Performance Study of Two Serial Interconnected Chemostats with Mortality

Abstract

The present work considers the model of two chemostats in series when a biomass mortality is considered in each vessel. We study the performance of the serial configuration for two different criteria which are the output substrate concentration and the biogas flow rate production, at steady state. A comparison is made with a single chemostat with the same total volume. Our techniques apply for a large class of growth functions and allow us to retrieve known results obtained when the mortality is not included in the model and the results obtained for specific growth functions in both the mathematical literature and the biological literature. In particular, we provide a complete characterization of operating conditions under which the serial configuration is more efficient than the single chemostat, i.e., the output substrate concentration of the serial configuration is smaller than that of the single chemostat or, equivalently, the biogas flow rate of the serial configuration is larger than that of the single chemostat. The study shows that the maximum biogas flow rate, relative to the dilution rate, of the series device is higher than that of the single chemostat provided that the volume of the first tank is large enough. This non-intuitive property does not occur for the model without mortality.

Appendices

Appendix A The Single Chemostat

In this section, we give a brief presentation of the mathematical model of the single chemostat with mortality rate.

The mathematical equations are given by

$$\begin{aligned} \begin{array}{lll} \dot{S} &{} = &{} D(S^\mathrm{in}-S)-f(S)x,\\ \dot{x} &{} = &{} -Dx+f(S)x-ax, \end{array} \end{aligned}$$

(A1)

where S and x denote, respectively, the substrate and the biomass concentration, $S^\mathrm{in}$ the input substrate concentration, a the mortality rate and $D=Q/V$ the dilution rate, with Q the input flow rate and V the volume of the tank. The specific growth rate f of the microorganisms satisfies Assumption 1. It is well known [see Harmand et al. (2017); Smith and Waltman (1995)] that, besides the washout $F_0=(S^\mathrm{in},0)$, this system can have a positive steady state

$$\begin{aligned} F_1=(S^*(D),x^*(S^\mathrm{in},D)), \end{aligned}$$

where

$$\begin{aligned} \textstyle S^*=\lambda (D+a) \quad \text{ and } \quad x^*=\frac{D}{D+a}(S^\mathrm{in}-\lambda (D+a)). \end{aligned}$$

See Fig. 11a for the plot of the function $D\mapsto S^*(D)$ and Fig. 11(b) for the plot of the function $D\mapsto x^*(S^\mathrm{in},D)$ for $0\le D\le \delta$, where $\delta =f(S^\mathrm{in})-a$.

The washout steady state $F_0$ always exists. It is GAS if and only if $D\ge \delta$. It is LES if and only if $D>\delta$. The positive steady state $F_1$ exists if and only if $D<\delta$. It is GAS and LES whenever it exists. Therefore, the curve $\Phi$ defined by

$$\begin{aligned} \Phi :=\{(S^\mathrm{in},D): D=f(S^\mathrm{in})-a\} \end{aligned}$$

(A2)

splits the set of operating parameters $(S^\mathrm{in},D)$ into two regions, denoted $I_0$ and $I_1$, as depicted in 11c. These regions are defined by

$$\begin{aligned} \begin{array}{l} I_0:=\{(S^\mathrm{in},D): D\ge f(S^\mathrm{in})-a\},\\ I_1:=\{(S^\mathrm{in},D): D<f(S^\mathrm{in})-a\}. \end{array} \end{aligned}$$

(A3)

The behavior of the system in each region is given in Table 2. Figure 11c, together with 2 is called the operating diagram of the single chemostat.

Fig. 11

a The map $D\mapsto S^\mathrm{out}(S^\mathrm{in},D)$ is increasing on $[0,\delta ]$, where $\delta =f(S^\mathrm{in})-a$. b The map $D\mapsto x^\mathrm{out}(S^\mathrm{in},D)$ with $f(S)=4S/(5+S)$, $S^\mathrm{in}=10$ and $a=0.6$. c The curve $\Gamma$ in the operating plane $(S^\mathrm{in},D)$ of the single chemostat (color figure online)

Table 2 Stability of steady states in the various regions of the operating diagram

The particularity of this operating diagram is that the curve limiting both regions $I_0$ and $I_1$ is translated from zero, unlike the case with mortality, as shown in Figure 2.5 of Harmand et al. (2017). Thus, with presence of mortality rate, the region where the washout is GAS, is larger.

The output substrate concentration of the single chemostat, at its stable steady state is given by

$$\begin{aligned} S^\mathrm {out}(S^\mathrm {in},D)= \left\{ \begin{array}{lcl} S^\mathrm {in}&{}{}\text { if }&{}{}D\ge \delta , \\ \lambda \left( D+a\right) &{}{}\text { if }&{}{}D< \delta . \end{array} \right. \end{aligned}$$

(A4)

Its output biomass concentration at steady state is then given by

$$\begin{aligned} x^\mathrm {out}(S^\mathrm {in},D)= \textstyle \frac{D}{D+a}(S^\mathrm {in}-S^\mathrm {out}(D,a)) \end{aligned}$$

(A5)

For all $S^\mathrm{in}>\lambda (a)$, one has

$$\begin{aligned} \textstyle \frac{\partial S^\mathrm {out}}{\partial D}(S^\mathrm {in},D)= \left\{ \begin{array}{lll} 0 &{}{} \text { if } &{}{} D> \delta , \\ \lambda '(D+a) &{}{} \text { if } &{}{} D< \delta . \end{array} \right. \end{aligned}$$

Thus, for any growth function satisfying Assumption 1 the function $D\mapsto S^\mathrm{out}(S^\mathrm{in},D)$ is increasing on $[0,\delta ]$, as shown in Fig. 11a. The function $D\mapsto x^\mathrm{out}(S^\mathrm{in},D)$ is illustrated in Fig. 11b for a Monod function.

The biogas flow rate of the single chemostat is defined, up to a multiplicative yield coefficient, by

$$\begin{aligned} G_\mathrm{chem}(S^\mathrm{in},D):=Vx^\mathrm{out}f(S^\mathrm{out}). \end{aligned}$$

(A6)

Using the expressions (A4) and (A5), respectively, of $S^\mathrm{out}$ and $x^\mathrm{out}$, the biogas flow rate of the single chemostat is given by:

$$\begin{aligned} G_\mathrm {chem}(S^\mathrm {in},D)= \left\{ \begin{array}{lll} 0 &{}{} \text { if } &{}{} D\ge \delta , \\ VD(S^\mathrm {in}-\lambda (D+a)) &{}{} \text { if } &{}{} D<\delta . \end{array} \right. \end{aligned}$$

(A7)

For a given $S^\mathrm{in}>\lambda (a)$, the function $D \mapsto G_\mathrm{chem}(S^\mathrm{in},D)$ is null for $D=0$ or $D\ge \delta$, and is positive for $D \in (0,\delta )$. Therefore it admits a maximum in $(0,\delta )$, which is assumed to be unique. A characterization of the growth functions for which this uniqueness is satisfied can be found in Sari (2022).

Proposition 9

Assume that for any $S^\mathrm{in}>\lambda (a)$, the maximum of $D \mapsto G_\mathrm{chem}(S^\mathrm{in},D)$ is unique, and define $\overline{D}(S^\mathrm{in})\in (0,\delta )$, such that

$$\begin{aligned} G_\mathrm{chem}\left( S^\mathrm{in},\overline{D}\left( S^\mathrm{in}\right) \right) =\max _{D \ge 0} G_\mathrm{chem}\left( S^\mathrm{in},D\right) . \end{aligned}$$

Then, the dilution rate $D=\overline{D}\left( S^\mathrm{in}\right)$ is the solution of the equation $S^\mathrm{in}=g(D)$, where the function $g:[0,m-a) \mapsto {\mathbb {R}}$ is given by

$$\begin{aligned} g(D):=\lambda (D+a)+D\lambda '(D+a)). \end{aligned}$$

(A8)

Proof

For any $S^\mathrm{in}>\lambda (a)$ and $D\in (0,\delta )$, we have

$$\begin{aligned} \textstyle \frac{\partial { G_\mathrm {chem}}}{\partial D}(S^\mathrm {in},D)=V\left( S^\mathrm {in}-\lambda (D+a)-D\lambda '(D+a)\right) . \end{aligned}$$

(A9)

Therefore, $\frac{\partial G_\mathrm{chem}}{\partial D}(S^\mathrm{in},D)=0$ if and only if $S^\mathrm{in}=g(D)$, where g is defined by (A8). $\square$

Notice that the function g defined by (A8) is the same as the function g, defined by (19), which was obtained as the limit, when r tends to 1, to the function $g_r$, defined by (9). Recall that $\Gamma$ is the curve of equation $S^\mathrm{in}=g(D)$, see (22). This curve is depicted in Fig. 11c. It is the set of operating conditions given the higher biogas of the single chemostat. More precisely, for any $S^\mathrm{in}>\lambda (a)$, the maximum $D=\overline{D}(S^\mathrm{in})$ of the biogas satisfies the condition $(S^\mathrm{in},D)\in \Gamma$.

