Shape, shear, search and strife; mathematical models of bacteria

生物模式的形成是自然界中最有趣的现象之一。这种模式的简单例子是由行波和平稳的周期性模式，发生在各种生物过程，包括形态发生和种群动态。这些模式的形成在运动微生物群体中，如盘基网柄菌和E.大肠杆菌已经在几个实验研究中显示。各种类型的模式的形成条件通常在包含扩散和平流项的动力系统的物理研究中得到解决。在项目中，我们展示了由不断增长的趋化活性细菌种群形成的时空模式。首先，我们展示了在一个物种种群中行波阵面和图灵模式的形成。我们表明，图灵模式形成时，细菌产生的化学物质，这是一个强大的引诱剂。然后，我们转向两个相互作用的细菌种群的系统，其中一个为另一个产生趋化剂。我们的分析和数值模拟表明，在这种情况下，图灵模式形成时，化学品作为一个排斥。利用线性稳定性分析研究了图灵斑图的形成条件，利用傅里叶级数分析了它们的特征（振幅和波长）。通过数值模拟验证了分析结果。

Biological pattern formation is one of the most intriguing phenomena in nature. Simplestexamples of such patterns are represented by travelling waves and stationary periodic patternswhich occur during various biological processes including morphogenesis and populationdynamics. Formation of these patterns in populations of motile microorganisms such asDictyostelium discoideum and E. coli have been shown in several experimental studies.Conditions for formation of various types of patterns are commonly addressed in mathematicalstudies of dynamical systems containing diffusion and advection terms. In project,we show spatio-temporal patterns formed by growing populations of chemotactically activebacteria. First, we show formation of travelling wavefronts and Turing patterns in a one-speciespopulation. We show that Turing patterns form when bacteria produce chemical which is astrong attractant. We then move on to a system of two interacting bacterial populations, one ofwhich produces a chemotactic agent for another. Our analysis and numerical simulations showthat in this case Turing patterns form when the chemical acts as a repellent. Study of conditionsfor formation of Turing patterns is done using linear stability analysis while their characteristics(amplitude and wavelength) are found using Fourier series. Analytical results are also supportedby means of numerical simulations.