Nonlocal Aggregation Models: A Primer of Swarm Equilibria

Bernoff, Andrew J.; Topaz, Chad M.

doi:10.1137/130925669

Cited by 72 publications

(74 citation statements)

References 51 publications

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“…Complex, co-dimension zero patterns also arise in biology, and have inspired researchers to develop mathematical models that can help explain, both evolutionarily and biologically, why and how these self-assembled patterns form [6,29,27,17,24,12,15,22]. Such models have proven fruitful in modeling locust swarms [3,21,36], where the techniques capture the unique swarm shapes of locusts. These models also help explain rings, annuli, and other complex, spotted patterns in bacterial colonies that form under stress in the lab [38,10,18,4].…”

Section: Introductionmentioning

confidence: 99%

Predicting Pattern Formation in Particle Interactions

Brecht

Uminsky

Kolokolnikov

et al. 2012

Math. Models Methods Appl. Sci.

123

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Large systems of particles interacting pairwise in d-dimensions give rise to extraordinarily rich patterns. These patterns generally occur in two types. On one hand, the particles may concentrate on a co-dimension one manifold such as a sphere (in 3D) or a ring (in 2D). Localized, space-filling, codimension zero patterns can occur as well. In this paper, we utilize a dynamical systems approach to predict such behaviors in a given system of particles. More specifically, we develop a non-local linear stability analysis for particles uniformly distributed on a d − 1 sphere. Remarkably, the linear theory accurately characterizes the patterns in the ground states from the instabilities in the pairwise potential. This aspect of the theory then allows us to address the issue of inverse statistical mechanics in selfassembly: given a ground state exhibiting certain instabilities, we construct a potential that corresponds to such a pattern.

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Section: Introductionmentioning

confidence: 99%

Predicting Pattern Formation in Particle Interactions

Brecht

Uminsky

Kolokolnikov

et al. 2012

Math. Models Methods Appl. Sci.

123

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“…which can be used to analyze the existence and stability of equilibria [8,9,39,30,61]. Despite its popularity in the literature, one shortfall of this model is that it does not produce so-called well-spaced groups.…”

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confidence: 99%

“…A moment expansion similar to the approximation (7) is used to derive a different model for the movement of ecological populations in [52]. A great deal is known about (9). As we will discuss below, it is related to both the CahnHilliard equation [56], which arises as a model of phase separation, and to thin-film equations, which model surface-tension driven wetting of a substrate by a viscous fluid [59,55,34].…”

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confidence: 99%

Biological Aggregation Driven by Social and Environmental Factors: A Nonlocal Model and Its Degenerate Cahn--Hilliard Approximation

Bernoff¹,

Topaz²

2016

SIAM J. Appl. Dyn. Syst.

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Abstract. Biological aggregations such as insect swarms and bird flocks may arise from a combination of social interactions and environmental cues. We focus on nonlocal continuum equations, which are often used to model aggregations, and yet which pose significant analytical and computational challenges. 1. Introduction. This paper has three primary aims. First, beginning with a partial integrodifferential equation model of biological aggregation, we show that a long-wave approximation yields a degenerate Cahn-Hilliard equation akin to models describing phase separation in materials science and evolution of thin fluid films. Second, we study solutions of the degenerate Cahn-Hillard model and demonstrate that they compare well with those of the original model. Using the Cahn-Hilliard model is advantageous because it eliminates many of the analytical and computational challenges of nonlocal equations, particularly in higher dimensions. Our methodology is to study the energy minimizers of this local equation, rather than studying the time evolution of the original nonlocal equation. Finally, we use the Cahn-Hilliard model as a testbed to assess whether imposing an external potential modeling the environment, e.g., food sources, can be used to control peak density in aggregations, which is of interest for locust swarms.Biological aggregations such as bird flocks, fish schools, and insect swarms are driven by social interactions between group members. Typical social forces include attraction, repulsion,