Therefore, a sufficient condition for the uniqueness of $\overline{D}(S^\mathrm{in})$ is that the mapping g is increasing. If, in addition, f is ${\mathcal {C}}^2$, then, deriving (A9) with respect of D, we have

$$\begin{aligned} \textstyle \frac{\partial ^2{ G_\mathrm{chem}}}{\partial D^2}(S^\mathrm{in},D)=-Vg'(D). \end{aligned}$$

Hence, a sufficient condition for Assumption 4 to be satisfied is that $g'(D)>0$ for $D\in [0,m-a)$. This last condition if satisfied whenever $f''\le 0$ on $(\lambda (a),+\infty )$, or $\left( \frac{1}{f-a}\right) ''>0$ on $(\lambda (a),+\infty )$, see Lemma 1 in Sari (2022). Therefore we can make the following remark.

Remark 3

Linear and Monod growth functions satisfy Assumption 4, since they satisfy $f''\le 0$ on $(0,+\infty )$. On the other hand the Hill function satisfies Assumption 4, since it satisfies $\left( \frac{1}{f-a}\right) ''>0$ on $(\lambda (a),+\infty )$, as shown in Proposition 8.

Appendix B The Serial Configuration

We consider a slight extension of system (1) with different mortality rates in the two tanks. Indeed, we assume that the growth environment differs from one tank to another one. This can lead to two different mortality rates in the tanks. We denote by $a_1$ and $a_2$ the mortality rates. The mathematical model is given by the following equations.

$$\begin{aligned} \begin{array}{lll} \dot{S}_{1} &{}{} = &{}{} \frac{D}{r}(S^\mathrm {in}-S_1)-f(S_1)x_{1},\\ \dot{x}_{1} &{}{} = &{}{} -\frac{D}{r}x_{1}+f(S_1)x_{1}-a_1x_1,\\ \dot{S}_{2} &{}{} = &{}{} \frac{D}{1-r}(S_1-S_2) -f(S_2)x_{2},\\ \dot{x}_{2} &{}{} = &{}{} \frac{D}{1-r}(x_{1}-x_{2})+f(S_2)x_{2}-a_2x_2. \end{array} \end{aligned}$$

(B10)

The following result is classical in the mathematical theory of the chemostat.

Lemma 7

For any non-negative initial condition, the solution of system (B10) $(S_1(t),x_1(t),S_2(t),x_2(t))$ is non-negative for any $t>0$ and positively bounded.

Proof

Since the vector field defined by (B10) is $C^1$, the uniqueness of the solution to an initial value problem holds. From (B10) and using $f(0)=0$, we have:

$$\begin{aligned} \begin{array}{rcl} \text { for } i=1,2,\quad S_i=0 &{}{}\Longrightarrow &{}{} {\dot{S}}_i> 0,\\ x_1=0 &{}{}\Longrightarrow &{}{} {\dot{x}}_1=0,\\ x_1\ge 0 \text { and } x_2=0 &{}{}\Longrightarrow &{}{} {\dot{x}}_2\ge 0. \end{array} \end{aligned}$$

Therefore, for $i=1,2$, $S_i(t)\ge 0$ and $x_i(t)\ge 0$, for all $t\ge 0$, for which they are defined, provided $S_i(0)\ge 0$ and $x_i(0)\ge 0$, for $i=1,2$, see Prop. B.7 in Smith and Waltman (1995). This proves that the solutions of non-negative initial conditions are always non-negative. Let $z_i=S_i+x_i$, $i=1,2$. From system (B10), we have

$$\begin{aligned} \textstyle \dot{z_1}=\frac{D}{r}(S^\mathrm{in}-z_1)-a_1x_1, \quad \dot{z_2}=\frac{D}{1-r}(z_1-z_2)-a_2x_2. \end{aligned}$$

Consequently, we have the differential inequality

$$\begin{aligned} \textstyle \dot{z_1}\le \frac{D}{r}(S^\mathrm{in}-z_1), \end{aligned}$$

It follows by comparison of solutions of ordinary differential equations (see, for instance, Walter (1998)) that one has

$$\begin{aligned} z_1(t)\le S^\mathrm {in}+(z_1(0)-S^\mathrm {in})e^{-\frac{D}{r}t}. \end{aligned}$$

Therefore, $z_1(t)\le Z_1$, where $Z_1=\max (S^\mathrm{in},z_1(0))$. Then, we also have the differential inequality

$$\begin{aligned} \textstyle \dot{z_2}\le \frac{D}{1-r}(Z_1-z_2). \end{aligned}$$

It follows again by comparison of solutions of ordinary differential equations that one has also

$$\begin{aligned} z_2(t)\le Z_1+(z_2(0)-Z_1)e^{-\frac{D}{1-r}t}. \end{aligned}$$

Therefore, $z_2(t)\le Z_2$, where $Z_2=\max (Z_1,z_2(0))$. Hence, the solutions of (B10) are positively bounded. Therefore, they are defined for all $t \ge 0$. $\square$

For the description of the steady states, we shall consider the following function h that will play a key role

$$\begin{aligned} \begin{array}{l} h\left( S_2\right) = \frac{D+(1-r)a_2}{1-r}\frac{S_1^*-S_{2}}{b-S_2}, \quad { S_2 \in (0,b)}\\ \text{ where } S_1^*=\lambda \left( \frac{D}{r}+a_1\right) ,~ b=\frac{D(S^\mathrm{in}-S_1^*)}{D+ra_1}+S_1^*. \end{array} \end{aligned}$$

(B11)

This function satisfies the following property.

Lemma 8

Assume that $D/r+a_1<f(S^\mathrm{in})$. The function h is decreasing from $h(0)>0$ to $h(S_1^*)=0$, where h(0) is given by

$$\begin{aligned} h\left( 0\right) =\textstyle \frac{D+(1-r)a_2}{1-r}\frac{({D+ra_1})S_1^*}{{DS^\mathrm{in}+ra_1S_1^*}}. \end{aligned}$$

(B12)

Proof

From the condition $D/r+a_1<f(S^\mathrm{in})$ it is deduced that $S_1^*<S^\mathrm{in}$. Note that

$$\begin{aligned} \textstyle b=\frac{DS^\mathrm{in}+ra_1S_1^*}{D+ra_1}. \end{aligned}$$

Hence, b is a convex combination of $S^\mathrm{in}$ and $S_1^*$, and we have $S_1^*<b<S^\mathrm{in}$. Therefore, the vertical asymptote $S_2=b$ of h is at right of $S_1^*$. The derivative of h is

$$\begin{aligned} \textstyle h'(S_2)=\frac{D+(1-r)a_2}{1-r}\frac{S_1^*-b}{(b-S_2)^2}. \end{aligned}$$

Hence, we have $h'(S_2)<0$ for all $S_2<b$. Therefore, h is defined on the interval $(0,S_1^*)$ and is decreasing from h(0), given by (B12) to $h(S_1^*)=0$. $\square$

Therefore, if $D/r+a_1<f(S^\mathrm{in})$, equation $f(S_2)=h\left( S_2\right)$ admits a unique solution, denoted by $S_2^*(S^\mathrm{in},D,r)$, as shown in Fig. 12a. This solution satisfies the following property.

Lemma 9

Considering $a_1=a_2=a$, for all $0< D<f(S^\mathrm {in})-a$, one has

$$\begin{aligned} \lim _{r\rightarrow 1}S_2^*(S^\mathrm{in},D,r)=\lambda (D+a). \end{aligned}$$

Proof

Let $0< D<f(S^\mathrm {in})-a$. Using (5), the condition $h(S_2)=f(S_2)$ is equivalent to

$$\begin{aligned} \begin{array}{l} (D+(1-r)a)(S_1^*-S_2^*)=\\ \qquad (1-r)\left( \frac{D}{D+ra}(S^\mathrm{in}-S_1^*)+S_1^*-S_2^*\right) f(S_2^*). \end{array} \end{aligned}$$

(B13)

For $r=1$, we have $S_1^*=\lambda (D+a)$. As $\lim _{r\rightarrow 1}f(S_2^*)<+\infty$ then, (B13) gives

$$\begin{aligned} D(\lambda (D+a)-\lim _{r\rightarrow 1}S_2^*(S^\mathrm{in},D,r))=0. \end{aligned}$$

Consequently, since D > 0, one has $\lim _{r\rightarrow 1}S_2^*(S^\mathrm{in},D,r))=\lambda (D+a)$. $\square$

The existence and stability of steady states of (B10) are given by the following result.

