Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 131, Issue 6, Pages 989-1021Publisher
SPRINGER
DOI: 10.1007/s10955-008-9529-8
Keywords
individual based model; fish behavior; Persistent Turning Walker model; Ornstein-Uhlenbeck process; kinetic Fokker-Planck equation; asymptotic analysis; diffusion approximation
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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.
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