Tenore, Alberto (2021) Free boundary problems for mixed-species biofilms: modelling and simulation. [Tesi di dottorato]

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This dissertation proposes a modelling study on biofilms, with a special interest in phototrophic biofilms, one of the most promising innovative biological technologies in the field of wastewater treatment. Novel mathematical models are derived and presented here, with the aim of describing exhaustively the formation and the growth of planar and granular biofilms, specifically phototrophic-heterotrophic biofilms, anaerobic granular biofilms and oxygenic photogranules. The introduction of novel mathematical formulations allows to describe crucial aspects and processes of these ecosystems, never or not exhaustively explored by mathematical models present in literature. Models presented here are formulated as one-dimensional free boundary problems, which describe the evolution of planar and granular biofilms. The processes taking place within the biofilm are modelled through systems of nonlinear partial differential equations (PDEs), derived using a continuum approach: non-linear hyperbolic PDEs model the advective transport and growth of sessile biomasses which constitute the biofilm, while quasi-linear parabolic PDEs govern the diffusive transport and conversion of soluble substrates, and the phenomena of microbial invasion by planktonic cells inhabiting the surrounding environment. The first proposed model consists of a free boundary problem describing the role of planktonic cells in the formation and evolution of planar multispecies biofilms, by modelling the phenomena of initial attachment and microbial invasion. Moreover, a theorem of existence and uniqueness of the solutions, based on the fixed-point theorem, is presented. The second model is formulated as a free boundary problem which describes the ecology of phototrophic-heterotrophic biofilms. It considers the processes of microbial invasion and focuses on the metabolic activities of phototrophic and heterotrophic species, their interactions and main factors involved in this biofilm ecosystem, such as light conditions and attenuation, photoinhibition, phototrophic release of organic matter, production of extracellular polymeric substances (EPS). In order to model the process of initial formation and growth of anaerobic granular biofilms, known as de novo granulation, a free boundary problem is formulated within a spherical domain with radial symmetry. In this case, a multiscale approach is used, by modelling both the evolution of granular anaerobic biofilms and the dynamics of the bioreactor where such biofilms develop. Hence, such model allows to simulate the global treatment process occurring in anaerobic granular systems and to draw engineering conclusions. Finally, the latest model is aimed to describe for the first time the oxygenic photogranules (OPGs), biofilm granules composed of cyanobacteria and microalgae, recently recognised as an attractive biological technology to remove polluting compounds from wastewaters. As the previous one, this multiscale model is formulated as a spherical free boundary problem with radial symmetry, and allows to accurately describe the genesis and growth processes of oxygenic photogranules within a sequencing batch reactor (SBR) and to investigate the treatment efficiency of this system. The model considers the main biotic and abiotic factors involved, the symbiotic and competitive microbial mechanisms driving the treatment process, the metabolic differences between cyanobacteria and microalgae and the key role that cyanobacteria play in the photogranulation. All models are integrated numerically through the development of original numerical softwares in MatLab platform. The main numerical methods used are the method of characteristics, the Euler explicit method and the method of lines. Furthermore, the models presented are used to carry out numerical studies of relevant engineering, biological and ecological interest.

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