Theorem 3

Assume that Assumption 1 is satisfied. The steady states of (B10) are:

• The washout steady state $E_0=(S^\mathrm{in}, 0,S^\mathrm{in},0)$ which always exists. It is GAS if and only if

  $$\begin{aligned} D\ge \max \{r(f(S^\mathrm{in})-a_1),(1-r)(f(S^\mathrm{in})-a_2)\}. \end{aligned}$$

  (B14)

  It is LES if and only if

  $$\begin{aligned} D>\max \{r(f(S^\mathrm{in})-a_1),(1-r)(f(S^\mathrm{in})-a_2)\}. \end{aligned}$$

• The steady state $E_1=(S^\mathrm{in},0,\overline{S}_2,\overline{x}_2)$ of washout in the first chemostat but not in the second one with

  $$\begin{aligned} \textstyle \overline{S}_2=\lambda \left( \frac{D}{1-r}+a_2\right) ,~\overline{x}_2=\frac{D}{D+(1-r)a_2}\left( S^\mathrm{in}-\overline{S}_2\right) . \end{aligned}$$

  (B15)

  It exists if and only if $D<(1-r)(f(S^\mathrm{in})-a_2)$. It is GAS if and only if

  $$\begin{aligned} r(f(S^\mathrm{in})-a_1)\le D \text{ and } D<(1-r)(f(S^\mathrm{in})-a_2). \end{aligned}$$

  (B16)

  It is LES if and only if

  $$\begin{aligned} r(f(S^\mathrm{in})-a_1)< D<(1-r)(f(S^\mathrm{in})-a_2). \end{aligned}$$

• The steady state $E_{2}=(S_{1}^{*},x_{1}^{*},S_{2}^{*},x_{2}^{*})$ of persistence of the species in both chemostats with

  $$\begin{aligned}&\textstyle S^{*}_{1}=\lambda \left( \frac{D}{r}+a_1\right) \text{, } \quad x^{*}_{1}= \frac{D}{D+ra_1}(S^\mathrm{in}-S_1^*) \text{, } \end{aligned}$$

  (B17)
  $$\begin{aligned}&\textstyle x^{*}_{2}=\frac{D}{D+(1-r)a_2}\left( \frac{D}{D+ra_1}(S^\mathrm{in}-S_1^*)+S_1^*-S_2^*\right) \end{aligned}$$

  (B18)

  and $S^{*}_{2}=S_2^ {*}(S^\mathrm{in},D,r)$ is the unique solution of the equation $h(S_2)=f(S_2)$ with h defined by (B11). This steady state exists and is positive if and only if $D<r(f(S^\mathrm{in})-a_1)$. It is GAS and LES whenever it exists and is positive.

Proof

The 4-dimensional system of ODEs (B10) has a cascade structure of two planar systems of ODEs, whose mathematical analysis is easy and well known in the mathematical theory of the chemostat (Harmand et al. 2017; Smith and Waltman 1995). Using this cascade structure, the global behavior of the system is deduced from the global behavior of planar systems and Thieme’s theory of asymptotically autonomous systems, all the solutions being bouded.

Fig. 12

a Existence and uniqueness of the solution $S_2^*$ of equation $f(S_2)=h(S_2)$. b Graphical illustration of Proposition 10: $S_2^*$ decreases when $S^\mathrm{in}$ increases (color figure online)

For the convenience of the reader the details of the proof are given in Appendix D.1. $\square$

Proposition 10

The function $S^\mathrm{in}\mapsto S_2^{*}(S^\mathrm{in},D,r)$ is decreasing.

Proof

D and r are fixed. Let $S^\mathrm{in,1}>S^\mathrm{in,2}$ and $h_i$ defined by (B11), with $S^\mathrm{in}=S^\mathrm{in,i}$, $i=1,2$. Let $S_2^{*i}$, the solution of equation $f(S_2)=h_i(S_2)$, $i=1,2$. Using Lemma 8, the graph of $h_i$ is a decreasing hyperbola from $h_i(0)$ defined by (B12), with $S^\mathrm{in}=S^\mathrm{in,i}$, to $h_i(S_1^*)=0$. Since $h_1(0)<h_2(0)$, we have $h_1(S_2)<h_2(S_2)$ for all $S_2\in (0,S_1^*)$. Therefore, $S_2^{*1}<S_2^{*2}$, see Fig. 12b. $\square$

This result means that the effluent steady-state concentration of substrate decreases when the influent concentration of substrate increases. This behavior is quite different from the single chemostat, where the effluent steady-state substrate concentration is independent of the influent substrate concentration.

Appendix C Operating Diagram

For the chemostat model, the operating diagram has as coordinates the input substrate concentration $S^\mathrm{in}$ and the dilution rate D, and shows how the solutions of the system behave for different values of these two parameters. The regions constituting the operating diagram correspond to different qualitative asymptotic behaviors. Indeed, the main interest of an operating diagram is to highlight the number and stability of the steady states for a given pair of parameters $(S^\mathrm{in},D)$. The input substrate concentration $S^\mathrm{in}$ and the dilution rate D are the usual parameters manipulated by the experimenter of a chemostat. Apart from these parameters, and the parameter r that can be also chosen by the experimenter but not easily changed as $S^\mathrm{in}$ and D, all other parameters have biological meaning and are fitted using experimental data from real measurements of concentrations of micro-organisms and substrates. Therefore the operating diagram is a bifurcation diagram, quite useful to understand the possible behaviors of the solutions of the system from both the mathematical and biological points of view.

Here, we fix $r \in (0,1)$ and we depict in the plane $(S^\mathrm{in},D)$ the regions in which the solution of system (B10) globally converges toward one of the steady state $E_0$, $E_1$ or $E_2$. From the results given in Theorem 3, it is seen that these regions are delimited by the curves $\Phi _r^1$ and $\Phi _{1-r}^2$ defined by:

$$\begin{aligned}&\Phi _r^1:=\left\{ (S^\mathrm{in},D) \in {\mathbb {R}}_+^2:D=r(f(S^\mathrm{in})-a_1)\right\} , \end{aligned}$$

(C19)

$$\begin{aligned}&\Phi _{1-r}^2:=\left\{ (S^\mathrm{in},D)\in {\mathbb {R}}_+^2:D=(1-r)(f(S^\mathrm{in})-a_2)\right\} . \end{aligned}$$

(C20)

Fig. 13

The operating diagram of (B10). The asymptotic behavior in each region is depicted in Table 3 (color figure online)

When $a_1=a_2=0$, as we have shown in Dali-Youcef et al. (2020), these curves croos only at one point (the origin) and merge when $r=1/2$. Therefore, in this case the curves $\Phi _r^1$ and $\Phi _{1-r}^2$ separate the operating plane $(S^\mathrm{in},D)$, in only three regions, see (Dali-Youcef et al. 2020, Figure 5). This property continues to hold when $a_1=a_2$, that is to say, the curves intersect only at $(\lambda (a_1),0)$ and merge when $r=1/2$. In this case the curves $\Phi _r^1$ and $\Phi _{1-r}^2$ separate the operating plane $(S^\mathrm{in},D)$, in only three regions, see Fig. 13c, d. The novelty when $a_1$ and $a_2$ are different and non-null, is that the intersection of the curves $\Phi _r^1$ and $\Phi _{1-r}^2$ can lie outside the $S^\mathrm{in}$ axis. Therefore there can be four regions in the operating plane, as depicted in Fig. 13a, f. For the description of the intersection of the curves $\Phi _r^1$ and $\Phi _{1-r}^2$, we need some definitions and notations. Let $\overline{r} \in (0,1)$ be defined by

$$\begin{aligned} \textstyle \overline{r}:=\frac{m-a_2}{2m-a_1-a_2}. \end{aligned}$$

(C21)

Note that if $a_1<a_2$, then $\overline{r}<1/2$, and if $a_1>a_2$, then $\overline{r}>1/2$. For $a_1<a_2$ and $0<r<\overline{r}$ (or $a_1>a_2$ and $\overline{r}<r<1$), we define the point $P=\left( S^\mathrm{in}_P,D_P\right)$ of the operating plane by:

$$\begin{aligned} \textstyle S^\mathrm{in}_P:=\lambda \left( \frac{ra_1-(1-r)a_2}{2r-1}\right) , \quad D_P:=\frac{r(1-r)(a_2-a_1)}{1-2r}. \end{aligned}$$

(C22)

Note that $S^\mathrm{in}_P>0$ and $D_P>0$. With these notations we can state the following result:

Proposition 11

• 1. If $a_1<a_2$, then for all $r\in (0,\overline{r})$, the curves $\Phi _r^1$ and $\Phi _{1-r}^2$ intersect at the point P and $\Phi _r^1$ is strictly below [resp. above] $\Phi _{1-r}^2$ for $S^\mathrm{in}>S^\mathrm{in}_P$ [resp. $S^\mathrm{in}<S^\mathrm{in}_P$], see Fig. 13a. For all $r\in (\overline{r},1)$, $\Phi _r^1$ is strictly above $\Phi _{1-r}^2$, see Fig. 13b.

• 2. If $a_1>a_2$ then for all $r\in (\overline{r},1)$, the curves $\Phi _r^1$ and $\Phi _{1-r}^2$ intersect at the point P and $\Phi _r^1$ is strictly above [resp. below] $\Phi _{1-r}^2$ for $S^\mathrm{in}>S^\mathrm{in}_P$ [resp. $S^\mathrm{in}<S^\mathrm{in}_P$], see Fig. 13f. For all $r\in (0,\overline{r})$, $\Phi _r^1$ is below $\Phi _{1-r}^2$, see Fig. 13e.

