Hydrodynamic model for a system of self-propelling particles with conservative kinematic constraints

Kulinskii, V. L.; Ratushnaya, V. I.; Zvelindovsky, Andrei; Bedeaux, Dick

doi:10.1209/epl/i2005-10086-2

Cited by 16 publications

(29 citation statements)

References 18 publications

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“…In Couzin et al (2002), interactions between individuals are restricted to localized "zones" of attraction, repulsion, and alignment, and yet mill formations emerge in certain parameter regimes. From a continuum perspective, in Topaz and Bertozzi (2004), Kulinskii et al (2005), Csahók and Czirók (2008) population density models have vortex-type formations as solutions.…”

Section: Discussionmentioning

confidence: 99%

A Conceptual Model for Milling Formations in Biological Aggregates

Lukeman

Edelstein‐Keshet

2008

Bull. Math. Biol.

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Collective behavior of swarms and flocks has been studied from several perspectives, including continuous (Eulerian) and individual-based (Lagrangian) models. Here, we use the latter approach to examine a minimal model for the formation and maintenance of group structure, with specific emphasis on a simple milling pattern in which particles follow one another around a closed circular path.We explore how rules and interactions at the level of the individuals lead to this pattern at the level of the group. In contrast to many studies based on simulation results, our model is sufficiently simple that we can obtain analytical predictions. We consider a Newtonian framework with distance-dependent pairwise interaction-force. Unlike some other studies, our mill formations do not depend on domain boundaries, nor on centrally attracting force-fields or rotor chemotaxis.By focusing on a simple geometry and simple distant-dependent interactions, we characterize mill formations and derive existence conditions in terms of model parameters. An eigenvalue equation specifies stability regions based on properties of the interaction function. We explore this equation numerically, and validate the stability conclusions via simulation, showing distinct behavior inside, outside, and on the boundary of stability regions. Moving mill formations are then investigated, showing the effect of individual autonomous self-propulsion on group-level motion. The simplified framework allows us to clearly relate individual properties (via model parameters) to group-level structure. These relationships provide insight into the more complicated milling formations observed in nature, and suggest design properties of artificial schools where such rotational motion is desired.

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Section: Discussionmentioning

confidence: 99%

A Conceptual Model for Milling Formations in Biological Aggregates

Lukeman

Edelstein‐Keshet

2008

Bull. Math. Biol.

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“…[26,36,47,48]). 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.…”

Section: Introductionmentioning

confidence: 99%

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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“…Before comparing this expression to the one we used in Refs. [21,22] (19), is satisfied for each density distribution which varies only in directions normal to the flow direction. It follows that the continuous analog of the CVA model has stationary linear and vortical solutions.…”

Section: Continuous Transport Equationsmentioning

confidence: 99%

Collective behavior of self-propelling particles with kinematic constraints: The relation between the discrete and the continuous description

Ratushnaya

Bedeaux

Kulinskii

et al. 2007

Physica A: Statistical Mechanics and its Applications

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In two papers we proposed a continuum model for the dynamics of systems of self propelling particles with kinematic constraints on the velocities and discussed some of its properties. The model aims to be analogous to a discrete algorithm used in works by T. Vicsek et al. In this paper we derive the continuous hydrodynamic model from the discrete description. The similarities and differences between the resulting model and the hydrodynamic model postulated in our previous papers are discussed. The results clarify the assumptions used to obtain a continuous description.

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Hydrodynamic model for a system of self-propelling particles with conservative kinematic constraints

Cited by 16 publications

References 18 publications

A Conceptual Model for Milling Formations in Biological Aggregates

A Conceptual Model for Milling Formations in Biological Aggregates

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

Collective behavior of self-propelling particles with kinematic constraints: The relation between the discrete and the continuous description

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