A Primer of Swarm Equilibria

Bernoff, Andrew J.; Topaz, Chad M.

doi:10.1137/100804504

Cited by 128 publications

(183 citation statements)

References 32 publications

(47 reference statements)

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“…This is the eigenvalue problem for the linear stability of the ring solution. Typically one measures the amplitude that the solution deviates either radially as |b 2 …”

Section: Power Force Lawmentioning

confidence: 99%

“…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 [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2].…”

Section: Introductionmentioning

confidence: 99%

“…One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2]. In the case of swarming, these nonlocal interactions between individuals usually consist of a shorter range repulsion to avoid collisions and medium to long range attraction to keep the swarm cohesive.…”

Section: Introductionmentioning

confidence: 99%

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Ring patterns and their bifurcations in a nonlocal model of biological swarms

Bertozzi

Kolokolnikov

Sun

et al. 2015

Communications in Mathematical Sciences

115

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Abstract. In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We review and expand upon recent developments of this class of problems as well as present new results. As in previous work, we leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence of the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the "designer potential" problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model. IntroductionMathematical 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 [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2]. In the case of swarming, these nonlocal interactions between individuals usually consist of a shorter range repulsion to avoid collisions and medium to long range attraction to keep the swarm cohesive. While some models include anisotropy in this communication (e.g. an organism's eyes may have a limited field of vision) simplified isotropic interactions have been shown to capture many important swarming behaviors including milling [20,10]. More recently it has been shown [17,38,37] that the competition between the desire to avoid collisions and the desire to remain in a cohesive swarm can sometimes result in simple radially symmetric patterns such as rings, annuli and uniform circular patches and other times result in exceedingly complex patterns. Moreover how modelers select the strength and form of the repulsion near the origin

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“…This is the eigenvalue problem for the linear stability of the ring solution. Typically one measures the amplitude that the solution deviates either radially as |b 2 …”

Section: Power Force Lawmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Ring patterns and their bifurcations in a nonlocal model of biological swarms

Bertozzi

Kolokolnikov

Sun

et al. 2015

Communications in Mathematical Sciences

115

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“…Systems with a large number of pairwise interacting particles pervade many disciplines, ranging from models of self-assembly processes in physics and chemistry [21][22][23]29] to models for biological swarming [1,8,16,28] to algorithms for the cooperative control of autonomous vehicles [33]. A simple example of these models employs a first order system of ordinary differential equations for the positions…”

Section: Introductionmentioning

confidence: 99%

“…The formal continuum limit of this system then yields the well-known aggregation equation Left: a "soccer ball" steady-state to the ODE model (1). Right: approximation of the steady state using the co-dimension one continuum model (8), i.e.…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness Theory for Aggregation Sheets

Brecht

Bertozzi

2012

Commun. Math. Phys.

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Abstract:In this paper, we consider distribution solutions to the aggregation equation ρ t + div(ρu) = 0, u = −∇V * ρ in R d , where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces. Moreover, we show that if the initial sheet is C 1 then the solution itself remains C 1 as long as it remains Lipschitz. Lastly, we provide conditions on the kernel g(s) = − dV ds that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail. BackgroundSystems with a large number of pairwise interacting particles pervade many disciplines, ranging from models of self-assembly processes in physics and chemistry [21][22][23]29] to models for biological swarming [1,8,16,28] to algorithms for the cooperative control of autonomous vehicles [33]. A simple example of these models employs a first order system of ordinary differential equations for the positionsThe interaction kernel g(s) describes the manner in which particles interact with one another, and therefore depends on the particular application for the model. The formal continuum limit of this system then yields the well-known aggregation equation for the density ρ of particles. This equation has received significant attention in recent years, and the majority of the analysis largely falls into two categories. More classical treatments focus on densities ρ that are absolutely continuous with respect to Lebesgue measure, such as those lying in an L p (R d ) space [2][3][4][5][6]9,10,14]. For densities that merely define a Borel measure on R d , such as point masses, ideas from optimal transport have proven fruitful for demonstrating the well-posedness of (2) for some classes of interaction kernels [7,12,13,19,20]. However, several recent studies [15,26,30,31] have found that rings, spheres and more complicated surface-like states naturally occur in the ODE systems (1) and the full PDE models. This suggests that a co-dimension one description of (2) might prove useful for studying such particle distributions (see Fig. 1). In this context, i.e. when the density must have support of co-dimension one, even the most basic well-posedness results do not yet exist. We therefore provide them in this paper.Specifically, we analyze distribution solutions to (2) that have support homeomorphic to the (d − 1) sphere S d−1 ⊂ R d , and so take the formThe map Φ(·, t) :, where ρ Φ (x, t) describes the density of particles along the manifold and dH Φ (x) denotes the surface measure on the manifold. By (3), we mean that ρ acts as a distribution on ψIn the usual manner, we then require thatWell-Posedness Theory for Aggregation Sheetshold for all...

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Analysis of Anisotropic Interaction Equations for Fingerprint Simulations

Kreußer

2018

Proc Appl Math and Mech

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Motivated by the simulation of fingerprints we consider a class of interaction models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. These anisotropic forces in our class of models cannot be derived from a potential and the underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the resulting stationary solutions both analytically and numerically by considering the particle model and its continuum counterpart.

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A Primer of Swarm Equilibria

Cited by 128 publications

References 32 publications

Ring patterns and their bifurcations in a nonlocal model of biological swarms

Ring patterns and their bifurcations in a nonlocal model of biological swarms

Well-Posedness Theory for Aggregation Sheets

Analysis of Anisotropic Interaction Equations for Fingerprint Simulations

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