• 3. If $a_1=a_2$ then, for $r=1/2$, $\Phi _{r}^1=\Phi _{1-r}^2$. Moreover, if $r<1/2$ then $\Phi _{r}^1$ is strictly below $\Phi _{1-r}^2$, see Fig. 13c and, if $r>1/2$, then $\Phi _{r}^1$ is strictly above $\Phi _{1-r}^2$, see Fig. 13d.

Proof

For $0<r<1$ and $S^\mathrm{in}>\lambda (a_i)$ we define the function $\varphi _i$, $i=1,2$, by

$$\begin{aligned} \begin{array}{l} \varphi _1(S^\mathrm{in},r)=r(f(S^\mathrm{in})-a_1), \\ \varphi _2(S^\mathrm{in},r)= (1-r)(f(S^\mathrm{in})-a_2). \end{array} \end{aligned}$$

(C23)

The curves $\Phi _{r}^1$ and $\Phi _{1-r}^2$, defined, respectively, by (C19) and (C20), intersect if and only if there exists $r\in (0,1)$ and $S^\mathrm{in}>\max \left( \lambda (a_1),\lambda (a_2)\right)$ such that $\varphi _1(S^\mathrm{in},r)=\varphi _2(S^\mathrm{in},r)$, that is to say

$$\begin{aligned} \textstyle f(S^\mathrm{in})=A(r), \quad \text{ with } \quad A(r):=\frac{ra_1-(1-r)a_2}{2r-1}. \end{aligned}$$

(C24)

This equation has a solution $S^\mathrm{in}>\max \left( \lambda (a_1),\lambda (a_2)\right)$ if and only if

$$\begin{aligned} \max \left( a_1,a_2\right)< A(r)<m, \end{aligned}$$

(C25)

where $m=\sup (f)$, as in (2). When these conditions are satisfied, the solution of (C24) is given by $S^\mathrm{in}=\lambda (A(r))$, where $\lambda$ is the inverse function of f, i.e., the break-even concentration defined by (3). Hence, $S^\mathrm{in}=S^\mathrm{in}_P$, given in (C22). The corresponding intersection point of $\Phi _{r}^1$ and $\Phi _{1-r}^2$ is given by $D_P=r\left( f(S^\mathrm{in}_P)-a_1\right)$, which is the value given in (C22).

Let us determine now for which value of r, the conditions (C25) are satisfied. The function A is a homographic function. Its graphical representation is a hyperbola, whose vertical asymptote is $r=1/2$. Its derivative is given by

$$\begin{aligned} A'(r)=\frac{a_2-a_1}{(2r-1)^2}. \end{aligned}$$

(C26)

Note that $A(r)=m$ if and only if $r=\overline{r}$, where $\overline{r}$ is defined by (C21). Therefore if $a_1<a_2$, then, according to (C26), A is increasing. Since $A(0)=a_2$, $A(\overline{r})=m$, and $\overline{r}<1/2$, the condition (C25) is satisfied if and only if $0<r<\overline{r}$. Similarly, if $a_1>a_2$, then, according to (C26), A is decreasing. Since $A(1)=a_1$, $A(\overline{r})=m$ and $\overline{r}>1/2$, the condition (C25) is satisfied if and only if $\overline{r}<r<1$. Finally, if $a_1=a_2$ then $A(r)=a_1$ and the condition (C25) cannot be satisfied.

Suppose that $a_1<a_2$. Note that for $0<r<1/2$, the condition $f(S^\mathrm{in})>A(r)$ [resp. $f(S^\mathrm{in})<A(r)$] is equivalent to $\varphi _1(S^\mathrm{in},r)<\varphi _2(S^\mathrm{in},r)$ [resp. $\varphi _1(S^\mathrm{in},r)>\varphi _2(S^\mathrm{in},r)$]. Thus:

• If $r\in (0,\overline{r})$, then $f(S^\mathrm{in})<A(r)$ if and only if $S^\mathrm{in}<S^\mathrm{in}_P$, where $S^\mathrm{in}_P$ is defined by (C22). Hence, the curves $\Phi _r^{1}$ and $\Phi _{1-r}^2$ intersect at $P=(S^\mathrm{in}_P,D_P)$ and the curve $\Phi _r^{1}$ is strictly below [resp. above] the curve $\Phi _{1-r}^{2}$, for all $S^\mathrm{in}>S^\mathrm{in}_P$ [resp. $S^\mathrm{in}<S^\mathrm{in}_P$].

• If $r\in [\overline{r},1/2)$, then $f(S^\mathrm{in})<A(r)$ for all $S^\mathrm{in}>0$, so that the curve $\Phi _r^{1}$ is strictly above the curve $\Phi _{1-r}^{2}$.

• If $r\in [1/2,1)$, then, using $r\ge 1-r$ and $a_1<a_2$, one has $\varphi _1(S^\mathrm{in},r)>\varphi _2(S^\mathrm{in},r)$. Therefore, the curve $\Phi _r^{1}$ is strictly above the curve $\Phi _{1-r}^{2}$.

If $a_1>a_2$, the proof is similar to the case $a_1<a_2$.

If $a_1=a_2$, then $\varphi _1(S^\mathrm{in},r)=\varphi _2(S^\mathrm{in},r)$ is equivalent to $r(f(S^\mathrm{in})-a_1)=(1-r)(f(S^\mathrm{in})-a_1)$. Therefore, $r=1-r$, that is $r=1/2$. In this case the curves $\Phi _{r}^1$ and $\Phi _{1-r}^2$ merge. In addition, if $r<1/2$ [resp. $r>1/2$], then $r<1-r$ [resp. $r>1-r$] and the curve $\Phi _r^1$ is strictly below [resp. above] the curve $\Phi _{1-r}^2$. This ends the proof of the proposition. $\square$

For any $r\in (0,1)$, the curves $\Phi _r^1$ and $\Phi _{1-r}^2$, defined by (C19) and (C20), respectively, split the plane $(S^\mathrm{in},D)$ in the regions denoted $I_0(r)$, $I_1(r)$, $I_2(r)$ and $I_3(r)$ defined in Table 3. These regions are depicted in Fig. 13 in the cases $a_1<a_2$, $a_1=a_2$ and $a_1>a_2$.

Table 3 The regions $I_k(r)$, $k=0,1,2,3$ of the operating diagram of (B10) and asymptotic behavior in these various regions

The behavior of the system in each region, when it is not empty, is given in Table 3. Notice that $E_1$ exists in both regions $I_1(r)$ and $I_2(r)$, but is stable only when $(S^\mathrm{in},D)$ is in $I_1(r)$.

When $a_1=a_2=0$, $\lambda (a_1)=\lambda (a_2)=0$ and the curves $\Phi _{r}^1$ and $,\Phi _{1-r}^2$ of the operating diagram start from the origin of the plane $(S^\mathrm{in},D)$ and merge for $r=1/2$. Therefore, the diagrams shown in panels (a), (b), (c), (d), (e) and (f) of Fig. 13 are reduced to only two different cases characterized by $0<r<1/2$ and $1/2<r<1$, as shown in Figure 5 of Dali-Youcef et al. (2020). There is no change in the stability of the steady states and in the number of the regions depicted in the operating diagram.

This result reveals an interplay between spatial heterogeneity (the ratio r of volume distribution between tanks) and the mortality heterogeneity (difference between $a_1$ and $a_2$). Indeed, panels (a) and (f) of Fig. 13 bring a particular feature when mortality rates are different: domains $I_1(r)$ and $I_3(r)$ can appear or disappear playing only with the spatial distribution r, a phenomenon which does not happens when mortality is identical in each tank. This shows that the existence of domains $I_1(r)$ and $I_3(r)$ is controlled by a relative toxicity in the tanks, and not only the spatial distribution as it is the case for identical mortality. This feature can have interest when practitioners can adjust pH or other abiotic parameters having impacts on the mortality rate, independently in each tank. Given operating parameters $S^\mathrm{in}$, D and r, panels (a) and (f) of Fig. 13 show that it is theoretically possible to pass from domain $I_3(r)$ to $I_2(r)$ when mortality parameter is diminished only in the second tank. In practice, being in domain $I_2(r)$ might be more desirable than $I_3(r)$ with respect to some dysfunctioning of the first tank that can drop suddenly its biomass to zero. Indeed, in $I_2(r)$, the second tank is no conducted to the wash-out differently to the $I_3(r)$ case.

When $a_1=a_2=a$, which is the case corresponding to the system (1) considered in Section 2, only panels (c,d) of Fig. 13 are encountered, as shown in Fig. 2. We describe hereafter the bifurcations that occur in this particular case. The general case, i.e., when $a_1\ne a_2$ is similar.

Remark 4

Transcritical bifurcations occur in the limit cases $D=r(f(S^\mathrm{in})-a)$ and $D=(1-r)(f(S^\mathrm{in})-a)$, for system (1). If $0<r<1/2$ then, we have a transcritical bifurcation of $E_0$ and $E_1$ when $D=(1-r)(f(S^\mathrm{in})-a)$ and a transcritical bifurcation of $E_1$ and $E_2$ when $D=r(f(S^\mathrm{in})-a)$. If $1/2<r<1$ then, we have a transcritical bifurcation of $E_0$ and $E_1$ when $D=(1-r)(f(S^\mathrm{in})-a)$ and a transcritical bifurcation of $E_0$ and $E_2$ when $D=r(f(S^\mathrm{in})-a)$. If $r=1/2$ and $D=(f(S^\mathrm{in})-a)/2$ then, we have transcritical bifurcations of $E_0$ and $E_1$, and $E_0$ and $E_2$, simultaneously.

