  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.
