Allen-Cahn-based model of bacterial growth in nutrient media: theoretical foundations and in silico studies of the Allee effect

Mathematical models designed for describing, predicting, and controlling states of microbial systems promote progress in solving vital problems at the forefront of microbiology. The paper presents a theoretical justification for the modeling and \textit{ in silico} studies of the evolution of bacterial communities cultivated in nutrient medium. We propose a coupled approach to formalize the formation of dendrite patterns of bacteria grown in nutrient medium and the corresponding characteristics of bacterial communication. The conceptualization includes the Allen-Cahn-based model of bacterial colony evolution combined with the model of changes in biomass-dependent nutrient concentration and the reaction-diffusion model of bacterial quorum sensing. The bacterial evolution model is associated with the Landau theory within the core framework of the existence of a growth threshold for bacteria during the growth process, known as the Allee effect. The unique solvability of the initial boundary value problem is proved for the mathematical model of nutrient-dependent bacterial growth. The theoretical results are based on the derivation of new \textit{a priori} estimates for the solution of the semilinear initial boundary value problem. The general mathematical model was implemented using the finite element method with the COMSOL Multiphysics platform. Various computation experiments attributed to \textit{Pseudomonas} bacterial strains were performed to examine different scenarios of the spatiotemporal dynamics of key substances of the biosystem. The results obtained indicate that the considered approach can be applied to simulate the growth of dendritic bacterial colonies and to control the optimal population size above which survival is possible despite the Allee effect.