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Analyzing fish movement as a persistent turning walker

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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.

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References

  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)

  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. 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)

    MathSciNet  Google Scholar 

  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. Balc T., Arkin R.: Behavior-based formation control for multi-robot teams. IEEE Trans. Robot. Autom. 14, 926–939 (1998)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. Bianchi C., Cleur E.M.: Indirect estimation of stochastic differential equation models: some computational experiments. Comput. Econ. 9, 257–274 (1996)

    Article  MATH  Google Scholar 

  8. Borenstein J., Koren Y.: The vector field histogram—fast obstacle avoidance for mobile robots. IEEE Trans. Robot. Autom. 7, 278–288 (1991)

    Article  Google Scholar 

  9. Brillinger D.R.: A particle migrating randomly on a sphere. J. Theor. Probab. 10, 429–443 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Caillol J-M.: Random walks on hyperspheres of arbitrary dimensions. J. Phys. A Math. Gen. 37, 3077–3083 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  15. Grégoire G., Chaté H., Tu Y.: Moving and staying together without a leader. Phys. D 181, 157–170 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  20. Degond P., Motsch S.: Large scale dynamics of the persistent turning walker model of fish behavior. J. Stat. Phys 131, 989–1021 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Do K.D., Jiang Z.P., Pan J.: Underactuated ship global tracking under relaxed conditions. IEEE Trans. Autom. Control 47, 1529–1536 (2002)

    Article  MathSciNet  Google Scholar 

  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. 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)

  24. Gautrais, J., Jost, C., Theraulaz, G.: Key behavioural factors in a self-organised fish school model. Annales Zoologici Fennici (2008, in press)

  25. Gazi V., Passino K.M.: Stability analysis of swarms. IEEE Trans. Autom. Control 48, 692–697 (2003)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  28. Hoffman G.: The random elements in the systematic search behavior of the desert isopod Hemilepistus reaumuri. Behav. Ecol. Sociobiol. 13, 81–92 (1983)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

  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)

    Article  Google Scholar 

  33. Justh E.W., Krishnaprasad P.S.: Equilibria and steering laws for planar formations. Syst. Control Lett. 52, 25–38 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kareiva P.M., Shigesada N.: Analyzing insect movement as a correlated random walk. Oecologia 56, 234–238 (1983)

    Article  Google Scholar 

  35. Kareiva P.: Habitat fragmentation and the stability of predator–prey interactions. Nature 326, 388–390 (1987)

    Article  Google Scholar 

  36. Keenleyside M.H.A.: Some aspects of the schooling behaviour of fish. Behaviour 8, 183–248 (1955)

    Article  Google Scholar 

  37. Komin N., Erdmann U., Schimansky-Geier L.: Random walk theory applied to daphnia motion. Fluct. Noise Lett. 4, 151–159 (2004)

    Article  Google Scholar 

  38. Latombe J.C.: Motion planning: a journey of robots, molecules, digital actors, and other artifacts. Int. J. Robot. Res. 18, 1119–1128 (1999)

    Article  Google Scholar 

  39. Laumond J.P., Risler J.J.: Nonholonomic systems: Controllability and complexity. Theor. Comput. Sci. 157, 101–114 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  41. Marshall J.A., Broucke M.E., Francis B.A.: Formations of vehicles in cyclic pursuit. IEEE Trans. Autom. Control 49, 1963–1974 (2004)

    Article  MathSciNet  Google Scholar 

  42. Murray R.M., Sastry S.S.: Nonholonomic motion planning: steering using sinusoids. IEEE Trans. Autom. Control 38, 700–716 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  44. Patlak C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)

    Article  MathSciNet  Google Scholar 

  45. Patlak C.S.: A mathematical contribution to the study of orientation of organisms. Bull. Math. Biophys. 15, 431–476 (1953)

    Article  MathSciNet  Google Scholar 

  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. Perrin F.: Etude mathématique du mouvement brownien de rotation. Annales scientifiques de l’ENS 45, 1–51 (1928)

    MathSciNet  Google Scholar 

  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)

  49. Radakov D.: Schooling in the Ecology of Fish. Wiley, New York (1973)

    Google Scholar 

  50. Reynolds C.W.: Flocks, herds, and schools: a distributed behavioural model. Comput. Graph. 21, 25–34 (1987)

    Article  Google Scholar 

  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. Schimansky-Geier L., Erdmann U., Komin N.: Advantages of hopping on a zig–zag course. Phys. A 351, 51–59 (2005)

    Article  Google Scholar 

  53. Soria M., Freon P., Chabanet P.: Schooling properties of an obligate and a facultative fish species. J. Fish Biol. 71, 1257–1269 (2007)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  56. Turchin P., Odendaal F.J., Rausher M.D.: Quantifying insect movement in the field. Environ. Entomol. 20, 955–963 (1991)

    Google Scholar 

  57. Turchin P.: Translating foraging movements in heterogeneous environments into the spatial distribution of foragers. Ecology 72, 1253–1256 (1991)

    Article  Google Scholar 

  58. Turchin P.: Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution in Animals and Plants. Sinauer Associates, Sunderland (1998)

    Google Scholar 

  59. Uhlenbeck G.E., Ornstein L.S.: On the theory of Brownian motion. Phys. Rev. 36, 823–841 (1930)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  61. Viscido S.V., Parrish J.K., Grünbaum D.: Factors influencing the structure and maintenance of fish schools. Ecol. Modell. 206, 153–165 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. C. R. Cognition Animale, CNRS UMR 5169, Univ. P. Sabatier, Toulouse, France

    Jacques Gautrais, Christian Jost & Guy Theraulaz

  2. Inst. de Recherche pour le Développement, La Réunion, France

    Marc Soria

  3. IRIDIA, Université Libre de Bruxelles, Brussels, Belgium

    Alexandre Campo

  4. Institut de Mathématiques de Toulouse, CNRS UMR 5219, Univ. P. Sabatier, Toulouse, France

    Sébastien Motsch

  5. LAPLACE, CNRS UMR 5213, Univ. P. Sabatier, Toulouse, France

    Richard Fournier & Stéphane Blanco

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  1. Jacques Gautrais
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Correspondence to Jacques Gautrais.

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Gautrais, J., Jost, C., Soria, M. et al. Analyzing fish movement as a persistent turning walker. J. Math. Biol. 58, 429–445 (2009). https://doi.org/10.1007/s00285-008-0198-7

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  • DOI: https://doi.org/10.1007/s00285-008-0198-7

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