Macroscopic limit of self-driven particles with orientation interaction

Degond, Pierre; Motsch, Sébastien

doi:10.1016/j.crma.2007.10.024

Cited by 72 publications

(68 citation statements)

References 16 publications

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“…In particular, it is generally usable in the more complex nonlinear cases (see e.g. [21,22]). A particular important complex situation is the case of many interacting fish.…”

Section: The Persistent Turning Walker Model: Presentation and Main Rmentioning

confidence: 99%

“…Several attempts to derive continuum models from the CVA model are also reported in the literature [34,44,45]. In [21,22], a derivation of a continuum model from a kinetic version of the CVA model is proposed. However, few Individual Based Models for fish have been validated against experimental data with a comparable care as in [27] for the PTW process.…”

Section: Introductionmentioning

confidence: 99%

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

2008

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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.Acknowledgements: The authors wish to thank Guy Théraulaz and Jacques Gautrais of the 'Centre de Recherches sur la Cognition Animale' in Toulouse, for introducing them to the model and for stimulating discussions.

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“…In particular, it is generally usable in the more complex nonlinear cases (see e.g. [21,22]). A particular important complex situation is the case of many interacting fish.…”

Section: The Persistent Turning Walker Model: Presentation and Main Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2008

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“…The detailed qualitative study of the system as well as numerical simulations will be the subject of future work. A summary of this work can be found in [13].…”

Section: Introductionmentioning

confidence: 99%

Continuum Limit of Self-Driven Particles With Orientation Interaction

Degond

Motsch

2008

Math. Models Methods Appl. Sci.

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331

616

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We consider the discrete Couzin-Vicsek algorithm (CVA) [1,9,19,36], which describes the interactions of individuals among animal societies such as fish schools. In this article, we propose a kinetic (mean-field) version of the CVA model and provide its formal macroscopic limit. The final macroscopic model involves a conservation equation for the density of the individuals and a non conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.Acknowledgements: The first author wishes to thank E. Carlen and M. Carvalho for their interest in this work and their helpful suggestions.

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“…In the biological context, the emergent behavior of bird flocks, fish schools or bacteria aggregations, among others, is a major research topic in population and behavioral biology and ecology [1,4,14,15,16,18,30,33,34]. Likewise, the coordination and cooperation among multiple mobile agents (robots or sensors) have been playing central roles in sensor networking, with broad applications in environmental control [22,13,28].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Flocking Dynamics for the Kinetic Cucker–Smale Model

Carrillo¹,

Fornasier²,

Rosado³

et al. 2010

SIAM J. Math. Anal.

461

398

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Abstract. In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [16] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.

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Macroscopic limit of self-driven particles with orientation interaction

Cited by 72 publications

References 16 publications

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

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

Continuum Limit of Self-Driven Particles With Orientation Interaction

Asymptotic Flocking Dynamics for the Kinetic Cucker–Smale Model

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