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Journal of Mathematical Biology

, Volume 58, Issue 3, pp 429–445 | Cite as

Analyzing fish movement as a persistent turning walker

  • Jacques GautraisEmail author
  • Christian Jost
  • Marc Soria
  • Alexandre Campo
  • Sébastien Motsch
  • Richard Fournier
  • Stéphane Blanco
  • Guy Theraulaz
Article
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Abstract

The trajectories of Kuhlia mugil fish swimming freely in a tank are analyzed in order to develop a model of spontaneous fish movement. The data show that K. mugil displacement is best described by turning speed and its auto-correlation. The continuous-time process governing this new kind of displacement is modelled by a stochastic differential equation of Ornstein–Uhlenbeck family: the persistent turning walker. The associated diffusive dynamics are compared to the standard persistent random walker model and we show that the resulting diffusion coefficient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other fish or the environment can be added to this spontaneous movement model we quantify the effect of tank walls on the turning speed and adequately reproduce the characteristics of the observed fish trajectories.

Keywords

Fish displacement model Stochastic model Nonlinear diffusion Ornstein–Uhlenbeck process 
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References

  1. 1.
    Alt, W.: Modelling of motility in biological systems. In: McKenna J., Temam R. (eds.) ICIAM’87 Proceedings of the First International Conference on Industrial and Applied Mathematics, pp. 15–30. Society for Industrial and Applied Mathematics, Philadelphia (1988)Google Scholar
  2. 2.
    Alt W.: Correlation analysis of two-dimensional locomotion paths. In: Alt, W., Hoffmann, G.(eds) Biological Motion, Lecture Notes in Biomathematics, vol 89. Springer (1990)Google Scholar
  3. 3.
    Aoki I.: An analysis of the schooling behavior of fish: internal organization and communication process. Bull. Ocean Res. Inst. Univ. Tokyo 12, 1–65 (1980)MathSciNetGoogle Scholar
  4. 4.
    Bai, H., Arcaka, M., Wen, J.T.: Adaptive design for reference velocity recovery in motion coordination. Syst. Control Lett. (2008, in press) doi: 10.1016/j.sysconle.2007.07.003
  5. 5.
    Balc T., Arkin R.: Behavior-based formation control for multi-robot teams. IEEE Trans. Robot. Autom. 14, 926–939 (1998)CrossRefGoogle Scholar
  6. 6.
    Benhamou S.: How to reliably estimate the tortuosity of an animal’s path: straightness, sinuosity, or fractal dimension?. J. Theor. Biol. 229, 209–220 (2004)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bianchi C., Cleur E.M.: Indirect estimation of stochastic differential equation models: some computational experiments. Comput. Econ. 9, 257–274 (1996)zbMATHCrossRefGoogle Scholar
  8. 8.
    Borenstein J., Koren Y.: The vector field histogram—fast obstacle avoidance for mobile robots. IEEE Trans. Robot. Autom. 7, 278–288 (1991)CrossRefGoogle Scholar
  9. 9.
    Brillinger D.R.: A particle migrating randomly on a sphere. J. Theor. Probab. 10, 429–443 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brillinger D.R., Preisler H.K., Ager A.A., Kie J.G., Stewart B.S.: Employing stochastic differential equations to model wildlife motion. Bull. Braz. Math. Soc. New Ser 33, 385–408 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Caillol J-M.: Random walks on hyperspheres of arbitrary dimensions. J. Phys. A Math. Gen. 37, 3077–3083 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Camazine S., Deneubourg J-L., Franks N., Sneyd J., Theraulaz G., Bonabeau E.: Self-Organization in Biological Systems. Princeton University Press, Princeton (2001)Google Scholar
  13. 13.
    Casellas E., Gautrais J., Fournier R., Blanco S., Combe M., Fourcassie V., Theraulaz G., Jost C.: From individual to collective displacements in heterogeneous environments. J. Theor. Biol. 250, 424–434 (2007)CrossRefGoogle Scholar
  14. 14.
    Challet M., Jost C., Grimal A., Lluc J., Theraulaz G.: How temperature influences displacements and corpse aggregation behaviors in the ant Messor Sancta. Ins. Soc. 52, 309–315 (2005)CrossRefGoogle Scholar
  15. 15.
