The flux limited Keller–Segel system; properties and derivation from kinetic equations

Abstract

The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyperbolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently.We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.

A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis

“…These and other issues motivated the introduction of models with flux saturation in this and related contexts; see [49] and citations therein. We also refer to [50,51] for formal and respectively rigorous derivations of chemotaxis models from KTEs, and to [25,52,53] for settings specifically relating to glioma invasion. Among the latter, ref.…”

Mathematical models for chemotaxis and their applications in self-organisation phenomena

“…In bacteria with well understood signalling and motion closely approximated by a velocity-jump process, such as E. coli, one can even link parameters and functions characterising molecular signalling and motor control to the parameters and functions that define diffusive/chemotactic sensitivity terms in a PKS model . Moreover, novel and interesting variations can emerge, such as the "perpendicular gradient following" that arises from swimming biases (Xue and Othmer, 2009), fractional operator terms due to non-Poisson type turning rate distributions (Estrada-Rodriguez et al, 2017) or "flux-limited" forms (Perthame et al, 2018). Noteworthy, the above derivations rely on ignoring interactions and Stevens (2000) is noted for providing the first rigorous derivation of a PKS equation for a population of stochastic (weakly) interacting particles.…”

Existence of spiky steady state of chemotaxis models with logarithm sensitivity

“…for the results of global existence and large-time behavior of solutions, we refer the reader to previous studies 21- 25 and Wang et al 26 Although there are many results for aggregation type models, the results of stationary solutions are few.…”

“…For aggregation type, Lin, Ni, and Takagi 2,9 studied the stationary solution; one can refer Fujie et al 10,11 and Winkler 12 to obtain global existence of time dependent solutions; Nagai and Senba 13 found the blow up phenomenon. For consumption type, the results of boundary layer problem can be found in Hou and Wang 14 and Hou et al 15 ; the results of traveling wave solutions are explored in previous studies 16‐20 and Wang 8 ; for the results of global existence and large‐time behavior of solutions, we refer the reader to previous studies 21‐25 and Wang et al 26 Although there are many results for aggregation type models, the results of stationary solutions are few.…”