Journal
PHYSICA D-NONLINEAR PHENOMENA
Volume 260, Issue -, Pages 65-76Publisher
ELSEVIER
DOI: 10.1016/j.physd.2012.10.009
Keywords
Biological swarms; Aggregation model; Newtonian potential
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Funding
- NSERC
- AARMS CRG grant on dynamical systems
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We consider the aggregation equation with an attractive-repulsive force law. Recent studies (Kolokolnikov et al. (2011) [22]; von Brecht et al. (2012) [23]; Balague et al. (2013) [15]) have demonstrated that this system exhibits a very rich solution structure, including steady states consisting of rings, spots, annuli, N-fold symmetries, soccer-ball patterns etc. We show that many of these patterns can be understood as singular perturbations off lower-dimensional equilibrium states. For example, an annulus is a bifurcation from a ring; soccer-ball patterns bifurcate off solutions that consist of delta-point concentrations. We apply asymptotic methods to classify the form and stability of many of these patterns. To characterize spot solutions, a class of semi-linear aggregation problems is derived, where the repulsion is described by a nonlinear term and the attraction is linear but non-symmetric. For a special class of perturbations that consists of a Newtonian repulsion, the spot shape is shown to be an ellipse whose precise dimensions are determined via a complex variable method. For annular shapes, their width and radial density profile are described using perturbation techniques. (c) 2012 Elsevier B.V. All rights reserved.
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