    Grégoire G., Chaté H., Tu Y.: Moving and staying together without a leader. Phys. D 181, 157–170 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Chowdhury, D.: 100 years of Einstein’s theory of Brownian motion: from pollen grains to protein trains. Resonance (Indian Academy of Sciences) 10, 63. arXiv:cond-mat/0504610 (2005)Google Scholar
  17. 17.
    Cleveland W.S., Grosse E., Shyu W.M.: Local regression models. In: Chambers, J., Hastie, T.J.(eds) Statistical Models in S, pp. 309–376. Wadsworth, Pacific Grove (1992)Google Scholar
  18. 18.
    Couzin I.D., Krause J.K., James R., Ruxton G.D., Franks N.R.: Collective memory and spatial sorting in animal groups. J. Theor. Biol 218, 1–11 (2002)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Couzin I.D., Krause J., Franks N.R., Levin S.A.: Effective leadership and decision making in animal groups on the move. Nature 433, 513–516 (2005)CrossRefGoogle Scholar
  20. 20.
    Degond P., Motsch S.: Large scale dynamics of the persistent turning walker model of fish behavior. J. Stat. Phys 131, 989–1021 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Do K.D., Jiang Z.P., Pan J.: Underactuated ship global tracking under relaxed conditions. IEEE Trans. Autom. Control 47, 1529–1536 (2002)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ferrando, S.E., Kolasa, L.A., Kovacevic, N.: Wave++: a C++ library of signal analysis tools (2007) http://www.scs.ryerson.ca/~lkolasa/CppWavelets.html
  23. 23.
    Fish, F.E.: Performance constraints on the maneuverability of flexible and rigid biological systems. In: Proceedings of the Eleventh International Symposium on Unmanned Untethered Submersible Technology, pp. 394–406. Autonomous Undersea Systems Institute, Durham (1999)Google Scholar
  24. 24.
    Gautrais, J., Jost, C., Theraulaz, G.: Key behavioural factors in a self-organised fish school model. Annales Zoologici Fennici (2008, in press)Google Scholar
  25. 25.
    Gazi V., Passino K.M.: Stability analysis of swarms. IEEE Trans. Autom. Control 48, 692–697 (2003)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Grünbaum D., Viscido S., Parrish J.K.: Extracting interactive control algorithms from group dynamics of schooling fish. In: Kumar, V., Leonard, N.E., Morse, A.S.(eds) Lecture Notes in Control and Information Sciences, pp. 103–117. Springer, Berlin (2004)Google Scholar
  27. 27.
    Hapca S., Crawford J.W., MacMillan K., Wilson M.J., Young I.M.: Modelling nematode movement using time-fractional dynamics. J. Theor. Biol. 248, 212–224 (2007)CrossRefGoogle Scholar
  28. 28.
    Hoffman G.: The random elements in the systematic search behavior of the desert isopod Hemilepistus reaumuri. Behav. Ecol. Sociobiol. 13, 81–92 (1983)CrossRefGoogle Scholar
  29. 29.
    Hoffman G.: The search behavior of the desert isopod Hemilepistus reaumuri as compared with a systematic search. Behav. Ecol. Sociobiol. 13, 93–106 (1983)CrossRefGoogle Scholar
  30. 30.
    Huepe C., Aldana M.: New tools for characterizing swarming systems: a comparison of minimal models. Phys. A 387, 2809–2822 (2008)Google Scholar
  31. 31.
    Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 988–1001 (2003)Google Scholar
  32. 32.
    Jeanson R., Blanco S., Fournier R., Deneubourg J-L., Fourcassié V., Theraulaz G.: A model of animal movements in a bounded space. J. Theor. Biol. 225, 443–451 (2003)CrossRefGoogle Scholar
  33. 33.
    Justh E.W., Krishnaprasad P.S.: Equilibria and steering laws for planar formations. Syst. Control Lett. 52, 25–38 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Kareiva P.M., Shigesada N.: Analyzing insect movement as a correlated random walk. Oecologia 56, 234–238 (1983)CrossRefGoogle Scholar
  35. 35.
    Kareiva P.: Habitat fragmentation and the stability of predator–prey interactions. Nature 326, 388–390 (1987)CrossRefGoogle Scholar
  36. 36.
    Keenleyside M.H.A.: Some aspects of the schooling behaviour of fish. Behaviour 8, 183–248 (1955)CrossRefGoogle Scholar
  37. 37.
    Komin N., Erdmann U., Schimansky-Geier L.: Random walk theory applied to daphnia motion. Fluct. Noise Lett. 4, 151–159 (2004)CrossRefGoogle Scholar
  38. 38.
    Latombe J.C.: Motion planning: a journey of robots, molecules, digital actors, and other artifacts. Int. J. Robot. Res. 18, 1119–1128 (1999)CrossRefGoogle Scholar
  39. 39.