Appendix D Proofs

1.1 D.1 Proof of Theorem 3

We begin by the existence of steady states. The steady states are the solutions of the set of equations ${\dot{S}}_1=0$, ${\dot{x}}_1=0$, ${\dot{S}}_2=0$, ${\dot{x}}_2=0$. From equation ${\dot{x}}_1=0$, it is deduced that $x_1=0$ or $f(S_1)=D/r+a_1$. Suppose first that $x_1=0$. Then, from equation ${\dot{S}}_1=0$ it is deduced that $S_1=S^\mathrm{in}$ and from equation ${\dot{x}}_2=0$ it is deduced that $x_2=0$ or $f(S_2)=D/(1-r)+a_2$. If $x_2=0$, then from equation $S_2=0$ it is deduced that $S_2=S^\mathrm{in}$. Hence we obtain the steady state $E_0=(S^\mathrm{in},0,S^\mathrm{in},0)$, which always exist. On the other hand, if $f(S_2)=D/(1-r)+a_2$, then $S_2=\overline{S}_2$, defined in (B15). From equation ${\dot{S}}_2=0$, it is deduced that $x_2=\overline{x}_2$, defined in (B15). Hence we obtain the steady state $E_1=(S^\mathrm{in},0,\overline{S}_2,\overline{x}_2)$. This steady state exists if and only if $S^\mathrm{in}>\overline{S}_2$, that is $D<(1-r)(f(S^\mathrm{in})-a_2)$.

Suppose now that $f(S_1)=D/r+a_1$. Then $S_1=S_1^*$, defined in (B17). From equation ${\dot{S}}_1=0$, it is deduced that $x_1= x_1^*$, defined in (B17). From equation ${\dot{S}}_2+{\dot{x}}_2=0$, it is deduced that

$$\begin{aligned} x_2=\frac{D}{D+(1-r)a_2}(S_1^*+x_1^*-S_2). \end{aligned}$$

(D27)

Replacing $x_2$ by this expression in the equation ${\dot{S}}_2=0$, it is deduced that $f(S_2)=h(S_2)$, where h is defined by (B11). Hence $S_2=S_2^*$, which is the unique solution of the equation $f(S_2)=h(S_2)$, as shown in Fig. 12a. Replacing $S_2$ by $S_2^*$ in (D27) gives $x_2=x_2^*$, defined by (B18). Consequently, we obtain the steady state $E_2=(S_1^*,x_1^*,S_2^*,x_2^*)$. This steady state is positive if and only if $S^\mathrm{in}>S_1^*$, which is equivalent to $D<r(f(S^\mathrm{in})-a_1)$.

Let us now study the local stability. Since the system has a cascade structure, the stability analysis reduces to the study of square $2\times 2$ matrices. Indeed, the Jacobian matrix associated to system (B10) is the lower triangular matrix by blocs, $J= \begin{pmatrix} A &{} 0\\ B &{} C\\ \end{pmatrix}$ where B is the diagonal matrix whose diagonal elements are ${D}/{(1-r)}$, and A and C are given by:

$$\begin{aligned} \begin{array}{l} A= \begin{pmatrix} -\frac{D}{r}-f'(S_{1})x_{1} &{} -f(S_{1}) \\ f'(S_{1})x_{1} &{} -\frac{D}{r}+f(S_{1})-a_1 \end{pmatrix}, \\ C= \begin{pmatrix} -\frac{D}{1-r}-f'(S_{2})x_{2} &{} -f(S_{2})\\ f'(S_{2})x_{2} &{} -\frac{D}{1-r}+f(S_{2})-a_2 \end{pmatrix}, \end{array} \end{aligned}$$

Hence, the eigenvalues of J are the ones of A and C.

For $E_0$, the eigenvalues are $-D/r$, $-D/r+f(S^\mathrm{in})-a_1$, $-D/(1-r)$ and $-D/(1-r)+f(S^\mathrm{in})-a_2$. They are negative if and only if $D>r(f(S^\mathrm{in})-a_1)$ and $D>(1-r)(f(S^\mathrm{in})-a_2)$. Therefore, $E_0$ is LES if and only if the condition in the theorem is satisfied.

For $E_1$, the eigenvalues of A are $-D/r+f(S^\mathrm{in})-a_1$ and $-D/r$. The first eigenvalue is negative if and only if $D>r(f(S^\mathrm{in})-a_1)$. On the other hand, since the determinant of C is positive, and its trace is negative, the eigenvalues of C have negative real parts. Therefore, $E_1$ is LES if and only if the condition in the theorem is satisfied.

For $E_2$, the determinant of A is positive and its trace is negative. On the other hand, using the notation $C_{E_2}$ for the matrix C evaluated at $E_2$, we have

$$\begin{aligned} \begin{array}{rl} \det (C_{E_2})&{}= \left( -\frac{D}{1-r}-f'(S^{*}_{2})x^{*}_{2}\right) \left( -\frac{D}{1-r}-a_2+f(S^{*}_{2})\right) +f(S_2^*)f'(S^{*}_{2})x_{2}^{*},\\ {{\,\mathrm{tr}\,}}(C_{E_2})&{}=-2\frac{D}{1-r}-a_2-f'(S^{*}_{2})x^{*}_{2}+f(S^{*}_{2}). \end{array} \end{aligned}$$

Note that $h(S_2)<D/(1-r)+a_2$ for all $S_2 \in (0,S_1^*)$. Therefore, from (B11), we have $f(S_2^*)=h(S_2^*)<D/(1-r)+a_2$. Consequently, $\det (C_{E_2})$ and ${{\,\mathrm{tr}\,}}(C_{E_2})$ are, respectively, positive and negative. Therefore, $E_2$ is LES whenever it exists, that is $D<r(f(S^\mathrm{in})-a_1)$.

For the study of the global stability we use the cascade structure of the system (B10) and Thieme’s theorem (see Theorem A1.9 of Harmand et al. (2017)). In the rest of the proof, we denote by $(S_1(t),x_1(t),S_2(t),x_2(t))$ the solution of (B10) with the initial condition $(S_1^0,x_1^0,S_2^0,x_2^0)$.

Then, $(S_1(t),x_1(t))$ is the solution of system

$$\begin{aligned} \begin{array}{lll} \dot{S}_{1} &{}{} = &{}{} \frac{D}{r}(S^\mathrm {in}-S_1)-f(S_1)x_1,\\ \dot{x}_{1} &{}{} = &{}{} -\frac{D}{r}x_{1}+f\left( S_1\right) x_{1}-a_1x_1, \end{array} \end{aligned}$$

(D28)

with initial condition $(S_1^0,x_1^0)$ and $(S_2(t),x_2(t))$ is the solution of the non-autonomous system of differential equations

$$\begin{aligned} \begin{array}{lll} \dot{S}_{2} &{}{} = &{}{} \frac{D}{1-r}\left( S_1(t)-S_2\right) -f\left( S_2\right) x_{2},\\ \dot{x}_{2} &{}{} = &{}{} \frac{D}{1-r}(x_{1}(t)-x_{2})+f\left( S_2\right) x_{2}-a_2x_2 ,\end{array} \end{aligned}$$

(D29)

with the initial condition $(S_2^0,x_2^0)$. The system (D28) is the classical model of a single chemostat. Its asymptotic behavior is well known (see, for instance, Proposition 2.2 of Harmand et al. (2017)). This system admits the steady states:

$$\begin{aligned} e_0^1=\left( S^\mathrm {in},0\right) \quad \text { and }\quad e_1^1=\left( S_1^*,x_1^*\right) , \end{aligned}$$

(D30)

where $S_1^*$ and $x_1^*$ are defined by (B17). Two cases must be distinguished.

Firstly, if $\lambda \left( D/r+a_1\right) \ge S^\mathrm{in}$, that is $D\ge r(f(S^\mathrm{in})-a_1)$ then, $e_0^1$, defined in (D30), is GAS for (D28) in the non-negative quadrant. Hence, for any non-negative initial condition $(S_1^0,x_1^0)$,

$$\begin{aligned} \lim _{t\rightarrow +\infty }(S_1(t),x_1(t))=(S^\mathrm{in},0). \end{aligned}$$

(D31)

Therefore, the system (D29) is asymptotically autonomous with the limiting system

$$\begin{aligned} \begin{array}{lll} \dot{S}_{2} &{}{} = &{}{} \frac{D}{1-r}(S^\mathrm {in}-S_2) -f\left( S_2\right) x_{2},\\ \dot{x}_{2} &{}{} = &{}{} -\frac{D}{1-r}x_{2} +f\left( S_2\right) x_{2}-a_2x_2. \end{array} \end{aligned}$$

(D32)

Recall that the solutions of (D29) are positively bounded. Therefore, we shall use Thieme’s results which apply for bounded solutions.

