Analysis and Optimization of the Chemostat Model with a Lateral Diffusive Compartment

Abstract

We consider the chemostat model with a side compartment connected by pure diffusion, and analyze its asymptotic properties. We investigate conditions under which this spatial structure is beneficial for species survival and conversion yield, compared to single chemostat. Under these conditions, we study the optimization problem for the best structure (volume distribution and diffusion rate), which minimizes the volume required to attain a desired conversion yield. The analysis reveals that particular configurations with a single tank connected by diffusion to the input stream can be the most efficient.

Appendix: Proofs of the Primary Results of Sect. 2

Proof of Lemma 2.2

Define $z_{i}=s_{\mathrm{in}}-s_{i}-x_{i}$ for each tank $i=1,2$ and consider the dynamics (1) in $(\mathbf {z},\mathbf {s})$ coordinates:

$$\begin{aligned} \dot{z}_{1}= & {} \displaystyle -\frac{Q}{V_{1}}z_{1}-\frac{d}{V_{1}}(z_{1}-z_{2}),\nonumber \\ \dot{s}_{1}= & {} \displaystyle -\mu (s_{1})(s_{\mathrm{in}}-s_{1}-z_{1}) + \frac{Q}{V_{1}}(s_{\mathrm{in}}-s_{1}) +\frac{d}{V_{1}}(s_{2}-s_{1}),\nonumber \\ \dot{z}_{2}= & {} \displaystyle -\frac{Q}{V_{2}}(z_{2}-z_{1}),\nonumber \\ \dot{s}_{2}= & {} \displaystyle -\mu (s_{2})(s_{\mathrm{in}}-s_{2}-z_{2}) +\frac{d}{V_{2}}(s_{1}-s_{2}). \end{aligned}$$

(18)

This system has a cascade structure with a first independent sub-system linear in $\mathbf {z}$

$$\begin{aligned} {\dot{\mathbf {z}}} = \underbrace{\left[ \begin{array}{cc} \displaystyle -\frac{Q+d}{V_{1}} &{} \displaystyle \frac{d}{V_{1}}\\ \displaystyle \frac{d}{V_{2}} &{} \displaystyle -\frac{d}{V_{2}} \end{array}\right] }_{\displaystyle \mathbf {M}} \mathbf {z}, \end{aligned}$$

(19)

where one has

$$\begin{aligned} \text{ tr }(\mathbf {M})=-\frac{Q+d}{V_{1}}-\frac{d}{V_{2}}<0 \quad \text{ and } \quad \text{ det }(\mathbf {M})=\frac{Qd}{V_{1}V_{2}}>0. \end{aligned}$$

Therefore, the matrix $\mathbf {M}$ is Hurwitz and any solution $\mathbf {z}$ of (19) converges exponentially to $\mathbf {0}$. Then, the solution $\mathbf {s}$ can be written as the solution of the non-autonomous dynamics

$$\begin{aligned} {\dot{\mathbf {s}}}=\mathbf {F}(t,\mathbf {s})=\left[ \begin{array}{c} \displaystyle \left( \frac{Q}{V_{1}}-\mu (s_{1})\right) (s_{\mathrm{in}}-s_{1}) +\frac{d}{V_{1}}(s_{2}-s_{1})+\mu (s_{1})z_{1}(t)\\ \displaystyle -\mu (s_{2})(s_\mathrm{in}-s_{2})+\frac{d}{V_{2}}(s_{1}-s_{2})+\mu (s_{2})z_{2}(t) \end{array}\right] . \end{aligned}$$

(20)

Notice that for any $(t,\mathbf {s})$, one has

$$\begin{aligned} \frac{\partial F_{1}(t,\mathbf {s})}{\partial s_{2}}=\frac{d}{V_{1}}> 0 \quad \text{ and } \quad \frac{\partial F_{2}(t,\mathbf {s})}{\partial s_{1}}=\frac{d}{V_{2}} > 0, \end{aligned}$$

and so the dynamics (20) is cooperative (see, e.g., [56]).

Define $\check{F}_{1}(t,\mathbf {s}):= -\frac{Q}{V_{1}}s_{1}-\mu (s_{1})(s_{\mathrm{in}}-s_{1}) +\frac{d}{V_{1}}(s_{2}-s_{1}) +\mu (s_{1})z_{1}(t)$, for which it follows that $F_{1}(t,\mathbf {s})>\check{F}_{1}(t,\mathbf {s})$ for any $(t,\mathbf {s})$. Proposition 2.1 in [56] allows to state that any solution of (20) with $s_{i}(0)\ge 0$ ($i=1,2$) satisfies $s_{i}(t)\ge \check{s}_{i}(t)$ ($i=1,2)$ for any $t>0$, where $\check{\mathbf {s}}$ is solution of the dynamics

$$\begin{aligned} {\dot{\check{\mathbf{s}}}}= {\check{\mathbf{F}}}(t,{\check{\mathbf{s}}})=\left[ \begin{array}{l} \check{F}_{1}(t,{\check{\mathbf{s}}})\\ F_{2}(t,{\check{\mathbf{s}}}) \end{array}\right] , \quad {\check{\mathbf{s}}}(0)=\mathbf {0} \end{aligned}$$

As one has ${\check{\mathbf{F}}}(t,\mathbf {0})=\mathbf {0}$ for any t, the solution ${\check{\mathbf{s}}}$ is identically null and one obtains that $s_{i}(t)$ ($i=1,2$) stays nonnegative for any positive t.

Similarly, $\mathbf {x}$ can be written as a solution of a non-autonomous cooperative dynamics

$$\begin{aligned} \dot{\mathbf {x}} = \mathbf {L}(t,\mathbf {x})=\left[ \begin{array}{c} \displaystyle \left( \mu (s_{1}(t))-\frac{Q+d}{V_{1}}\right) x_{1}+\frac{d}{V_{1}}x_{2}\\ \displaystyle \frac{d}{V_{2}}x_{1}+\left( \mu (s_{2}(t))-\frac{d}{V_{2}}\right) x_{2} \end{array}\right] \end{aligned}$$

with $\mathbf {L}(t,\mathbf {0})=\mathbf {0}$, which allows to conclude that $x_{i}(t)$ ($i=1,2$) stays nonnegative for any positive t.

Finally, the convergence of $\mathbf {z}$ to $\mathbf {0}$ provides the boundedness of the solutions $\mathbf {s}(t)$, $\mathbf {x}(t)$. $\square$

Proof of Proposition 2.1

From the two last equations of (1), one has $s_{1}+x_{1}=s_{2}+x_{2}$ at steady state, and from the two first ones $s_{1}+x_{1}=s_{\mathrm{in}}$. The values $s_{1}$, $s_{2}$ at steady state are then solutions of the system of two equations

$$\begin{aligned} 0= & {} \left( \frac{Q}{V_{1}}-\mu (s_{1})\right) (s_\mathrm{in}-s_{1}) +\frac{d}{V_{1}}(s_{2}-s_{1}) \end{aligned}$$

(21)

$$\begin{aligned} 0= & {} -\mu (s_{2})(s_\mathrm{in}-s_{2})+\frac{d}{V_{2}}(s_{1}-s_{2}) \end{aligned}$$

(22)

and $x_{1}$, $x_{2}$ at steady state are uniquely defined from each solution $(s_{1},s_{2})$ of (21)–(22). Clearly, $(s_{\mathrm{in}},s_{\mathrm{in}})$ is a solution of (21)–(22). We look for (positive) solutions different to $(s_{\mathrm{in}},s_\mathrm{in})$. Posit

$$\begin{aligned} \lambda _{1}(s_{\mathrm{in}}):=\max \left\{ s_{1} \in [0,s_{\mathrm{in}}] \text{: } \mu (s_{1})\le \frac{Q}{V_{1}} \right\} \end{aligned}$$

From Eqs. (21)–(22), a solution different to $(s_{\mathrm{in}},s_{\mathrm{in}})$ has to satisfy $s_{1}>s_{2}>0$ and then from Eq. (21), one has also $s_{1}<\lambda _{1}(s_\mathrm{in})$. Define then the functions:

$$\begin{aligned} \phi _{1}(s_{1}):= & {} s_{1}-\frac{Q-V_{1}\mu (s_{1})}{d}(s_\mathrm{in}-s_{1}) =s_{1}-\frac{Q}{d}(s_{\mathrm{in}}-s_{1})+\frac{V_{1}}{d}\beta (s_{1}),\\ \phi _{2}(s_{2}):= & {} s_{2}+\frac{V_{2}\mu (s_{2})}{d}(s_\mathrm{in}-s_{2}) =s_{2}+\frac{V_{2}}{d}\beta (s_{2}). \end{aligned}$$

so that any solution of (21)–(22) fulfills $s_{2}=\phi _{1}(s_{1})$ and $s_{1}=\phi _{2}(s_{2})$. One has

$$\begin{aligned} \phi _{1}'(s_{1})=1+\frac{V_{1}}{d}\mu '(s_{1})(s_\mathrm{in}-s_{1})+\frac{Q-V_{1}\mu (s_{1})}{d}. \end{aligned}$$

