This article studies a parabolic-elliptic system modelling the pattern formation in E. coli bacteria in response to a chemoattractant known as acylhomoserine lactone concentration (AHL). The system takes into account certain bacterial strains with motility regulation, and the parameters of the equations represent the bacterial logistic growth, AHL diffusion and the rates of production and degradation of AHL. We consider the numerical solution to the system using the Generalized Finite Difference (GFD) Method, a meshless method known to effectively compute numerical solutions to nonlinear problems. The paper is organized to first explain the derivation of the explicit formulae of the method, followed by the study of the convergence of the explicit scheme. Then, several examples over regular and irregular meshes are given.

翻译：本文研究了一个抛物-椭圆型系统，该系统模拟大肠杆菌在响应被称为酰基高丝氨酸内酯（AHL）浓度的趋化因子时的模式形成。该系统考虑了具有迁移调节功能的特定细菌菌株，方程中的参数代表细菌逻辑增长、AHL扩散以及AHL的产生与降解速率。我们采用广义有限差分（GFD）方法对该系统进行数值求解，这是一种已知能有效计算非线性问题数值解的无网格方法。本文首先阐述了该方法的显式公式推导，随后研究了显式格式的收敛性。最后，给出了若干基于规则网格和不规则网格的数值算例。