The system (D32) represents the classical model of a single chemostat. It admits the two steady states $e_0^2=(S^\mathrm{in},0)$ and $e_1^2=(\overline{S}_2,\overline{x}_2)$, with $(\overline{S}_2,\overline{x}_2)$ defined by (B15). Two subcases must be distinguished.

• If $\lambda \left( D/(1-r)+a_2\right) \ge S^\mathrm{in}$, that is $D\ge (1-r)(f(S^\mathrm{in})-a_2)$, then $e_0^2$ is GAS in the non-negative quadrant. Using Thieme’s theorem, we deduce that for any non-negative $(S_2^0,x_2^0)$, the solution $(S_2(t),x_2(t))$ of (D29) converges toward $e_0^2=(S^\mathrm{in},0)$. Using (D31) we deduce that when $D\ge \max (r(f(S^\mathrm{in})-a_1),(1-r)(f(S^\mathrm{in})-a_2))$, the solution $(S_1(t),x_1(t),S_2(t),x_2(t))$ of (B10) converges toward $E_0=(S^\mathrm{in},0,S^\mathrm{in},0)$, which proves (B14).

• In contrast, if $\lambda (D/(1-r)+a_2)< S^\mathrm{in}$, that is $D<(1-r)(f(S^\mathrm{in})-a_2)$ then, both steady states $e_0^2$ and $e_1^2$ exist and $e_1^2$ is GAS in the positive quadrant. Although system (D29) has the saddle point $e_0^2$, no polycycle can exist. Using Thieme’s theorem, for any positive $(S_2^0,x_2^0)$, the solution $(S_2(t),x_2(t))$ of (D29) converges toward $e_1^2=(\overline{S}_2,\overline{x}_2)$. Using (D31) we deduce that, if $r(f(S^\mathrm{in})-a_1)\le D$ and $D<(1-r)(f(S^\mathrm{in})-a_2)$, then the solution $(S_1(t),x_1(t),S_2(t),x_2(t))$ of (B10) converges toward $E_1=\left( S^\mathrm{in},0,\overline{S}_2,\overline{x}_2\right)$, which proves (B16).

Secondly, if $\lambda \left( D/r+a_1\right) <S^\mathrm{in}$, that is $D<r(f(S^\mathrm{in})-a_1)$, then $e_1^1$, defined in (D30), is GAS for (D28) in the positive quadrant. Hence, for any positive initial condition $(S_1^0,x_1^0)$

$$\begin{aligned} \lim _{t\rightarrow +\infty }(S_1(t),x_1(t))=\left( S_1^*,x_1^*\right) . \end{aligned}$$

(D33)

Therefore, the system (D29) is asymptotically autonomous with the limiting system

$$\begin{aligned} \begin{array}{lll} \dot{S}_{2} &{}{} = &{}{} \frac{D}{1-r}(S_1^{*}-S_2) -f\left( S_2\right) x_{2},\\ \dot{x}_{2} &{}{} = &{}{} \frac{D}{1-r}(x_{1}^{*}-x_{2})+f\left( S_2\right) x_{2}-a_2x_2. \end{array} \end{aligned}$$

(D34)

The system (D34) represents the classical model of a single chemostat with an input biomass. In this case, there is no washout and the system (D34) always admits one LES steady state $e_2=(S^*_2,x_2^*)$ with positive biomass defined by (B18) and $S_2^*$ the unique solution of $h(S_2)=f(S_2)$.

Let us show that this steady state is GAS for (D34). Assume that $x_2>0$. Consider the change of variable $\xi =\ln (x_2)$. The system (D34) becomes as

$$\begin{aligned} \begin{array}{lll} \dot{S}_{2} &{}{} = &{}{} \frac{D}{1-r}(S_1^{*}-S_2) -f\left( S_2\right) e^\xi , \\ {\dot{\xi }} &{}{} = &{}{} \frac{D}{1-r}(x_1^*e^{-\xi }-1)+f\left( S_2\right) -a_2. \end{array} \end{aligned}$$

(D35)

The divergence of the vector field

$$\begin{aligned} \psi (S_2, \xi )=\left[ \begin{array}{lll} \frac{D}{1-r}(S_1^{*}-S_2) -f\left( S_2\right) e^\xi \\ \frac{D}{1-r}(x_1^*e^{-\xi }-1)+f\left( S_2\right) -a_2 \end{array} \right] \end{aligned}$$

associated to (D35) is $\mathrm {div}\psi (S_2, \xi )=-\frac{D}{1-r}(1+x_1^*e^\xi )-f'(S_2)e^\xi$. It is negative. Thus, using Bendixson–Dulac criterion, system (D35) cannot have a periodic solution. Hence, system (D34) has no cycle in the positive quadrant. For any non-negative initial condition $(S_2^0,x_2^0)$, the solution of (D34) is bounded. Hence, the $\omega$-limit set of $(S_2^0,x_2^0)$, denoted $\omega (S_2^0,x_2^0)$, is non-empty and included in the positive quadrant. If $e_2\not \in \omega (S_2^0,x_2^0)$ then, using Poincaré–Bendixson theorem, $\omega (S_2^0,x_2^0)$ is a limit cycle, but the system does not present any, due to the divergence property. One then deduces $e_2\in \omega (S_2^0,x_2^0)$ and, as $e_2$ is LES, then $\omega (S_2^0,x_2^0)=\{e_2\}$. Consequently, $e_2$ is GAS for (D34) in the positive quadrant.

Using again Thieme’s theorem, for any positive $(S_2^0,x_2^0)$, the solution $(S_2(t),x_2(t))$ of (D29) converges toward $e_2=(S^*_2,x_2^*)$. Using (D33) we deduce that, if $D< r(f(S^\mathrm{in})-a_1)$, then the solution $(S_1(t),x_1(t),S_2(t),x_2(t))$ of (B10) converges toward $E_2=(S^*_1,x_1^*,S^*_2,x_2^*)$. This ends the proof of the theorem.

1.2 D.2 Proof of Lemma 4

Let us fix $S^\mathrm{in}$ such that $\delta :=f(S^\mathrm{in})-a >0$. The proof consists in showing that the function $(D,r) \mapsto G_2(S^\mathrm{in},D,r)$ can be formally extended as a $C^2$ function for values of r larger than 1 (although such values have no physical meaning). Recall first that for any $D \in (0,\delta )$, one has $G_2(S^\mathrm{in},D,1)=G_\mathrm{chem}(S^\mathrm{in},D)$. As $G_2(S^\mathrm{in},{\overline{D}}(1),1)>0$ and $G_2(S^\mathrm{in},0,1)=0$, there exists by continuity of the function $G_2$, numbers ${\underline{D}} \in (0,{\overline{D}}(1))$, ${\underline{r}} \in (0,1)$ such that

$$\begin{aligned} G_2(S^\mathrm{in},D,r)<\max _{d \in (0,r\delta )} G_2(S^\mathrm{in},d,r), \; (D,r) \in [0,{\underline{D}}]\times [{\underline{r}},1]. \end{aligned}$$

(D36)

Let $\varepsilon >0$ be such that

$$\begin{aligned} D_\varepsilon := \epsilon \left( a+ \max _{s \in [0,S^\mathrm{in}]} f'(s)(S^\mathrm{in}-s) \right) < {\underline{D}} \end{aligned}$$

(D37)

and consider the domain

$$\begin{aligned} {{{\mathcal {D}}}}_\varepsilon :=\left\{ (D,r) ; \; D \in (D_\varepsilon ,\delta ), \; r \in \left( \max \left( {\underline{r}},\frac{D}{\delta }\right) ,1+\varepsilon \right) \right\} . \end{aligned}$$

Note that for any $(D,r) \in {{{\mathcal {D}}}}_\varepsilon$, the number $\lambda (D/r+a)=f^{-1}(D/r+a)$ is well defined. Posit the function

$$\begin{aligned}\textstyle \varphi (S_2,D,r)=&\left( D+(1-r)a\right) (\lambda (D/r+a)-S_2)\\ {}&\textstyle -(1-r)f(S_2)\left( \frac{DS^\mathrm {in}+ra\lambda (D/r+a)}{D+ra}-S_2\right) , \end{aligned}$$

where $(S_2,D,r) \in (0,S^\mathrm{in})\times {{{\mathcal {D}}}}_\varepsilon$. As f is $C^2$, $\varphi$ is $C^2$ on $(0,S^\mathrm{in})\times {{{\mathcal {D}}}}_\varepsilon$.

