Standing pulse-like solutions of a spatially aggregating population model

A Nonlocal Continuum Model for Biological Aggregation

“…This model is related to the one-dimensional models proposed by Mimura and Yamaguti (1982), Nagai and Mimura (1983), Ikeda (1985) and Ikeda and Nagai (1987) (those models may be put in the same form as ours by using commutativity of differentiation and convolution). In those models, the dispersal rate is a (tunable) power of the density, and the sensing kernels K grow linearly in space with a possible cutoff at a finite range (Ikeda, 1985;Ikeda and Nagai, 1987). Our model is also similar to that in Mogilner and Edelstein-Keshet (1999).…”

“…On the other hand, earlier extensions of the work of Kawasaki (1978) by Mimura and Yamaguti (1982), Nagai and Mimura (1983), Alt (1985), Ikeda (1985) and Ikeda and Nagai (1987) include density dependent (rather than constant) diffusion, still coupled to long-range advective attraction. The model of Alt (1985) is inspired by chemotactic locomotion, and includes nonlinear diffusion which is degenerate for finite ρ.…”

“…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.…”

“…The usual method involves a Fokker-Planck approximation which relates the distribution of jump distances made by individuals to terms in the advection-diffusion equation (Okubo and Levin, 2001). Since the social communications between organisms often take place at large distances via sight, sound, or smell, models may be nonlocal in space (Kawasaki, 1978;Alt, 1985;Ikeda, 1985;Satsuma and Mimura, 1985;Ikeda and Nagai, 1987;Hosono and Mimura, 1989;Grünbaum and Okubo, 1994;Flierl et al, 1999;Mogilner and Edelstein-Keshet, 1999;Okubo et al, 2001;Topaz and Bertozzi, 2004).…”

“…The model of Hosono and Mimura (1989) is similar, but adds reaction-type terms modeling logistic growth and predation. Depending on p and ℓ, the nonlinear diffusion models have stationary (Ikeda, 1985;Ikeda and Nagai, 1987;Hosono and Mimura, 1989) and traveling (Mimura and Yamaguti, 1982;Nagai and Mimura, 1983) clump solutions. Although this class of models exhibits the desirable behavior of clump formation, its limitation arises from the biologically unrealistic assumption of strong attractive interactions between individuals over arbitrarily large distances.…”