A Mixed-Culture Biofilm Model with Cross-Diffusion

Abstract

We propose a deterministic continuum model for mixed-culture biofilms. A crucial aspect is that movement of one species is affected by the presence of the other. This leads to a degenerate cross-diffusion system that generalizes an earlier single-species biofilm model. Two derivations of this new model are given. One, like cellular automata biofilm models, starts from a discrete in space lattice differential equation where the spatial interaction is described by microscopic rules. The other one starts from the same continuous mass balances that are the basis of other deterministic biofilm models, but it gives up a simplifying assumption of these models that has recently been criticized as being too restrictive in terms of ecological structure. We show that both model derivations lead to the same PDE model, if corresponding closure assumptions are introduced. To investigate the role of cross-diffusion, we conduct numerical simulations of three biofilm systems: competition, allelopathy and a mixed system formed by an aerobic and an anaerobic species. In all cases, we find that accounting for cross-diffusion affects local distribution of biomass, but it does not affect overall lumped quantities such as the total amount of biomass in the system.

Analysis of a Degenerate and Singular Volume-Filling Cross-Diffusion System Modeling Biofilm Growth

“…These effects are extremely useful in wastewater engineering, where different processes (like aerobic and anoxic processes or simultaneous sulfate reduction and nitrogen removal) take place simultaneously. It has been pointed out [22] that when two colonies of different species merge, spatial biomass gradients can be observed, leading to a spatially heterogeneous distribution of biomass. These phenomena can be described by cross diffusion, which models how the diffusion of one species is influenced by the concentration gradient of the other species in diffusive multi-species systems.…”

“…These phenomena can be described by cross diffusion, which models how the diffusion of one species is influenced by the concentration gradient of the other species in diffusive multi-species systems. Recently, a cross-diffusion biofilm model (see (3)) was introduced by Rahman, Sudarsan and Eberl [22], which reflects the same properties as the single-species nonlinear diffusion model [9] (see (6)) constructed from experiments, namely a porous-medium type degeneracy when the local biomass vanishes, which leads to a finite speed of propagation of the interface, and a singularity when the biomass reaches the maximum capacity, which guarantees the boundedness of the total mass. It can be formally derived from a space-discrete random-walk lattice model [20,22,26] (see Appendix).…”

“…Recently, a cross-diffusion biofilm model (see (3)) was introduced by Rahman, Sudarsan and Eberl [22], which reflects the same properties as the single-species nonlinear diffusion model [9] (see (6)) constructed from experiments, namely a porous-medium type degeneracy when the local biomass vanishes, which leads to a finite speed of propagation of the interface, and a singularity when the biomass reaches the maximum capacity, which guarantees the boundedness of the total mass. It can be formally derived from a space-discrete random-walk lattice model [20,22,26] (see Appendix). Due to the cross-diffusion structure, standard techniques like maximum principles and regularity theory cannot be used, and since the diffusion matrix is generally neither symmetric nor positive definite, even the local-in-time existence and boundedness of solutions is hard to prove.…”

“…These boundary conditions describe well the behavior of the biofilm at its border, but also homogeneous Neumann boundary conditions can be handled, see Remarks 10,11,12. In order to close the model, the functions p and q need to be chosen appropriately. Following the approach in [22], the idea is to derive p and q in (3) from the single-species biofilm model [9,17] (here without reaction)…”

“…In order to compensate the disadvantages of the model classes described above, Rahman, Sudarsan, and Eberl [22] introduced a two-species diffusion model which captures the quantitative amount of local mixing effects between the species within a biofilm colony with the help of cross-diffusion terms. The derivations of this system from mass balances or from discrete lattice models do not (a priori) impose the condition that the volume fractions must add up to unity.…”