Analysis of a Biofilm Model in a Continuously Stirred Tank Reactor with Wall Attachment

Abstract

We investigate a mathematical model for a bacterial population in a continuously stirred tank reactor with wall attachment. The model couples a free-boundary value problem for substrate diffusion in the one-dimensional biofilm with a system of nonlinear ODEs for biofilm thickness, suspended biomass, and free substrate concentration. We establish global well-posedness and analyze the long-term dynamics. In particular, we characterize the local and global stability of the trivial (washout) equilibrium, prove the existence of a nontrivial equilibrium, and, under additional structural assumptions, establish its uniqueness and derive conditions for its local stability.

Figures (4)

• Figure 1: Schematic view of a CSTR.
• Figure 2: Convergence to the washout equilibrium $(0,\,S^*,\,0)=(0,\,0.5,\,0)$. The conditions of Corollary \ref{['C10']} are satisfied with $d(0)=0$, $g(r(S^*))=-2/3<0$, and $\nu(S^*)=2/3<k_2=2$. Initial conditions: $(h_0,S_0,Q_0)=(0.5,\,0.3,\,0.3)$.
• Figure 3: Convergence to the unique nontrivial equilibrium for $S^*=5$. The washout equilibrium is unstable since $g(r(S^*))=4/3>0$ (cf. Remark \ref{['R1']} with $d(0)=0$). Existence and uniqueness follow from Theorem \ref{['T5']} with $\alpha+k_2=3>\nu(S^*)=5/3$. Initial conditions: $(h_0,S_0,Q_0)=(0.1,\,5.0,\,0.1)$. Dashed lines indicate the numerically determined equilibrium $(h_\star,S_\star,Q_\star)\approx (0.923,2.118,0.518)$.
• Figure 4: Five trajectories from widely separated initial conditions in $(0,\infty)^3$ all converge to the unique nontrivial equilibrium $(h_\star,S_\star,Q_\star)$ (gold dashed lines). This is consistent with the boundedness of orbits in Proposition \ref{['PX']} and provides numerical evidence that the locally stable equilibrium of Theorem \ref{['T6']} is in fact globally attracting. Parameters and nontrivial equilibrium as in Figure \ref{['fig:case2']} with $S^*=5$.

Theorems & Definitions (33)

• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Theorem 2.3
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof