Anisotropic interactions in a first-order aggregation model : a proof of concept

J.H.M. Evers, R.C. Fetecau, L. Ryzhik

Research output: Book/ReportReportAcademic

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Abstract

We extend a well-studied ODE model for collective behaviour by considering anisotropic interactions among individuals. Anisotropy is modelled by limited sensorial perception of individuals, that depends on their current direction of motion. Consequently, the first-order model becomes implicit, and new analytical issues, such as non-uniqueness and jump discontinuities in velocities, are being raised. We study the well-posedness of the anisotropic model and discuss its modes of breakdown. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter e, which can be interpreted as a small amount of inertia or response time. We show that the limit e ¿ 0 can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as e ¿ 0 is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions. Keywords: Anisotropy; visual perception; aggregation models; implicit equations; regularization; relaxation time; uniqueness criteria; singular perturbation.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Universiteit Eindhoven
Number of pages28
Publication statusPublished - 2014

Publication series

NameCASA-report
Volume1420
ISSN (Print)0926-4507

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