Beyond the Keller-Segel model

Active Brownian particles

“…Here we discuss briefly models of (active) Brownian particles interacting via a self-generated chemoattractant, as an alternative to a pure PDEs approach [310,72,292].…”

“…This behavior has significant impact on the macroscopic pattern formation. At large S any formation of clusters may be inhibited and at moderate values of S a "smoothing" of spikes in the concentration profile towards flat spots can be observed [292].…”

“…Only if κ 0 ≥ κ c pattern formation on the macroscopic scale can be observed. Although this result was derived for overdamped dynamics it can be also applied for actively moving particles with a Rayleigh-Helmholtz friction function, by taking the friction coefficient from the Stokes friction as a single fit parameter [292].…”

“…In the following Ebeling, Schweitzer and others used this term in the context of self-propelled particles far from equilibrium (see e.g. [307,101,111,306,292]). In general we will refer to "Active Brownian Particles" in the latter context as Brownian particles performing active motion, which may be accounted for by an internal energy depot and/or a (nonlinear) velocity-dependent friction function.…”

Swarming and pattern formation due to selective attraction and repulsion

“…Most of the mathematical models for collective motion proposed in the literature contain some sort of explicit velocity-alignment mechanisms, which tends to align the velocity of a focal individual with the velocity of its neighbours [2,6,[20][21][22][23]. However, recent experimental studies of collective behaviour in fish do not find any clear evidence for the existence of explicit velocity -alignment interactions [24,25].…”