Mathematical Biosciences Institute, Ohio State University, Columbus, USA.
Bull Math Biol. 2011 Aug;73(8):1695-733. doi: 10.1007/s11538-010-9586-4. Epub 2010 Oct 16.
Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically.
细菌群体的数学模型通常被写成细菌密度和细胞外（信号）化学物质浓度的偏微分方程组。自 20 世纪 70 年代 Keller 和 Segel 的开创性工作以来，一直采用这种方法（Keller 和 Segel，J. Theor. Biol. 30：235-248，1971）。该系统已被证明允许传播波解，这些解对应于细菌菌落中的传播带形成，但仅在特定条件下，例如趋化敏感性函数在信号接近零时出现奇点。这种奇点会产生无限大的宏观速度，这在生物学上是不现实的。在本文中，我们提出了一个模型，该模型考虑了细胞内过程的相关细节，同时避免了趋化敏感性中的奇点。我们证明了解的整体存在性，然后通过数值和解析方法证明了传播波解的存在性。
