Mathematical modeling of biofilm development

Abstract

We perform mathematical anaysis of the biofilm development process. A model describing biomass growth is proposed: It arises from coupling three parabolic nonlinear equations: a biomass equation with degenerate and singular diffusion, a nutrient tranport equation with a biomass-density dependent diffusion, and an equation of the Navier-Stokes type, describing the fluid flow in which the biofilm develops. This flow is subject to a biomass-density dependent obstacle. The model is treated as a system of three inclusions, or variational inequalities; the third one causes major difficulties for the system's solvability. Our approach is based on the recent development of the theory on Navier-Stokes variational inequalities. on the other hand u ε := ρ ε * u is the local spatial-average of u(x, t) by means of the usual mollifier ρ ε (x) (see Section 2 for details).(ii) The nutrient concentration w(x, t) is non-negative and has the threshold value 1, i.e. 0 ≤ w(x, t) ≤ 1. Also, we suppose that there is no nutrient supply from the exterior. The diffusion coefficient d(u) depends on the biomass density u and c d ≤ d(r) ≤ c d , |d(r 1 ) − d(r 2 )| ≤ L(d)|r 1 − r 2 |, ∀r 1 , r 2 ∈ R, (1.2)

Analysis of a finite-volume scheme for a single-species biofilm model

“…A biofilm growth model that well describes the spatial spreading mechanism for biomass and the dependency on the nutrient was suggested in [14]. The model was analyzed in [15,21] and numerically solved in [2,11]. Up to our knowledge, there does not exist any analysis for the numerical approximations in the literature.…”

“…The existence and uniqueness of a global weak solution to (1)-( 5) was shown in [15], while the original model was analyzed in [21] by formulating it as a system of variational inequalities. The model of [14] was extended in [13] by taking into account nutrient taxis, which forces the biofilm to move up a nutrient concentration gradient.…”

Parabolic Quasi-Variational Inequalities (I) -- Semimonotone Operator Approach

“…In this context, many mathematical models, arising in biochemical/mechanical problems (cf. [1,2,8,11,13,14,16,19,26,27], have been discussed.…”

“…In particular, in Section 7.2. we give the concrete setting for the superconductivity type II problem, in a description inspired by [1,2,19,27]. As for the description of the flow problem, see our previous work [11,12] and references therein.…”

Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm

“…For instance, multidimensional models can be implemented for specific applications when micro-scale outputs are required [7]. The spatial distribution of diffusing compounds and microbial species within the biofilm, and the physical structure of the biofilm at a micro-scale level can be investigated by using complex 2D and 3D mathematical models [23][24][25][26]. If a macro-scale output is required, as in the case of engineering biofilm reactors, 1D formulations have been recognized as efficient tools to analyze bioreactor performances in terms of biomass accumulation and degradation of substrates [27].…”