ANALYSIS AND SIMULATIONS OF A FREE BOUNDARY PROBLEM MODELLING PHOTOTROPHIC GRANULAR BIOFILMS

A model simulating the initial formation and growth of a phototrophic granular biofilm around wireless light emitter is proposed. The work focuses on the well-posedness and quantitative analysis of the model describing the initiation of the biofilm growth induced by cell attachment in the case of spherical geometry. Biofilm granule generates from the agglomeration of free living cells which lead to the formation of spherical-shaped microbial consortia. Under the hypothesis of radial symmetry, the granular biofilm is modelled as a free boundary domain, where the free boundary is the radius of the granule. The attached species proliferate by consuming dissolved substrates diffusively transported from the bulk liquid within the granule. Granule size evolves over time due to attachment flux, microbial growth and invasion. The evolution of the state variables, that is the concentrations of the microbial cells both in sessile and suspended form and the dissolved substrates, is governed by a nonlinear system of hyperbolic-elliptic differential equations, while the free boundary evolution is governed by an ordinary differential equation. Light intensity provided by the emitter is accounted in the model as a further state variable, as it affects the attachment velocity and metabolic activity of light-dependent microorganisms. By using the method of characteristics, the equations are converted into an equivalent integral system. Existence and uniqueness of solutions are discussed and proved for the attachment regime using fixed point strategies. A numerical study is performed to explore and investigate the role of light-dependent species on the granulation process and how light conditions and concentration of nutrients affect the attachment and granule initiation.