2014/06/04 by Joep H. M. Evers, Evers, Joep H. M., Razvan C. Fetecau +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Physics and Astronomy · #34A09 #34A12 #37M05 #65L11 #Classical Analysis and ODEs (math.CA) #Complex Network Analysis Techniques #Diffusion and Search Dynamics #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Opinion Dynamics and Social Influence #math.CA #msc:34A09 #msc:34A12 #msc:37M05 #msc:65L11
paper · pdf · doi:10.48550/arxiv.1406.0967
arxiv created 2014/06/04 · openalex publication_date 2014/06/04 · arxiv updated 2014/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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 ε, which can be interpreted as a small amount of inertia or response time. We show that the limit ε → 0 can be used as a jump criterion to select the physically correct velocities. In smooth regimes, the convergence of the relaxation system as ε → 0 is guaranteed by a theorem due to Tikhonov. We illustrate the results with numerical simulations in two dimensions.