On solutions of spatio-temporal patterns 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。例如，菌落模式表现为扩散限制聚集(DLA)模式或密集分枝形态(DBM)模式，在本研究中，我试图通过数学模型来解释这种模式在细菌群体中的形成机制。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。斑点图案的复制取决于趋化系数的程度。此外，通过采用具有非线性扩散系数的扩散反应方程，还可以得到依赖于初始营养物质浓度和细菌扩散系数的更复杂的密集分枝形态。随着营养物初始浓度的增加，菌落模式的扩张速度呈抛物线增加。当初始浓度较高时，数值模拟得到的膨胀率与理论计算的膨胀率基本一致。

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. For instance, the colony pattern shows DLA (diffusion limited aggregation) patterns or DBM (dense branching morphology) patterns, In this study I have attempted 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. The reproduction of the spotty patterns depends on the degree of chemotactic coefficient. Furthermore, by employing a diffusion reaction equation with a nonlinear diffusion coefficient, I could also obtain a more complex dense branched morphology, which depends on initial concentration of nutrient and diffusion coefficient of bacteria. The rate of expansion of colony pattern parabolically increases, as initial concentration of nutrient increases. Expansion rate obtained by numerical simulations mostly agrees with one given by theoretical calculations, when the inticial concentration of nutrient is high.