On solutions of spatial pattern formation in nonlinear reaction-diffusion systems

许多生物表现出趋化性聚集，以响应引诱剂分子的浓度梯度，形成各种空间格局。正如在细菌菌落、细胞纤细霉菌和成群的昆虫中常见的那样，众所周知，细菌菌落可以根据不同的培养条件表现出非常复杂的空间格局，例如营养浓度和琼脂平板的稠度。在这项研究中，我试图通过数学模型来解释细菌菌落中这种模式的形成机制。细菌的趋化运动由一个非线性偏微分方程式描述。它的数值解再现了同心斑点图案，其前沿随着时间的推移向外推进。此外，通过采用具有非线性扩散系数的扩散反应方程，我还可以得到更复杂的致密支化形态。如果考虑细菌密度的随机波动的影响，这种分枝形态被发现是被强调的。

Many organisms display chemotactic aggregation in response to the concentration gradient of attractant molecules to form variety of spatial patterns. As typically seen in bacterial colonies, cellular slim molds and swarming insects, it has been known that bacterial colonies could show very complex spatial pattern depending on various culture conditions such as nutrient concentration and the solidity of agar plate. In this study I have attemped to elucidate the mechanism of such pattern formation in bacterial colony by means of mathematical modeling. The chemotactic movement of bacteria were formulated by a nonlinear partial differential equation. Its numerical solution reproduces concentric spotty patterns, whose front advances outward with time. Furthermore, by employing a diffusion reaction equation with a nonlinear diffusion coefficient, I could also obtain a more complex dense branched morphology. This branching morphology was found to be emphasized if the effect of random fluctuation in bacterial density are taken into consideration.