The run and tumble process is well established in order to describe the movement of bacteria in response to a chemical stimulus. However, the relation between the tumbling rate and the internal state of bacteria is poorly understood. This study aims at deriving macroscopic models as limits of the mesoscopic kinetic equation in different regimes. In particular, we are interested in the roles of the stiffness of the response and the adaptation time in the kinetic equation. Depending on the asymptotics chosen both the standard Keller–Segel equation and the flux-limited Keller–Segel (FLKS) equation can appear. An interesting mathematical issue arises with a new type of equilibrium equation leading to solution with singularities.

中文翻译：

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具有内部状态的动力学方程的多重渐近

为了描述细菌响应化学刺激的运动，已经很好地建立了运行和翻滚过程。然而，人们对翻滚率与细菌内部状态之间的关系知之甚少。本研究旨在推导宏观模型作为不同状态下的介观动力学方程的限制。特别是，我们对响应刚度和适应时间在动力学方程中的作用感兴趣。根据所选择的渐近线，可以出现标准的 Keller-Segel 方程和通量受限的 Keller-Segel (FLKS) 方程。一个有趣的数学问题出现在一种新型的平衡方程导致具有奇异性的解中。