Increasing the dilution rate can globally stabilize two-step biological systems

Highlights

• Conditions are given to stabilize a 2-step bioprocess in increasing the dilution rate.

• Simulations are realized to illustrate the new control strategy.

Abstract

We revisit two-step mass-balance models of biological processes as met to describe numerous biological systems including the anaerobic digestion or the nitrification process in view of its global stabilization. We show that when a bi-stability occurs, it can be possible to globally stabilize the dynamics toward an unique positive equilibrium by increasing the dilution rate. We give sufficient conditions on the growth functions of the model for this situation to appear. This illustrates that for biological multi-step reactional systems, increasing the residence time (e.g. decreasing the input flow rate) may not be the only way to stabilize the dynamics.

Introduction

In most of continuous cultures, it is well known that increasing the dilution rate (or equivalently reducing the residence time inside the reactors) can destabilize the dynamics, in the sense that it enlarges the attraction basin of the wash-out equilibrium. This can be easily shown on the classical mathematical model of the chemostat, whatever the kinetics includes inhibition or not (see [1]). For growth inhibited by the substrate, bi-stability systematically occurs for large values of the input concentration of substrate. This feature has practical impacts on positive equilibrium (when it exists) because it cannot be globally stable, and the dynamics can conduct the system to the wash-out of the biomass, when the state belongs to the attraction basin of the washout equilibrium. Ways to guarantee a global stability is either to fix a lower dilution rate, which is penalizing for the performance of the process, or to control the dilution rate with a feedback loop, which temporarily diminishes the dilution rate when the state is far from the positive steady state [2], [3], [4], [5]. In any case, the removal has to be reduced at a certain stage.

Here we consider a more complex reaction scheme that are two-step systems, as met for instance in many models representing the anaerobic digestion process [6] or the nitrification process [7]. For these systems we show that there exist situations presenting a bi-stability for which increasing (and not decreasing) the dilution rate also leads the system to a globally asymptotically stable steady state, in opposition to classical stabilizing practices. In such a way, we can treat more matter per unit time during the transient than when decreasing the dilution rate.

In the paper, we denote by ${\mathbb{R}}_{+}$ the set of non-negative numbers and by ${\mathbb{R}}_{+,\star }$ the set of positive numbers.

Let us consider the general mathematical model of a two-step mass-balance biological process, given by the following equations:

$$\left\{\begin{array}{ccc}{\dot{x}}_1\hfill & =\hfill & {\mu }_1\left(s_1\right)x_1-\alpha D{x}_1\hfill \\ {\dot{s}}_1\hfill & =\hfill & -{\mu }_1\left(s_1\right)x_1+D(s_1^{in}-s_1)\hfill \\ {\dot{x}}_2\hfill & =\hfill & {\mu }_2\left(s_2\right)x_2-\alpha D{x}_2\hfill \\ {\dot{s}}_2\hfill & =\hfill & -{\mu }_2\left(s_2\right)x_2+{\mu }_1\left(s_1\right)x_1+D(s_2^{in}-s_2)\hfill \end{array}\right.$$

where the parameter $D$ denotes the dilution rate.

This model is presented under the original form proposed in [6]. The first reaction involves a microbial species of concentration $x_1$ which grows on a substrate of concentration $s_1$ with a monotonic specific rate ${\mu }_1$. The incoming flow fed the culture with substrate of concentration $s_1^{in}$. The second reaction involves a second microbial species of concentration $x_2$ which grows on another substrate of concentration $s_2$, with a specific growth rate denotes ${\mu }_2$. This reaction is also fed by the first one which produces the second substrate. In addition, the incoming flow rate may contain (or not) substrate of concentration $s_2^{in}$. The parameter $\alpha \in (0,1]$ reflects the fact that the effective dilution rate of the biomass is impacted by a retention inside the tank, differently to the abiotic resource. Here, the yield coefficients of the transformations of substrate $s_i$ into biomass $x_i$ ($i=1,2$), and of the production of substrate for the second reaction by the first one, have been all kept equal to $1$ (this is always possible without any loss of generality, by a right choice of the concentration units).

In many biological systems, such as the anaerobic digestion or the nitrification processes, it is often reported in the literature that the second reaction is inhibited by large values of $s_2$, which amounts to consider the following hypotheses.

Assumption 1

The functions ${\mu }_1$, ${\mu }_2$ belong to $C^1({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ and fulfill the following properties.

• ${\mu }_1$ is increasing on ${\mathbb{R}}_{+}$ with ${\mu }_1\left(0\right)=0$.

• There exists ${\hat{s}}_2>$ such that ${\mu }_2$ is increasing on $[0,{\hat{s}}_2)$ and decreasing on $({\hat{s}}_2,+\infty )$, with ${\mu }_2\left(0\right)=0$ and ${\mu }_2(+\infty )=0$.

The model (1) has a cascade structure: the first reaction is independent of the second one and the $(x_1,s_1)$ sub-system follows the classical (mono-specific) chemostat model. However, the $(x_2,s_2)$ sub-system is more complex to study as it receives substrate from the first reaction and ${\mu }_2$ is non-monotonic. This model and some of its variants has been already well studied in the literature [6], [8], [9], depending on the operating parameters $(s_1^{in},s_2^{in},D)$. In particular, it has been shown that the dynamics may exhibit a multiple-stability, and the complete operating diagram has been established in [7], [10]. The purpose of the present work is to complement those studies, investigating how to adapt the value of the dilution rate $D$ to ensure a global stability of the dynamics. For sake of completeness, we first recall in the next section the set of possible asymptotic behaviors of the model.