Traveling Wave and Aggregation in a Flux-Limited Keller-Segel Model

TL;DR

This paper investigates traveling waves and aggregation phenomena in a flux-limited Keller-Segel model for bacterial chemotaxis, revealing novel backward traveling waves, a new analytic speed formula, and the impact of chemotactic response stiffness.

Contribution

It introduces a comprehensive numerical and theoretical analysis of traveling waves in the FLKS model, including backward waves and a new speed formula, advancing understanding of chemotactic population dynamics.

Findings

Discovery of backward traveling waves transitioning to localized spikes

Derivation of a novel analytic minimum traveling speed formula

Identification of an unstable unimodal traveling wave solution

Abstract

Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term. Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an unstable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response. In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counterbalancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.