Modelling Social Animal Aggregations

Grünbaum, Daniel; Ōkubo, Akira

doi:10.1007/978-3-642-50124-1_18

Cited by 158 publications

(119 citation statements)

References 71 publications

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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%

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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%

A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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“…Another interesting example in this line is non-local aggregation [21][22][23], e.g. appearing in animal swarms looking for a density, but also in macroscopic models of consensus formation.…”

Section: Spatial Pattern Formation By Consensus and Herdingmentioning

confidence: 99%

Partial differential equation models in the socio-economic sciences

Burger

Caffarelli

Markowich

2014

Phil. Trans. R. Soc. A.

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Mathematical models based on partial differential equations (PDEs) have become an integral part of quantitative analysis in most branches of science and engineering, recently expanding also towards biomedicine and socio-economic sciences. The application of PDEs in the latter is a promising field, but widely quite open and leading to a variety of novel mathematical challenges. In this introductory article of the Theme Issue, we will provide an overview of the field and its recent boosting topics. Moreover, we will put the contributions to the Theme Issue in an appropriate perspective.

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“…In the spatial approach the space (environment) is either explicitly or implicitly present in the model and the analysis. It can be divided into two distinct frameworks which are individual-based (or Lagrangian) framework and continuum (or Eulerian) framework [6]. In the individual-based models the basic description is the motion equation of each (separate) individual and therefore it is a natural approach for modeling and analysis of complex social interactions and aggregations.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Social Foraging Swarms

Gazi

Passino

2004

IEEE Trans. Syst., Man, Cybern. B

548

341

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Abstract-In this article we specify an -member "individual-based" continuous time swarm model with individuals that move in an -dimensional space according to an attractant/repellent or a nutrient profile. The motion of each individual is determined by three factors: i) attraction to the other individuals on long distances; ii) repulsion from the other individuals on short distances; and iii) attraction to the more favorable regions (or repulsion from the unfavorable regions) of the attractant/repellent profile. The emergent behavior of the swarm motion is the result of a balance between inter-individual interactions and the simultaneous interactions of the swarm members with their environment. We study the stability properties of the collective behavior of the swarm for different profiles and provide conditions for collective convergence to more favorable regions of the profile.

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Modelling Social Animal Aggregations

Cited by 158 publications

References 71 publications

A Nonlocal Continuum Model for Biological Aggregation

A Nonlocal Continuum Model for Biological Aggregation

Partial differential equation models in the socio-economic sciences

Stability Analysis of Social Foraging Swarms

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