A non-local model for a swarm

A Nonlocal Continuum Model for Biological Aggregation

“…We now develop a kinematic model (Edelstein-Keshet et al, 1998;Mogilner and Edelstein-Keshet, 1999;Topaz and Bertozzi, 2004) which assumes that the attractive and dispersive velocities v a and v d depend only on properties of ρ at the current time.…”

“…These swarms are not stable over long periods of time, having a tendency to break down by losing individuals at the rear of the swarm (Mogilner and Edelstein-Keshet, 1999). However, the authors note (via analysis and numerical experiments) that nonlinear diffusion has a tendency to stabilize the swarms.…”

“…The Lagrangian approach treats each organism as a particle obeying a nonlinear difference or differential equation (Sakai, 1973;Suzuki and Sakai, 1973;Okubo et al, 1977;Vicsek et al, 1995;Levine et al, 2001;Schweitzer et al, 2001;Couzin et al, 2002;Erdmann et al, 2002;Parrish et al, 2003;Aldana and Huepe, 2003;Erdmann and Ebeling, 2003;Mogilner et al, 2003). Alternatively, the Eulerian approach describes the local flux of individuals with an advectiondiffusion equation for a continuum population density field (Kawasaki, 1978;Okubo, 1980;Mimura and Yamaguti, 1982;Passo and Demottoni, 1984;Ikeda, 1984;Alt, 1985;Ikeda, 1985;Satsuma and Mimura, 1985;Ikeda and Nagai, 1987;Hosono and Mimura, 1989;Grünbaum and Okubo, 1994;Edelstein-Keshet et al, 1998;Toner and Tu, 1998;Flierl et al, 1999;Mogilner and Edelstein-Keshet, 1999;Topaz and Bertozzi, 2004). A variety of methods can be used to connect the two formulations.…”

“…We refer to this phenomenon as clumping. While clumping has been observed in two-dimensional numerical numerical (Levine et al, 2001) and analytical (Topaz and Bertozzi, 2004) models, recent mathematical analyses of swarming behaviors (Mogilner and Edelstein-Keshet, 1999) were unable to find the same kind of clumping from biologically reasonable one-dimensional models. As we will describe below, earlier one-dimensional models which support clumping behavior include assumptions about biological interactions which are unlikely to be met in nature.…”

“…One describes nonlinear degenerate diffusion arising from anti-crowding behavior, and the other describes attractive nonlocal social interactions. In contrast to previous studies (Mimura and Yamaguti, 1982;Nagai and Mimura, 1983;Edelstein-Keshet et al, 1998;Mogilner and Edelstein-Keshet, 1999), we do not seek traveling solutions, which would correspond to cohesive group movement. Rather, we are concerned with stationary solutions.…”