For $r<1$ and $(D,r) \in {{{\mathcal {D}}}}_\varepsilon$, one has

$$\begin{aligned}\textstyle \varphi (S_2,D,r)=(1-r)\left( \frac{DS^\mathrm{in}+ra\lambda (D/r+a)}{D+ra}-S_2\right) \left( h(S_2)-f(S_2)\right) \end{aligned}$$

where h is the function defined in (5). According to Lemma 8, h is positive decreasing on $(0,\lambda (D/r+a)$, and $h-f$ admits an unique zero $S_2^\star =S_2^\star (S^\mathrm{in},D,r)$ on $(0,\lambda (D/r+a)$. Then, one can write

$$\begin{aligned} \partial _{S_2}\varphi \Big \vert _{S_2=S_2^\star }= (1-r)\left( \frac{DS^\mathrm{in}+ra\lambda (D/r+a)}{D+ra}-S_2\right) \left( \partial _{S_2}h-f'\right) \Big \vert _{S_2=S_2^\star } <0. \end{aligned}$$

For $r \in [1,1+\varepsilon )$ and $(D,r)\in {{{\mathcal {D}}}}_\varepsilon$, on has

$$\begin{aligned} \partial _{S_2}\varphi =&-\left( D+(1-r)a\right) -(1-r)f'(S_2)\Big (\frac{DS^\mathrm{in}+ra\lambda (D/r+a)}{D+ra}-S_2\Big )\\&\qquad +(1-r)f(S_2), \end{aligned}$$

which is negative for any $S_2 \in (0,S^\mathrm{in})$ thanks to condition (D37). As $\varphi (0,D,r)>0$ and $\varphi (S^\mathrm{in},D,r)<0$, we deduce the existence of a unique $S_2^\star =S_2^\star (S^\mathrm{in},D,r)$ in $(0,S^\mathrm{in})$ such that $\varphi (S_2^\star ,D,r)=0$, which also verifies $\partial _{S_2}\varphi <0$ at $S_2=S_2^\star$.

Then, by the implicit function theorem, the function $(D,r) \mapsto S_2^\star (S^\mathrm{in},D,r)$ is $C^2$ on ${{{\mathcal {D}}}}_\varepsilon$. Recall that for $r<1$ and $D<r\delta$, on has the expression $G_2(S^\mathrm{in},D,r)=VD(S^\mathrm{in}-S_2^\star (S^\mathrm{in},D,r))$ (see Proposition 4). We extend now the function $(D,r) \mapsto G_2(S^\mathrm{in},D,r)$ with this last $C^2$ expression on ${{{\mathcal {D}}}}_\varepsilon$. As $G_2(S^\mathrm{in},D,1)=G_\mathrm{chem}(S^\mathrm{in},D)$ for any $D \in (0,\delta )$, one deduces, by continuity of the partial derivatives of $G_2$ with respect to D and property (D36), the existence of ${{{\mathcal {V}}}}_D$, ${{{\mathcal {V}}}}_r$ as neighborhoods, respectively, of ${\overline{D}}(1)$ and 1 with ${{{\mathcal {V}}}}_D\times {{{\mathcal {V}}}}_r \subset {{{\mathcal {D}}}}_\varepsilon$ such that for any $r \in {{{\mathcal {V}}}}_r$, the function $D \mapsto G_2(S^\mathrm{in},D,r)$ possesses the following properties

• 1. it is strictly concave on ${{{\mathcal {V}}}}_D$,

• 2. it is increasing on $(D_\varepsilon ,{\overline{D}}(1)) \setminus {{{\mathcal {V}}}}_D$ and decreasing on $({\overline{D}}(1),r\delta )\setminus {{{\mathcal {V}}}}_D$,

• 3. its maximum over $(0,r\delta )$ is not reached for $D\le D_\varepsilon$.

We thus deduce that $D \mapsto G_2(S^\mathrm{in},D,r)$ admits a unique maximum ${\overline{D}}(r)$ on $(0,r\delta )$, for any $r \in {{{\mathcal {V}}}}_r$.

Finally, for any $r \in {{{\mathcal {V}}}}_r$, ${\overline{D}}(r)$ is characterized as the zero of the map $D \mapsto F(D,r)$ where F is the $C^1$ function

$$\begin{aligned} F(D,r) := \partial _D G_2(S^\mathrm{in},D,r) \end{aligned}$$

From property 1. above, one obtains

$$\begin{aligned} \partial _D F({\overline{D}}(r),r)= \partial ^2_{DD} G_2(S^\mathrm{in},{\overline{D}}(r),r) < 0, \quad r \in {{{\mathcal {V}}}}_r \end{aligned}$$

and by the implicit function theorem, there exists a neighborhood ${{{\mathcal {V}}}}_1 \subset {{{\mathcal {V}}}}_r$ of 1 such that ${\overline{D}}$ is $C^1$ on ${{{\mathcal {V}}}}_1$, which ends the proof of the lemma.

1.3 D.3 Proof of Proposition 6

$S^\mathrm{in}$ being fixed, we shall drop the $S^\mathrm{in}$ dependency in the expressions of $S^*_i$, $x_i^*$ $(i=1,2)$ and $G_2$. Thus, let us define

$$\begin{aligned} \begin{array}{l} G(D,r):=G_2(S^\mathrm{in},D,r),\\ F_i(D,r):=f(S_i^*(D,r))x_i^*(D,r),\quad i=1,2, \end{array} \end{aligned}$$

as functions of $D\ge 0$ and $r\in {\mathcal {V}}_1 \cap \{ r < 1\}$. Remark from the expression of $F_1$, that it is well defined as well as its partial derivatives at $r=1$. In addition, for the limiting case $r=1$, using Lemma 9, for all $D\ge 0$, one has

$$\begin{aligned} \begin{array}{l} S_2^*(D,1)=S_1^*(D,1)=\lambda (D+a)\\ x_2^*(D,1)=x_1^*(D,1)=\frac{D}{D+a}(S^\mathrm{in}-\lambda (D+a)). \end{array} \end{aligned}$$

(D38)

Thus, for all $D\ge 0$, one has

$$\begin{aligned} F_1(D,1)=F_2(D,1), \end{aligned}$$

(D39)

and $F_2$ is also well defined for $r=1$. Thus, according to (37), for all $D\ge 0$ and $r\in {\mathcal {V}}_1\cap \{r\le 1\}$, one has

$$\begin{aligned} G(D,r)=r F_1(D,r)+(1-r)F_2(D,r), \end{aligned}$$

and from Lemma 4, for $r\in {\mathcal {V}}_1\cap \{r<1\}$, one has

$$\begin{aligned} {\overline{G}}(r)=G(\overline{D}(r),r), \end{aligned}$$

(D40)

with $\overline{G}$ defined by (42). For convenience, for a function E of (D, r) that is differentiable, we shall define the three following functions: $\overline{E}(r):=E({\overline{D}}(r),r)$ and

$$\begin{aligned} \partial _r E(r):= \frac{\partial E}{\partial r}({\overline{D}}(r),r), \quad \partial _D E(r):=\frac{\partial E}{\partial D}({\overline{D}}(r),r). \end{aligned}$$

Therefore, the function ${\overline{G}}$ writes

$$\begin{aligned} {\overline{G}}(r)=r{\overline{F}}_1(r)+(1-r){\overline{F}}_2(r), \text{ for } r\in {\mathcal {V}}_1\cap \{r<1\}. \end{aligned}$$

(D41)

As the functions $F_i, i=1,2,$ are differentiable and as ${\overline{D}}(r)$ is a maximizer of $D \mapsto rF_1(D,r)+(1-r)F_2(D,r)$ on the interior of the interval $[0,f(S^\mathrm{in})-a]$, one has

$$\begin{aligned} r\partial _D F_1(r)+(1-r)\partial _D F_2(r)=0, \text{ for } r\in {\mathcal {V}}_1\cap \{r<1\}, \end{aligned}$$

(D42)

and $\partial _D F_1(1)=0.$

As f is ${\mathcal {C}}^2$ and $\overline{D}$ is assumed to be differentiable on ${\mathcal {V}}_1\cap \{r<1\}$, ${\overline{G}}$ is differentiable and from (D41), for all $r\in {\mathcal {V}}_1\cap \{r<1\}$, one has

$$\begin{aligned} \begin{array}{lr} {\overline{G}}'(r)&{}=\overline{F}_1(r)-\overline{F}_2(r)+r\partial _r F_1(r)+(1-r)\partial _r F_2(r)\\ &{}+(r\partial _D F_1(r)+(1-r)\partial _D F_2(r)){\overline{D}}'(r), \end{array} \end{aligned}$$

and with (D42), for all $r\in {\mathcal {V}}_1\cap \{r<1\}$, one has simply

$$\begin{aligned} {\overline{G}}'(r)=\overline{F}_1(r)-\overline{F}_2(r)+r\partial _r F_1(r)+(1-r)\partial _r F_2(r). \end{aligned}$$

(D43)

Let us now determine the limits of the terms of the right side of this last equality when r tends to 1. Firstly, according to (D39), one has in particular

$$\begin{aligned} \overline{F}_1(1)=\overline{F}_2(1). \end{aligned}$$

(D44)

Secondly, remark that the dynamics of the first tank is parameterized by the single dilution rate $D_1=D/r$, the other parameters being fixed (see the expression (B17)). The function $F_1$ takes then the form $F_1(D,r)={\tilde{F}}_1\left( D/r\right)$ where ${\tilde{F}}_1$ is a smooth function. Therefore, one has

$$\begin{aligned} \partial _D F_1(r)=-\frac{r}{\overline{D}(r)}\partial _r F_1(r). \end{aligned}$$

