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

By Pierre Degond, Sébastien Motsch. 2007

198 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6bff481c6bc1172256a09fa6342caae10e8a4733

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In Condensed Matter Physics

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Citations in the corpus

A Kinetic Flocking Model with Diffusion (2010, background): “1 Main results Description of the collective and interactive motion of multi-agents such as school of fish, flocking of birds or swarm of bacteria became recently a major research topic in population and behavioral biology and ecology [23, 4, 19, 20, 5, 6].”
A New Model for Self-organized Dynamics and Its Flocking Behavior (2011, background): “2), we can derive the macroscopic limit of the dynamics [10, 11, 15].”
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system (2009, background): “Recently several mathematical models for flocking were introduced and analyzed [2, 3, 4, 5, 6, 16].”
Swarming, schooling, milling: phase diagram of a data-driven fish school model (2013, methodology): “The large-scale properties of this model have been extensively studied by Degond and Motsch [21, 22] In the above equation, ω(t) is the stimulus/response function based on the individuals response to the proximity of the tank wall—irrelevant in the following as we shall consider groups evolving in a”
Ring patterns and their bifurcations in a nonlocal model of biological swarms (2015, background): “Introduction Mathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10, 6, 2, 26, 25, 14, 7, 13, 27, 19, 33, 32, 23, 11, 17, 37, 38, 34, 3”
From particle to kinetic and hydrodynamic descriptions of flocking (2008, background): “We refer to Degond and Motsch [7, 8, 9] for recent kinetic description of Vicsek type model of flocking.”
Continuum limit of self-driven particles with orientation interaction (2007, unknown): “…Scientific Publishing Company http://www.worldscinet.com/m3as/m3as.shtml Key words: Individual based model, Fish behavior, Couzin-Vicsek algorithm, asymptotic analysis, orientation interaction, hydrodynamic limit, collision invariants AMS Subject classification: 35Q80, 35L60, 82C22, 82C70, 92D50”
A simple proof of the Cucker-Smale flocking dynamics and mean-field limit (2009, background): “We refer to Degond and Motsch’s works [4, 5, 6] for recent kinetic description of Viscek type model for flocking.”
On collision-avoiding initial configurations to Cucker-Smale type flocking models (2012, background): “Introduction Flocking phenomena [3, 11, 12, 13, 14, 32, 33, 34] such as flocking of birds, schooling of fishes, and swarming of bacteria are often observed in complex biological systems, and are some of the collective self-organizing phenomena in biological, mechanical, and social groups of self-pro”
Double milling in self-propelled swarms from kinetic theory (2009, background): “While several continuum models based on heuristic derivations have been presented in the literature, few attempts have been made at deriving such models starting from principles of dynamics of individual agents [5, 9, 20, 13, 14].”
Central limit theorems for additive functionals of ergodic Markov diffusions processes (2011, background): “Degond and Motsch (2008)) is to consider P (t, ν, dx, dv) = ε P (”