Therefore, $\phi _{1}$ is increasing on $[0,\lambda _{1}(s_\mathrm{in})]$, with $\phi _{1}(0)=-(Q/d)s_{\mathrm{in}}<0$ and $\phi _{1}(\lambda _{1}(s_{\mathrm{in}}))=\lambda _{1}(s_{\mathrm{in}})>0$. Thus, $\phi _{1}$ is invertible on $[-(Q/d)s_{\mathrm{in}},\lambda _{1}(s_\mathrm{in})]$ with

$$\begin{aligned} \phi _{1}^{-1}(0) \in ]0,\lambda _{1}(s_{\mathrm{in}})[. \end{aligned}$$

From Lemma 2.1, it follows that $\phi _{1}$ and $\phi _{2}$ are strictly concave functions on $[0,s_{\mathrm{in}}]$. Consider then the function

$$\begin{aligned} \gamma (s_{2})=\phi _{2}(s_{2})-\phi _{1}^{-1}(s_{2}) \quad s_{2} \in [0,s_{\mathrm{in}}], \end{aligned}$$

which is also strictly concave on $[0,s_{\mathrm{in}}]$. Then, a solution $(s_{1},s_{2})$ can be written as a solution of

$$\begin{aligned} \gamma (s_{2})=0 , \quad s_{1}=\phi _{2}(s_{2}) \quad \text{ with } s_{2} \in [0,\lambda _{1}(s_{\mathrm{in}})]. \end{aligned}$$

Notice that one has $\gamma (s_{\mathrm{in}})=0$, and as $\gamma$ is strictly concave, it cannot have more than two zeros. Therefore, there is at most one solution $(s_{1},s_{2})$ different to $(s_\mathrm{in},s_{\mathrm{in}})$. Furthermore, one has $\gamma (0)=-\phi _{1}^{-1}(0)<0$. Now, distinguish two different cases:

• When $\lambda _{1}(s_{\mathrm{in}})<s_{\mathrm{in}}$ (or equivalently $\mu (s_{\mathrm{in}})>Q/V_{1}$), one has

  $$\begin{aligned} \gamma (\lambda _{1}(s_{\mathrm{in}}))=\frac{QV_{2}}{dV_{1}}(s_\mathrm{in}-\lambda _{1}(s_{\mathrm{in}}))>0. \end{aligned}$$

  By using the mean value theorem, one concludes that there exists $s_{2} \in (0,\lambda _{1}(s_{\mathrm{in}}))$ such that $\gamma (s_{2})=0$.

• When $\lambda _{1}(s_{\mathrm{in}})=s_{\mathrm{in}}$ (that is when $\mu (s_\mathrm{in})\le Q/V_{1}$), the function $\gamma$ takes positive values on the interval $[0,s_{\mathrm{in}}]$ if and only if $\gamma '(s_{\mathrm{in}})<0$ ($\gamma$ being strictly concave on $[0,s_{\mathrm{in}}]$), or equivalently when the condition

  $$\begin{aligned} \phi _{2}'(s_{\mathrm{in}})<\frac{1}{\phi _{1}'(s_{\mathrm{in}})} \end{aligned}$$

  is fulfilled. Notice that one has $\phi _{1}'(s_{\mathrm{in}})>0$ because $\lambda _{1}(s_{\mathrm{in}})=s_{\mathrm{in}}$. So the condition can be also written as $\phi _{1}'(s_{\mathrm{in}})\phi _{2}'(s_{\mathrm{in}})<1$. From the expressions of $\phi _{1}$ and $\phi _{2}$, one can write this condition as

  $$\begin{aligned} \frac{(d+Q-V_{1}\mu (s_{\mathrm{in}}))(d-V_{2}\mu (s_{\mathrm{in}}))}{d^2}<1, \end{aligned}$$

  and check that this exactly amounts to require $s_{\mathrm{in}}$ to satisfy $P(\mu (s_{\mathrm{in}}))<0$.

