Hybrid approach to modeling and evaluation of the structural features for patterns of cultured bacteria

Mathematical models and simulation of the growth kinetics of microorganisms under certain external conditions provide the basis for diagnostics, control and prediction of chemical substances characterizing the state of systems of various natures. The paper is devoted to the approach development for modeling and numerical specification of the structural and dynamic characteristics of the patterning during the surface cultivation of bacteria on nutrient media. The mathematical model is formalized as an initial-boundary problem for a system of reaction-diffusion equations that determine the concentrations of bacterial biomass and nutrient substrate. For model description of naturalistic patterns, the stochastic procedure was introduced to provide the evolutionary deformation of the bacterial population with colonization potential and the formation of colonies with different incubation periods. An algorithm for the numerical solution of a nonlinear differential problem is constructed using the Yanenko finite-difference scheme, supplemented by an iterative procedure. The technique for assessing the geometric characteristics of bacterial patterns is based on the calculation of the fractal dimension of the boundaries of cluster structures. The software implementation of the algorithms was performed using the Matlab software. The results of a series of computational experiments are presented to visualize the spatiotemporal distributions of bacterial biomass and nutrient substrate under variation of the control parameters of the hybrid model. The bacterial pattern for mation of a branching morphological type was specified depending on changes in the level of initial nutrient concentration, diffusion parameter, and stochastic growth parameter. The simulation results indicated that an increase in the initial concentration of nutrient leads to a geometric phase transition, namely the transformation of the structure from a dendrite, through an irregular structure with a floral pattern, to a homogeneous cluster with a rather smooth boundary.