On solutions of spatio-temporal patterns in nonlinear reaction-diffusion systems

许多生物对引诱剂分子的浓度梯度有趋化聚集反应，形成各种空间格局。如在细菌菌落、细胞细霉菌和群集昆虫中通常所见，已知细菌菌落可以显示非常复杂的空间模式，这取决于各种培养条件，例如营养物浓度和琼脂平板的固体度。例如，菌落的殖民地模式表现为DLA（扩散限制聚集）模式或DBM（密集分支形态）模式。用非线性偏微分方程描述了细菌的趋化运动。它的数值解再现了同心斑点图案，其前沿随着时间的推移而向外推进。斑点图案的再现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，我还可以获得更复杂的密集分支形态，这取决于初始营养浓度和细菌的扩散系数。随着营养盐初始浓度的增加，菌落形态的扩展速率呈抛物线增加。当初始营养盐浓度较高时，数值模拟得到的膨胀率与理论计算结果基本一致。

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.