    Laumond J.P., Risler J.J.: Nonholonomic systems: Controllability and complexity. Theor. Comput. Sci. 157, 101–114 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Lin Z., Broucke M.E., Francis B.A.: Local control strategies for groups of mobile autonomous agents. IEEE Trans. Autom. Control 49, 622–629 (2004)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Marshall J.A., Broucke M.E., Francis B.A.: Formations of vehicles in cyclic pursuit. IEEE Trans. Autom. Control 49, 1963–1974 (2004)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Murray R.M., Sastry S.S.: Nonholonomic motion planning: steering using sinusoids. IEEE Trans. Autom. Control 38, 700–716 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Nagy M., Darukab I., Vicsek T.: New aspects of the continuous phase transition in the scalar noise model (SNM) of collective motion. Phys. A 373, 445–454 (2007)CrossRefGoogle Scholar
  44. 44.
    Patlak C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Patlak C.S.: A mathematical contribution to the study of orientation of organisms. Bull. Math. Biophys. 15, 431–476 (1953)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Parrish J.K., Turchin P.: Individual decisions, traffic rules, and emergent pattern in schooling fish. In: Parrish, J.K., Hammer, W.M.(eds) Animal Groups in Three Dimensions, pp. 126–142. Cambridge University Press, London (1997)Google Scholar
  47. 47.
    Perrin F.: Etude mathématique du mouvement brownien de rotation. Annales scientifiques de l’ENS 45, 1–51 (1928)MathSciNetGoogle Scholar
  48. 48.
    Preisler, H.K., Brillinger, D.R., Ager, A.A., Kie, J.G., Akers, R.P.: Stochastic differential equations: a tool for studying animal movement. In: Proceedings of International Union Forest Research Organization (2001)Google Scholar
  49. 49.
    Radakov D.: Schooling in the Ecology of Fish. Wiley, New York (1973)Google Scholar
  50. 50.
    Reynolds C.W.: Flocks, herds, and schools: a distributed behavioural model. Comput. Graph. 21, 25–34 (1987)CrossRefGoogle Scholar
  51. 51.
    Scharstein H.: Paths of carabid beetles walking in the absence of orienting stimuli and the time structure of their motor output. In: Alt, W., Hoffmann, G.(eds) Biological Motion. Lecture Notes in Biomathematics, vol. 89, Springer, Heidelberg (1990)Google Scholar
  52. 52.
    Schimansky-Geier L., Erdmann U., Komin N.: Advantages of hopping on a zig–zag course. Phys. A 351, 51–59 (2005)CrossRefGoogle Scholar
  53. 53.
    Soria M., Freon P., Chabanet P.: Schooling properties of an obligate and a facultative fish species. J. Fish Biol. 71, 1257–1269 (2007)CrossRefGoogle Scholar
  54. 54.
    Sfakiotakis M., Lane D.M., Davies J.B.C.: Review of fish swimming modes for aquatic locomotion. IEEE J. Ocean. Eng. 24, 237–252 (1999)CrossRefGoogle Scholar
  55. 55.
    Tourtellot M.K., Collins R.D., Bell W.J.: The problem of movelength and turn definition in analysis of orientation data. J. Theor. Biol 150, 287–297 (1991)CrossRefGoogle Scholar
  56. 56.
    Turchin P., Odendaal F.J., Rausher M.D.: Quantifying insect movement in the field. Environ. Entomol. 20, 955–963 (1991)Google Scholar
  57. 57.
    Turchin P.: Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology 72, 1253–1256 (1991)CrossRefGoogle Scholar
  58. 58.
    Turchin P.: Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer Associates, Sunderland (1998)Google Scholar
  59. 59.
    Uhlenbeck G.E., Ornstein L.S.: On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930)CrossRefGoogle Scholar
  60. 60.
    Umeda T., Inouye K.: Possible role of contact following in the generation of coherent motion of Dictyostelium cells. J. Theor. Biol 291, 301–308 (2002)CrossRefMathSciNetGoogle Scholar
  61. 61.
    Viscido S.V., Parrish J.K., Grünbaum D.: Factors influencing the structure and maintenance of fish schools. Ecol. Modell. 206, 153–165 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Jacques Gautrais
    • 1
    Email author
  • Christian Jost
    • 1
  • Marc Soria
    • 2
  • Alexandre Campo
    • 3
  • Sébastien Motsch
    • 4
  • Richard Fournier
    • 5
  • Stéphane Blanco
    • 5
  • Guy Theraulaz
    • 1
  1. 1.C. R. Cognition Animale, CNRS UMR 5169Univ. P. SabatierToulouseFrance
  2. 2.Inst. de Recherche pour le DéveloppementLa RéunionFrance
  3. 3.IRIDIAUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Institut de Mathématiques de Toulouse, CNRS UMR 5219Univ. P. SabatierToulouseFrance
  5. 5.LAPLACE, CNRS UMR 5213Univ. P. SabatierToulouseFrance

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