We conclude that there exists a positive steady state if and only if $\mu (s_{\mathrm{in}})>Q/V_{1}$ or $P(\mu (s_{\mathrm{in}}))<0$ and that this steady state (when it exists) is unique.

Let us study now the stability of the steady states. Due to the cascade structure of the dynamics (1) that is made explicit in the proof of Lemma 2.2, the Jacobian matrix in the $(\mathbf {z},\mathbf {s})$ coordinates depends only on $\mathbf {s}$ and is equal to

$$\begin{aligned} \mathbf {J}(\mathbf {s})= & {} \left[ \begin{array}{ll} \mathbf {M} &{} \mathbf {0}\\ \mathbf {N}(\mathbf {s}) &{} \mathbf {J}_{a}(\mathbf {s}) \end{array}\right] \quad \text{ with } \quad \mathbf {J}_{a}(\mathbf {s})=\left[ \begin{array}{ll} \displaystyle -\frac{d}{V_{1}}\phi _{1}'(s_{1}) &{} \displaystyle \frac{d}{V_{1}}\\ \displaystyle \frac{d}{V_{2}} &{} \displaystyle -\frac{d}{V_{2}}\phi _{2}'(s_{2}) \end{array}\right] ,\\ \mathbf {N}(\mathbf {s})= & {} \left[ \begin{array}{ll} \mu (s_{1}) &{} 0\\ 0 &{} \mu (s_{2}) \end{array}\right] \end{aligned}$$

where the matrix $\mathbf {M}$ defined in (19) is Hurwitz. Accordingly to Proposition 2.1, the equilibrium $\mathbf {E}^\star \ne \mathbf {E}^0$ exists when $P(\mu (s_\mathrm{in}))>0$ or $\mu (s_{\mathrm{in}})>Q/V_{1}$.

• When $P(\mu (s_{\mathrm{in}}))>0$, one has $\phi _{1}'(s_\mathrm{in})\phi _{2}'(s_{\mathrm{in}})<1$ or equivalently $\text{ det }(\mathbf {J}_{a}(\mathbf {s}^0))<0$. Then, $\mathbf {E}^0$ is a saddle point (with a stable manifold of dimension one).

• When $\mu (s_{\mathrm{in}})>Q/V_{1}$, notice that the equilibrium $\mathbf {E}^0$ is not necessarily hyperbolic (as one can have $P(\mu (s_{\mathrm{in}}))=0$ which implies then $\text{ det }(\mathbf {J}_{a}(\mathbf {s}^0))=0$) and we cannot conclude its stability properties directly.

As already mentioned in Lemma 2.2, the dynamics is cooperative in the $(\mathbf {z},\mathbf {s})$ coordinates. Moreover, it is irreducible when $\mu (s_{1})$ or $\mu (s_{2})$ is non-null. But one has

$$\begin{aligned} s_{i}=0 \; \Rightarrow \; \dot{s}_{i} \ge \frac{Q}{V_{i}}s_\mathrm{in}>0 \quad (i=1,2) \end{aligned}$$

The domain ${{\mathcal {D}}}={\mathbb {R}}^2\times ({\mathbb {R}}_{+}\setminus \{0\})^2$ is thus invariant and one can consider without loss of generality initial conditions in ${{\mathcal {D}}}$. Then, the dynamics is strongly monotone on ${{\mathcal {D}}}$. As any forward orbit of (18) in ${{\mathcal {D}}}$ is bounded (see Lemma 2.2), we can use the property of strongly monotone systems (see for instance Theorem C.8 in [7]) to conclude that for any initial condition of (1) in ${\mathbb {R}}_{+}^4$, except on a set of null measure, the trajectory solution converges asymptotically to an equilibrium. Finally, when the equilibrium $\mathbf {E}^\star$ exists, the analysis conducted in the proof of Proposition 2.1 allows us to deduce the inequalities $\phi _{1}'(s_{1}^\star )>0$ and $\gamma '(s_{2}^\star )>0$, which in turn imply $\phi _{1}'(s_{1}^\star )\phi _{2}'(s_{2}^\star )>1$, and so $\phi _{2}'(s_{2}^\star )>0$. Then, one has $\text{ tr }(\mathbf {J}_{a}(\mathbf {s}^\star ))<0$ and $\text{ det }(\mathbf {J}_{a}(\mathbf {s}^\star ))>0$ i.e., $\mathbf {J}(\mathbf {s}^\star )$ is Hurwitz, which proves that the attractive equilibrium $\mathbf {E}^\star$ is also locally exponentially stable.