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.
Highlights
Aggregations and traveling waves are ubiquitous in collective dynamics of chemotactic cells
Aggregation stems from the chemotaxis of motile cells, where cells are attracted to migrate toward a higher-concentration region of chemical cues produced by themselves
The numerical simulations uncover the variety of solution types in the Flux-limited Keller-Segel (FLKS) system, i.e., (i) monotonic traveling wave for a small stiffness, (ii) nonmonotonic traveling wave with a positive propagation speed for a small modulation amplitude, i.e., χ ≤ 2, and a sufficiently large stiffness under the linear stability condition, (iii) backward traveling wave for a large modulation amplitude, i.e., χ > 2, and a sufficiently large stiffness under the linear stability condition, (iv) sequential strip pattern formation with a propagating front for a small modulation amplitude, i.e., χ ≤ 2, in the linear unstable condition, and (v) the stationary localized spikes for a large modulation amplitude, i.e., χ > 2, in the linear unstable condition
Summary
Aggregations and traveling waves are ubiquitous in collective dynamics of chemotactic cells. It is well known that chemotactic bacteria as E. Coli extend their habitat as creating patterns with localized aggregations of population [1, 2, 3, 4]. Aggregation stems from the chemotaxis of motile cells, where cells are attracted to migrate toward a higher-concentration region of chemical cues produced by themselves. A challenge is to understand the interaction between aggregation. Flux-limited Keller-Segel model, Chemotaxis, Traveling wave, Aggregation, Population growth
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