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J. Math. Biol. (2009) 58:429445
DOI 10.1007/s00285-008-0198-7
Received: 13 February 2008 / Revised: 30 May 2008 / Published online: 28 June 2008 Springer-Verlag 2008
Abstract The trajectories of Kuhlia mugil sh swimming freely in a tank are analyzed in order to develop a model of spontaneous sh 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 OrnsteinUhlenbeck 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 coefcient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other sh 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 sh trajectories.
Keywords Fish displacement model Stochastic model Nonlinear diffusion
OrnsteinUhlenbeck process
J. Gautrais (B) C. Jost G. Theraulaz
C. R. Cognition Animale, CNRS UMR 5169, Univ. P. Sabatier, Toulouse, France e-mail: [email protected]
M. Soria
Inst. de Recherche pour le Dveloppement, La Runion, France
A. Campo
IRIDIA, Universit Libre de Bruxelles, Brussels, Belgium
S. Motsch
Institut de Mathmatiques de Toulouse, CNRS UMR 5219, Univ. P. Sabatier, Toulouse, France
R. Fournier S. Blanco
LAPLACE, CNRS UMR 5213, Univ. P. Sabatier, Toulouse, France
Mathematical Biology
Analyzing sh movement as a persistent turning walker
Jacques Gautrais Christian Jost Marc Soria
Alexandre Campo Sbastien Motsch Richard Fournier
Stphane Blanco Guy Theraulaz
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1 Introduction
The highly coordinated displacement of hundreds or thousands of sh in so-called sh schools has been the focus of many theoretical and some experimental studies. The spatial group cohesion, unless it is ensured by a conning environment, must be the result of interactions between the animals. As in any collective behaviour, these interactions should be considered as individual decision processes that synchronise the behavioural outputs [12].
Many authors have tried to understand these collective behaviours from a theoretical perspective. They propose biologically plausible (but nevertheless hypothetical) interactions that lead to a synchronization of the sh headings (moving directions), see [18,61] and references therein. The interactions are implemented...