(D45)

As $\partial _D F_1(1)=0$ then one deduces

$$\begin{aligned} \partial _r F_1(1)=0. \end{aligned}$$

(D46)

Finally, from ${\dot{S}}_2=0$, for all $r\in {\mathcal {V}}_1\cap \{r<1\}$, one gets

$$\begin{aligned} F_2(D,r)= \frac{D}{1-r}(S_1^*(D,r)-S_2^*(D,r)). \end{aligned}$$

(D47)

Differentiating (D47) with respect to r gives

$$\begin{aligned} \frac{\partial F_2}{\partial r}(D,r)=\frac{D}{1-r}\left( \frac{\partial S_1^*}{\partial r}(D,r)-\frac{\partial S_2^*}{\partial r}(D,r)\right) +\frac{D}{(1-r)^2}(S_1^*(D,r)-S_2^*(D,r)) \end{aligned}$$

which can be written equivalently as

$$\begin{aligned} (1-r)\frac{\partial F_2}{\partial r}(D,r)=D\left( \frac{\partial S_1^*}{\partial r}(D,r)-\frac{\partial S_2^*}{\partial r}(D,r)\right) +F_2(D,r). \end{aligned}$$

Thus, for $D=\overline{D}(r)$, one has

$$\begin{aligned} (1-r)\partial _r F_2(r)=\overline{D}(r)(\partial _r S_1^*(r)-\partial _r S_2^*(r))+\overline{F}_2(r). \end{aligned}$$

Notice that for $D=\overline{D}(r)$, (D47) gives

$$\begin{aligned} \overline{F}_2(r)=\frac{\overline{D}(r)}{1-r}(\overline{S}_1^*(r)-\overline{S}_2^*(r)), \quad \text{ for } \text{ all } r\in {\mathcal {V}}_1\cup \{r<1\}. \end{aligned}$$

(D48)

Using L’Hôpital’s rule in (D48) when r tends to 1, one gets

$$\begin{aligned} \overline{F}_2(1)=\lim _{r\rightarrow 1^-}\textstyle \frac{\overline{D}^\prime (r)(\overline{S}_1^*(r)-\overline{S}_2^*(r))+\overline{D}(r)(\partial _r S_1^*(r)-\partial _r S_2^*(r))}{-1} \end{aligned}$$

and using (D38) and (D44), one obtains

$$\begin{aligned} \overline{F}_1(1)=\lim _{r\rightarrow 1^-}-\overline{D}(r)(\partial _r S_1^*(r)-\partial _r S_2^*(r)). \end{aligned}$$

Consequently, one has

$$\begin{aligned} \lim _{r\rightarrow 1^-}(1-r)\partial _r F_2(r)=0. \end{aligned}$$

(D49)

With (D44), (D46) and (D49), expression (D43) gives the existence of the limit of ${\overline{G}}'$ when r tends to 1 with $r<1$, which is

$$\begin{aligned} \overline{G}^\prime (1^-)=0. \end{aligned}$$

(D50)

Note that $\overline{G}''(1^-)$ exists if and only if $\lim _{r\rightarrow 1^-} \frac{\overline{G}'(r)-\overline{G}'(1)}{r-1}$ exists. Using (D50) and (D43), one has

$$\begin{aligned} \begin{array}{l} \frac{\overline{G}'(r)-\overline{G}'(1^-)}{r-1} =-\frac{\overline{G}^\prime (r)}{1-r} =-\frac{\overline{F}_1(r)-\overline{F}_2(r)+r\partial _r F_1(r)+(1-r)\partial _r F_2(r)}{1-r} \end{array} \end{aligned}$$

(D51)

On the one hand, using L’Hôpital’s rule, one has

$$\begin{aligned} \lim _{r\rightarrow 1^-} \frac{\overline{F}_1(r)-\overline{F}_2(r)}{1-r}=\displaystyle \lim _{r\rightarrow 1^-} \frac{\overline{F}^\prime _1(r)-\overline{F}_2^\prime (r)}{-1}. \end{aligned}$$

Recall that $\partial _r F_1(1)=0$ and thus one has ${\overline{F}}_1'(1)=0$. Consequently, one has

$$\begin{aligned}&\displaystyle \lim _{r\rightarrow 1^-} \frac{\overline{F}_1(r)-\overline{F}_2(r)}{1-r}= \displaystyle \lim _{r\rightarrow 1^-}\overline{F}^\prime _2(r)\nonumber \\ {}&\quad =\displaystyle \lim _{r\rightarrow 1^-} \partial _rF_2(r)+\partial _DF_2(r) {\overline{D}}'(r). \end{aligned}$$

(D52)

On the other hand, using (D42) and (D45), one has

$$\begin{aligned} \frac{r}{1-r}\partial _rF_1(r)=\frac{\overline{D}(r)}{r}\partial _DF_2(r). \end{aligned}$$

(D53)

Thus, according to (D51), (D52) and (D53), one gets

$$\begin{aligned} \begin{array}{l} \displaystyle \lim _{r\rightarrow 1^-} \frac{\overline{G}'(r)-\overline{G}'(1^-)}{r-1}=\displaystyle \lim _{r\rightarrow 1^-}-2\partial _r F_2(r)-\partial _D F_2(r)\left( \bar{D}^\prime (r) +\frac{\bar{D}(r)}{r}\right) \end{array} \end{aligned}$$

(D54)

Let us show now that the limit of $\partial _DF_2(r)$ is 0 when r tends to 1. One has

$$\begin{aligned}\textstyle \frac{\partial F_2}{\partial D}=f'(S_2^*) \frac{\partial S_2^*}{\partial D}x_2^*+f(S_2^*)\frac{\partial x_2^*}{\partial D}. \end{aligned}$$

Let us use the expression $G(D,r)=D(S^\mathrm{in}-S_2^*(D,r))$ given by Proposition 4. As ${\overline{D}}(r)$ is a maximizer, then one has

$$\begin{aligned} \partial _DG(r)=S^\mathrm{in}-\overline{S}_2^*(r)-\overline{D}(r)\partial _DS_2^*(r)=0. \end{aligned}$$

Using (D38), one then deduces

$$\begin{aligned} \partial _DS_2^*(1^-)=\frac{S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) }{\overline{D}(1)}. \end{aligned}$$

In addition, using expressions (B18) and (D38), one gets

$$\begin{aligned} \partial _Dx_2^*(1^-)=-\frac{\overline{D}(1)}{\left( \overline{D}(1)+a\right) ^2}\left( S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) \right) , \end{aligned}$$

and hence the limit of $\partial _DF_2$ when r tends to 1 exists:

$$\begin{aligned} \partial _DF_2(1^-)=\frac{S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) }{\overline{D}(1)+a}f'\left( \lambda \left( \overline{D}(1)+a\right) \right) A, \end{aligned}$$

where $A=S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) -\frac{\overline{D}(1)}{f'\left( \lambda \left( \overline{D}(1)+a\right) \right) }$. Thus, one has

$$\begin{aligned} \partial _DF_2(1^-)=\frac{S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) }{\overline{D}(1)+a}f'\left( \lambda \left( \overline{D}(r)+a\right) \right) \left( S^\mathrm{in}-g\left( \overline{D}(1)\right) \right) , \end{aligned}$$

with g defined by (A8). According to Proposition 9, one has $S^\mathrm{in}-g\left( \overline{D}(1)\right) =0$. Consequently, one has $\partial _DF_2(1^-)=0$.

Finally, it remains to calculate the limit of $\partial _r F_2(r)$ when r tends to 1. One has

$$\begin{aligned} \frac{\partial F_2}{\partial r}=f'(S_2^*) \frac{\partial S_2^*}{\partial r}x_2^*+f(S_2^*)\frac{\partial x_2^*}{\partial r}. \end{aligned}$$

Let us use again the expression $G(D,r)=D(S^\mathrm{in}-S_2^*(D,r))$. According to (D41), one has

$$\begin{aligned} {\overline{G}}'(r)=\partial _r G(r)+\partial _D G(r){\overline{D}}'(r) \end{aligned}$$

where $\partial _DG(r)=0$. According to (D50), we have $\partial _r G(1^-)=0$, and thus $\partial _r S_2^*(1^-)=0.$ Using expression (B18), one gets

$$\begin{aligned} \partial _r x_2^*(1^-)=-a\overline{D}(1)\frac{S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) }{\left( \overline{D}(1)+a\right) ^2}, \end{aligned}$$

and then the limit of $\partial _r F_2$ when r tends to 1 exists:

$$\begin{aligned} \partial _r F_2(1^-)= -a\overline{D}(1)\frac{S^\mathrm{in}-\lambda \left( \overline{D}(1)+a\right) }{\overline{D}(1)+a}. \end{aligned}$$

As ${\overline{D}}'$ is assumed to be bounded on ${\mathcal {V}}_1\cup \{r<1\}$, we thus obtain from (D54) the existence of $\overline{G}''(1^-)$ with

$$\begin{aligned} \overline{G}''(1^-)=-2\partial _r F_2(1^-) \end{aligned}$$

which is given by expression (43).