, A. S. R. S. & Rao, C. R.) Ch. 13 (Elsevier, 2018).

  24. Ramaswamy, S. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323–45 (2010).

    ADS  Article  Google Scholar 

  25. Barré, J., Chétrite, R., Muratori, M. & Peruani, F. Motility-induced phase separation of active particles in the presence of velocity alignment. J. Stat. Phys. 158, 589–600 (2015).

    ADS  MathSciNet  Article  Google Scholar 

  26. Bertin, E. et al. Mesoscopic theory for fluctuating active nematics. New J. Phys. 15, 085032 (2013).

    ADS  Article  Google Scholar 

  27. Dean, D. S. Langevin equation for the density of a system of interacting Langevin processes. J. Phys. A 29, L613 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  28. Laighléis, E. Ó., Evans, M. R. & Blythe, R. A. Minimal stochastic field equations for one-dimensional flocking. Phys. Rev. E 98, 062127 (2018).

    ADS  MathSciNet  Article  Google Scholar 

  29. Chatterjee, P. & Goldenfeld, N. Three-body interactions drive the transition to polar order in a simple flocking model. Phys. Rev. E 100, 040602 (2019).

    ADS  Article  Google Scholar 

  30. Sumpter, D. J. Collective Animal Behavior (Princeton University Press, 2010).

  31. Ioannou, C. C., Guttal, V. & Couzin, I. D. Predatory fish select for coordinated collective motion in virtual prey. Science 337, 1212–1215 (2012).

    ADS  Article  Google Scholar 

  32. Jhawar, J. & Guttal, V. Noise-induced effects in collective dynamics and inferring local interactions from data. Phil. Tran. R. Soc. B (2020).

  33. Lukeman, R., Li, Y. X. & Edelstein-Keshet, L. Inferring individual rules from collective behavior. Proc. Natl Acad. Sci. USA 107, 12576–12580 (2010).

    ADS  Article  Google Scholar 

  34. Katz, Y., Tunstrøm, K., Ioannou, C. C., Huepe, C. & Couzin, I. D. Inferring the structure and dynamics of interactions in schooling fish. Proc. Natl Acad. Sci. USA 108, 18720–18725 (2011).

    ADS  Article  Google Scholar 

  35. Puckett, J. G., Ni, R. & Ouellette, N. T. Time-frequency analysis reveals pairwise interactions in insect swarms. Phys. Rev. Lett. 114, 258103 (2015).

    ADS  Article  Google Scholar 

  36. Calovi, D. S. et al. Disentangling and modeling interactions in fish with burst-and-coast swimming reveal distinct alignment and attraction behaviors. PLoS Comput. Biol. 14, e1005933 (2018).

    Article  Google Scholar 

  37. Gardiner, C. Handbook of Stochastic Methods 3rd edn (Springer, 2003).

  38. Kirman, A. Ants, rationality, and recruitment. Q. J. Econ. 108, 137–156 (1993).

    Article  Google Scholar 

  39. Alfarano, S. & Lux, T. A noise trader model as a generator of apparent financial power laws and long memory. Macroecon. Dyn. 11, 80–101 (2007).

    Article  Google Scholar 

  40. Minors, K., Rogers, T. & Yates, C. A. Noise-driven bias in the non-local voter model. Europhys. Lett. 122, 10004 (2018).

    ADS  Article  Google Scholar 

  41. Mathematica Version 10.0 (Wolfram Research Inc., 2014).

  42. Gillespie, D. T. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976).

    ADS  MathSciNet  Article  Google Scholar 

  43. Gillespie, D. T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977).

    Article  Google Scholar 

  44. Goldberg, D. E. Genetic Algorithms in Search, Optimization & Machine Learning (Addison-Wesley, 1989).

Download references


V.G. thanks C. Jayaprakash for introducing him to the fascinating world of noise-induced phenomena. We acknowledge assistance from S. Chakraborty, E. M. Jos, A. Nabeel, A. Karichannavar and T. Goel. We thank Binoy V. V. for suggestions on schooling fish species native to India and their hatcheries. We also thank S. Ramaswamy for a critical reading of the manuscript and C. C. Ioannou for discussions. J.J. acknowledges support by the CSIR, India, through a research scholarship. R.G.M. acknowledges both the Simons Foundation (USA) and EMBL Australia for funding. M.D.R. acknowledges a DST India INSPIRE faculty award for funding. T.R. acknowledges the Royal Society (UK) for funding. V.G. acknowledges support from the DBT-IISc partnership programme, SERB (DST) and infrastructure support from DST-FIST.

Author information

Authors and Affiliations



V.G. conceived of and oversaw the project. J.J., U.R.A.-K. and H.R. performed experiments. J.J. and R.G.M. analysed and interpreted the data. J.J. and M.D.R. performed simulations. T.R. and R.G.M. performed calculations. R.G.M. wrote the manuscript, with input from J.J., M.D.R. and V.G. J.J. and R.G.M. contributed equally to the manuscript.

Corresponding authors

Correspondence to Jitesh Jhawar, Richard G. Morris or Vishwesha Guttal.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Guy Theraulaz and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Auto-correlation of group polarisation.

Blue, yellow and green solid lines represent the data for N=15, 30, and 60, respectively. Two characteristic timescales are apparent; τ, which encapsulates the rate of initial decay of correlations to zero (solid black lines) and τenv, which is rate of decay of the envelope of quasi-periodic correlations (dotted grey lines).

Extended Data Fig. 2 Numerical check of the extracted SDE.

Milstein-method simulations of the SDE that was extracted from the data [Eq. (3) of the main manuscript]. The results are qualitatively in-line with experimental observations.

Extended Data Fig. 3 Second jump-moments: pairwise vs. ternary.

In agreement with theoretically derived expressions, the simulation-generated diagonal (non-zero) second jump-moments take a similar form for both pairwise and ternary models. Data points are generated by Gillespie simulation (using the stated parameter values), whilst both surfaces and the analytical expressions to which they correspond are taken from theory (Methods Sections II A & B).

Extended Data Fig. 4 Optimisation of higher-order copying interaction models.

Using a Genetic Algorithm in the context of repeated Gillespie simulations (Methods Section III C), we optimise a given model’s specific rates against the experimental data. The results – specifically, large values of r2 and negligible values of ri where i > 2 – imply that pairwise copying is the dominant mode of interaction and that higher order interactions are likely negligible.

Extended Data Fig. 5 Optimisation of higher-order Vicsek-like interaction models.

Using a Genetic Algorithm in the context of repeated Gillespie simulations for higher-order Vicsek-like interaction models (Methods section III D), we optimise a given model’s specific rates against the experimental data. The results confirm that direction-averaging (represented by ri where i > 2) is not represented by the data; this can be inferred from the values of the DKL corresponding to the optimized rates of interaction.

Supplementary information

Supplementary Information

Supplementary experimental details, boundary controls, discussion of one-dimensional toy models, configuration space mixing analysis and spatial schooling model with boundary.

Reporting Summary

Supplementary Video 1

A sample video of the experiments with schools of fish, E. suratensis, in groups of sizes 15, 30 and 60.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jhawar, J., Morris, R.G., Amith-Kumar, U.R. et al. Noise-induced schooling of fish. Nat. Phys. 16, 488–493 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing