The Allen–Cahn-Based Approach to Cross-Scale Modeling Bacterial Growth Controlled by Quorum Sensing

This study, grounded in traveling wave theory, develops a cross-scale reaction-diffusion model to describe nutrient-dependent bacterial growth on agar surfaces and applies it to in silico investigations of microbial population dynamics. The approach is based on the coupling of a modified Allen–Cahn equation with the formulation of quorum sensing signal dynamics, incorporating a nutrient-dependent regulatory threshold and stochastic diffusion. A closed-loop model of bacterial growth regulated by quorum sensing is developed through theoretical analysis, numerical simulations, and computational experiments.The model is implemented using Yanenko’s computational scheme, which incorporates corrective refinement via Heun’s method to account for nonlinear components. Numerical simulations are carried out in MATLAB, allowing for accurate computation of spatio-temporal patterns and facilitating the identification of key mechanisms governing the collective behavior of bacterial communities.