We present a multi-dimensional continuum mathematical model for modeling the growth of a symbiotic biofilm system. We take a dual-species namely, the Streptococcus–Veillonella sp. biofilm system as an example for numerical investigations. The presented model describes both the cooperation and competition between these species of bacteria. The coupled partial differential equations are solved by using an integrative finite element numerical strategy. Numerical examples are carried out for studying the evolution and distribution of the bio-components. The results demonstrate that the presented model is capable of describing the symbiotic behavior of the biofilm system. However, homogenized numerical solutions are observed locally. To study the homogenization behavior of the model, numerical investigations regarding on how random initial biomass distribution influences the homogenization process are carried out. We found that a smaller correlation length of the initial biomass distribution leads to faster homogenization of the solution globally, however, shows more fluctuated biomass profiles along the biofilm thickness direction. More realistic scenarios with bacteria in patches are also investigated numerically in this study.

我们提出了一个多维连续统数学模型，用于对共生生物膜系统的生长进行建模。我们采用双物种，即链球菌-韦永氏菌sp. 生物膜系统作为数值研究的一个例子。所提出的模型描述了这些细菌物种之间的合作和竞争。耦合偏微分方程通过使用综合有限元数值策略求解。数值例子用于研究生物成分的演变和分布。结果表明，所提出的模型能够描述生物膜系统的共生行为。然而，在局部观察到均匀的数值解。为了研究模型的均质化行为，进行了关于随机初始生物量分布如何影响均质化过程的数值研究。我们发现，初始生物量分布的较小相关长度会导致溶液在全球范围内更快地均匀化，但是，沿着生物膜厚度方向显示出更多波动的生物量分布。在这项研究中，还对贴片中细菌的更现实场景进行了数值研究。