Traveling Wave and Aggregation in a Flux-Limited Keller-Segel Model
Traveling Wave and Aggregation in a Flux-Limited Keller-Segel Model
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DOI：
10.3934/krm.2018035
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发表时间：
2017-09
期刊：
影响因子：
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通讯作者：
V. Calvez;B. Perthame;S. Yasuda
中科院分区：
文献类型：
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作者：
V. Calvez;B. Perthame;S. Yasuda
通量限制的Keller-Segel(FLKS)模型是最近从细菌趋化的动力学传输模型中推导出来的，并被证明能更好地代表实验中观察到的集体运动。最近，与动力学模型相关的一种新的不稳定性形式被发现与僵硬的趋化反应有关。这促使我们基于带种群增长项的FLKS模型来研究趋化细胞种群动力学中的行波和聚集。我们的研究包括数值和理论两方面的贡献。在数值部分，除了标准的Fisher/KPP类型的行波外，我们还发现了一维FLKS模型中的各种解类型。显著的结果是一种反直觉的反向行波，在局部人口动力学中，最初处于稳定状态的人口密度饱和向不稳定状态过渡。出乎意料的是，我们还发现，随着趋化反应的刚性增加，反向行波解转变为局域尖峰解。在理论部分，我们得到了一个新的最小传播速度的解析公式，它包含了传播前沿中趋化漂移与复制/扩散的平衡效应。除了局域尖峰解外，数值结果的前沿传播速度与最小传播速度仅略有偏离，即使是反向行波也是如此。我们还发现了单峰行波在大刚度极限下的解析解，它当然是不稳定的，但存在于一定的参数范围内。
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.