Hybrid stochastic fractional-based approach to modeling bacterial quorum sensing

Bacterial communication is a complex process, which can be formalized by a deterministic approach and then explored by means of methods of mathematical modeling and computer simulation. To describe bacterial cooperative behavior in the special case of quorum sensing we propose a new mathematical model based on a hybrid stochastic fractional concept that is found examining self-similar dynamic processes of bacterial nucleation and growth. The mathematical model is specified as an initial-boundary value problem for a system of fractional order partial differential equations. We derive a numerical algorithm based on the combination of an implicit finite difference scheme for solving fractional partial differential equations and Monte-Carlo simulation of bacterial dynamics. The computational experiments are performed to simulate reaction-diffusion processes in biological systems related to the self-similar behavior of heterogeneous structure and time memory effect.