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This paper considers a new model of individual displacement, based on fishmotion, the so-called Persistent Turning Walker (PTW) model, which involves anOrnstein-Uhlenbeck process on the curvature of the particle trajectory. Thegoal is to show that its large time and space scale dynamics is of diffusivetype, and to provide an analytic expression of the diffusion coefficient. Twomethods are investigated. In the first one, we compute the large timeasymptotics of the variance of the individual stochastic trajectories. Thesecond method is based on a diffusion approximation of the kinetic formulationof these stochastic trajectories. The kinetic model is a Fokker-Planck typeequation posed in an extended phase-space involving the curvature among thekinetic variables. We show that both methods lead to the same value of thediffusion constant. We present some numerical simulations to illustrate thetheoretical results.

Large scale dynamics of the Persistent Turning Walker model of fish behavior