Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups

Topaz, Chad M.; Bertozzi, Andrea L.

doi:10.1137/s0036139903437424

Cited by 452 publications

(350 citation statements)

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“…The Lagrangian approach treats each organism as a particle obeying a nonlinear difference or differential equation (Sakai, 1973;Suzuki and Sakai, 1973;Okubo et al, 1977;Vicsek et al, 1995;Levine et al, 2001;Schweitzer et al, 2001;Couzin et al, 2002;Erdmann et al, 2002;Parrish et al, 2003;Aldana and Huepe, 2003;Erdmann and Ebeling, 2003;Mogilner et al, 2003). Alternatively, the Eulerian approach describes the local flux of individuals with an advectiondiffusion equation for a continuum population density field (Kawasaki, 1978;Okubo, 1980;Mimura and Yamaguti, 1982;Passo and Demottoni, 1984;Ikeda, 1984;Alt, 1985;Ikeda, 1985;Satsuma and Mimura, 1985;Ikeda and Nagai, 1987;Hosono and Mimura, 1989;Grünbaum and Okubo, 1994;Edelstein-Keshet et al, 1998;Toner and Tu, 1998;Flierl et al, 1999;Mogilner and Edelstein-Keshet, 1999;Topaz and Bertozzi, 2004). A variety of methods can be used to connect the two formulations.…”

Section: Introductionmentioning

confidence: 99%

“…The usual method involves a Fokker-Planck approximation which relates the distribution of jump distances made by individuals to terms in the advection-diffusion equation (Okubo and Levin, 2001). Since the social communications between organisms often take place at large distances via sight, sound, or smell, models may be nonlocal in space (Kawasaki, 1978;Alt, 1985;Ikeda, 1985;Satsuma and Mimura, 1985;Ikeda and Nagai, 1987;Hosono and Mimura, 1989;Grünbaum and Okubo, 1994;Flierl et al, 1999;Mogilner and Edelstein-Keshet, 1999;Okubo et al, 2001;Topaz and Bertozzi, 2004).…”

Section: Introductionmentioning

confidence: 99%

“…We refer to this phenomenon as clumping. While clumping has been observed in two-dimensional numerical numerical (Levine et al, 2001) and analytical (Topaz and Bertozzi, 2004) models, recent mathematical analyses of swarming behaviors (Mogilner and Edelstein-Keshet, 1999) were unable to find the same kind of clumping from biologically reasonable one-dimensional models. As we will describe below, earlier one-dimensional models which support clumping behavior include assumptions about biological interactions which are unlikely to be met in nature.…”

Section: Introductionmentioning

confidence: 99%

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A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady state clumps are approached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions, as well, and is demonstrated via numerical simulations in two dimensions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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“…Remark: We note here that the similarity transformation has resulted in our new evolution equations (6) and (7) to have a repulsion-attraction interaction kernel, β|y − y | − K (|y − y |). This has the effect of fixing the collapsing S d−1 solutions to be frozen and we can then study the stability of these constant states.…”

Section: Weak Formulation Of the Problemmentioning

confidence: 99%

Stability and clustering of self-similar solutions of aggregation equations

Sun

Uminsky

Bertozzi

2012

Journal of Mathematical Physics

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In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K * ρ) in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd, d ⩾ 2, where K(r) = rγ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math. 70, 2582–2603 (2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ > 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two-dimensional (in-)stability implies n-dimensional (in-)stability.

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“…. .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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Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups

Cited by 452 publications

References 23 publications

A Nonlocal Continuum Model for Biological Aggregation

A Nonlocal Continuum Model for Biological Aggregation

Stability and clustering of self-similar solutions of aggregation equations

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

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