Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior

Abstract

This paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results.Acknowledgements: The authors wish to thank Guy Théraulaz and Jacques Gautrais of the 'Centre de Recherches sur la Cognition Animale' in Toulouse, for introducing them to the model and for stimulating discussions.

A jump persistent turning walker to model zebrafish locomotion

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A simple proof of the Cucker-Smale flocking dynamics and mean-field limit

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Ring patterns and their bifurcations in a nonlocal model of biological swarms

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