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“…In order to incorporate fluid couplings into discrete particle models, one must first identify the physical origin of the interactions between particles. In typical models of swarming [9,12,16,17,20,35], the propensity of agents to self-propel themselves towards or away from others is modulated by an effective "social" interaction potential. When immersed in 2470-0045/2016/93(4)/043112 (12) 043112-1 ©2016 American Physical Society YAO-LI CHUANG, TOM CHOU, AND MARIA R. D'ORSOGNA PHYSICAL REVIEW E 93, 043112 (2016) low-Reynolds-number Newtonian fluids, particle selfpropulsion is force free.…”

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confidence: 99%

“…. .PHYSICAL REVIEW E 93, 043112 (2016) dimensions [9,17,19,20,88]. Generally, particles are subject to two tendencies: changing their separations to minimize and adjusting their velocities to match the characteristic speed.…”

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confidence: 99%

Swarming in viscous fluids: Three-dimensional patterns in swimmer- and force-induced flows

2016

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We derive a three-dimensional theory of self-propelled particle swarming in a viscous fluid environment. Our model predicts emergent collective behavior that depends critically on fluid opacity, mechanism of self-propulsion, and type of particle-particle interaction. In "clear fluids" swimmers have full knowledge of their surroundings and can adjust their velocities with respect to the lab frame, while in "opaque fluids" they control their velocities only in relation to the local fluid flow. We also show that "social" interactions that affect only a particle's propensity to swim towards or away from neighbors induces a flow field that is qualitatively different from the long-ranged flow fields generated by direct "physical" interactions. The latter can be short-ranged but lead to much longer-ranged fluid-mediated hydrodynamic forces, effectively amplifying the range over which particles interact. These different fluid flows conspire to profoundly affect swarm morphology, kinetically stabilizing or destabilizing swarm configurations that would arise in the absence of fluid. Depending upon the overall interaction potential, the mechanism of swimming ( e.g., pushers or pullers), and the degree of fluid opaqueness, we discover a number of new collective three-dimensional patterns including flocks with prolate or oblate shapes, recirculating pelotonlike structures, and jetlike fluid flows that entrain particles mediating their escape from the center of mill-like structures. Our results reveal how the interplay among general physical elements influence fluid-mediated interactions and the self-organization, mobility, and stability of new three-dimensional swarms and suggest how they might be used to kinetically control their collective behavior. DOI: 10.1103/PhysRevE.93.043112 The collective behavior of self-propelled agents in natural and artificial systems has been extensively studied . Many of the lessons learned from experimental and theoretical work conducted on organisms as diverse as bacteria, ants, locusts, and birds [23][24][25][26][27][28][29][30][31][32][33][34][35][36] have been successfully applied to engineered robotic systems to help frame decentralized control strategies through ad hoc algorithms [37][38][39][40][41][42][43][44]. In most mathematical "swarming" models, particles are assumed to be self-driven by internal mechanisms that impart a characteristic speed. A pairwise short-ranged repulsion and a long-ranged decaying attraction are typically employed as the most realistic choices when modeling aggregating particles [9,11,45]. The interplay between self-propulsion, particle interactions, initial conditions, and number of particles is key in determining the large scale patterns that dynamically arise. In two dimensions, rotating mills and translating flocks are often observed, the latter configuration also arising in three dimensions [9,10,17,19,46,47]. It is possible to classify swarm morphology in terms of interaction strength and length scales, as shown for particles coupled via conserved f...

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Nonlocal Aggregation Models: A Primer of Swarm Equilibria

Cited by 72 publications

References 51 publications

Predicting Pattern Formation in Particle Interactions

Predicting Pattern Formation in Particle Interactions

Biological Aggregation Driven by Social and Environmental Factors: A Nonlocal Model and Its Degenerate Cahn--Hilliard Approximation

Swarming in viscous fluids: Three-dimensional patterns in swimmer- and force-induced flows

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