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

Pierre Degond (1) , Sébastien Motsch (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: Ornstein-Uhlenbeck process; kinetic Fokker-Planck equation; asymptotic analysis; diffusion approximation.; diffusion approximation; Individual based model; Fish behavior; Persistent Turning Walker model

Domaines: Physique mathématique [math-ph]

Pierre Degond, Sébastien Motsch. Large scale dynamics of the Persistent Turning Walker model of fish behavior. Journal of Statistical Physics, 2008, 131, pp.989-1021. ⟨hal-00182431⟩