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

TL;DR

This paper extends a collective behavior model by incorporating anisotropic interactions based on limited perception, analyzing its mathematical properties, and proposing a relaxation system to handle discontinuities, supported by numerical simulations.

Contribution

It introduces a new anisotropic first-order aggregation model, studies its well-posedness, and develops a relaxation approach to manage velocity discontinuities.

Findings

The anisotropic model exhibits non-uniqueness and jump discontinuities.

The relaxation system converges to the original model as the response time parameter approaches zero.

Numerical simulations demonstrate the model's behavior in two dimensions.

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 $\varepsilon$, which can be interpreted as a small amount of inertia or response time. We show that the limit $\varepsilon \to 0$ can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as $\varepsilon \to 0$ is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions.