A chemostat model with variable dilution rate due to biofilm growth

Abstract

In many real life applications, a continuous culture bioreactor may cease to function properly due to bioclogging which is typically caused by the microbial overgrowth. This is a problem that has been largely overlooked in the chemostat modeling literature, despite the fact that a number of models explicitly accounted for biofilm development inside the bioreactor. In a typical chemostat model, the physical volume of the biofilm is considered negligible when compared to the volume of the fluid. In this paper, we investigate the theoretical consequences of removing such assumption. Specifically, we formulate a novel mathematical model of a chemostat where the increase of the biofilm volume occurs at the expense of the fluid volume of the bioreactor, and as a result the corresponding dilution rate increases reciprocally. We show that our model is well-posed and describes the bioreactor that can operate in three distinct types of dynamic regimes: the washout equilibrium, the coexistence equilibrium, or a transient towards the clogged state which is reached in finite time. We analyze the multiplicity and the stability of the corresponding equilibria. In particular, we delineate the parameter combinations for which the chemostat never clogs up and those for which it clogs up in finite time. We also derive criteria for microbial persistence and extinction. Finally, we present a numerical evidence that a multistable coexistence in the chemostat with variable dilution rate is feasible.

Figures (6)

• Figure 1: The aggregations of the species $u$ form the larger and heavier biopolymer which is not influenced by the dilution. Hence, these polymers will stay on the wall as the adherent, which is corresponding to the shaded part of the second figure. The third figure describes the relationships between all variables in the chemostat model and the dilution rate is influenced by the growth of wall-attached species $w$.
• Figure 2: This Figure shows the simulated trajectories of the original system (\ref{['dilution.model']}) (shown in blue) and the rescaled system (\ref{['rescaled.model']}) (shown in red) corresponding to the same initial condition $(S(0), u(0), w(0)) = (0.2, 3, 14) \in \Omega$. The parameters and the growth functions are given by $\alpha = 1, \beta = 1.4, w_{0} = 14.7$, $f(S) = \dfrac{1.03S}{3.7+S}$, and $g(S) = \dfrac{0.6S}{0.8+S}$, respectively.
• Figure 3: This Figure shows a numerically simulated trajectory of the rescaled system (\ref{['rescaled.model']})corresponding to the initial condition $(S(0), u(0), w(0)) = (0.2, 3, 5) \in \Omega$. The parameters and the growth functions are given by $\alpha = 1, \beta = 0.1, w_{0} = 14.7$, $f(S) = \dfrac{1.03S}{3.7+S}$, and $g(S) = \dfrac{0.6S}{0.8+S}$, respectively. The lower right graph describes the relationship between the physical time $t$ and the rescaled time $\tau$. Numerically, the clogged time is $t_{\max} \sim 3.72$.
• Figure 4: In this Figure, we illustrate the scenario in which the clogged state $E_w$ is globally attractive. The parameter values and the growth functions are given by $\alpha = 1, \beta = 0.1, w_{0} = 14.7$, $f(S) = \dfrac{1.03S}{3.7+S}$, and $g(S) = \dfrac{0.6S}{0.8+S}$, respectively. With these parameter values, the system admits no positive equilibria. Since $g(1) =\frac{1}{3} >\beta$, $E_w$ is stable. This Figure shows 5 orbits corresponding to the initial conditions chosen randomly from $\Omega$. All of these orbits are attracted by $E_w$.
• Figure 5: In this Figure, we illustrate the bistable scenario in the case when the clogged state $E_w$ is locally stable, that is, in the absence of the uniform persistence. The parameter values and the growth functions are given by $\alpha = 3.4, \beta = 0.94, w_{0} = 6.5$, $f(S) = \dfrac{5S}{2+S}$, and $g(S) = \dfrac{1.8S}{0.9+S}$, respectively. With these parameter values, the system admits two positive equilibria: $E_{c} \sim (0.328,0.672,4.975)$ and $E_{c1} \sim (0.5,0.5,5.721)$. The equilibria $E_{c}$ and $E_{w}$ are locally stable, and $E_{c1}$ is a saddle point whose stable manifold separates the basins of attraction of $E_{c}$ and $E_{w}$, respectively. The figure shows 5 red orbits and 5 blue orbits. The red orbits correspond to initial conditions that are randomly chosen from the set $\left\{(S,u,w)|0.8 \leq S \leq 1, 0\leq u \leq 1+\frac{\beta w_0}{4}, 0.9 w_0 \leq w < w_0\right\}$, and all red orbits are attracted by $E_{c}$. The blue orbits correspond to initial conditions that are randomly chosen from the set $\left\{(S,u,w)|0.8 \leq S \leq 1, 0\leq u \leq 1+\frac{\beta w_0}{4}, 0.9 w_0 \leq w < w_0\right\}$, and all blue orbits are attracted by $E_{w}$.

Theorems & Definitions (36)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Lemma 4.1
• proof