Anisotropic interactions in a first-order aggregation model

作者：Evers Joep H M^*; Fetecau Razvan C; Ryzhik Lenya

来源：Nonlinearity, 2015, 28(8): 2847-2871.

DOI：10.1088/0951-7715/28/8/2847

• anisotropy
• visual perception
• aggregation models
• implicit equations
• regularization
• relaxation time
• uniqueness criteria

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

• 出版日期2015-8