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

TL;DR

This paper introduces a new fish movement model called the Persistent Turning Walker (PTW), analyzes its large-scale diffusive behavior, and derives an explicit diffusion coefficient using two different mathematical approaches.

Contribution

The paper provides the first analytical derivation of the diffusion coefficient for the PTW fish movement model using variance analysis and kinetic diffusion approximation.

Findings

Large time behavior of PTW trajectories is diffusive.

Both methods yield the same diffusion coefficient.

Numerical simulations confirm theoretical predictions.

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.