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1 Introduction

We consider a well-known anaerobic digestion model for biological treatment of wastewater in a continuously stirred tank bioreactor (cf. for example [2, 3]). Here we include discrete time delays in the equations to model the delay in the conversion of nutrient consumed by the viable biomass. For more detailed motivation see [13, 14] and the references therein. The model is described by the following nonlinear differential equations:

$$\begin{aligned} \begin{array}{lcl} \displaystyle \frac{d}{dt} s_1(t) &{}=&{} u(s_1^i - s_1(t)) - k_1 \mu _1(s_1(t)) x_1(t) \\[6pt] \displaystyle \frac{d }{dt} x_1(t) &{}=&{} \displaystyle e^{- \alpha u \tau _1} \mu _1 (s_1(t-\tau _1)) x_1(t-\tau _1) - \alpha u x_1 (t) \\[6pt] \displaystyle \frac{d }{dt} s_2(t) &{}=&{} u(s_2^i - s_2(t)) + k_2 \mu _1 (s_1(t)) x_1(t) - k_3 \mu _2(s_2(t)) x_2(t) \\[6pt] \displaystyle \frac{d }{dt} x_2(t) &{}=&{} \displaystyle e^{- \alpha u \tau _2}\mu _2 (s_2(t-\tau _2)) x_2(t-\tau _2) - \alpha u x_2 (t). \end{array} \end{aligned}$$
(1)

The state variables \(s_1\), \(s_2\) and \(x_1\), \(x_2\) denote substrate and biomass concentrations, respectively: \(s_1\) is the organic substrate, characterized by its chemical oxygen demand (COD), \(s_2\) denotes the volatile fatty acids (VFA), \(x_1\) and \(x_2\) are the acidogenic and methanogenic bacteria respectively; \(s_1^i\) and \(s_2^i\) are the input substrate concentrations. The constants \(\tau _j\ge 0\), \(j=1,2\), stand for the time delay in conversion of the corresponding substrate to viable biomass for the jth bacterial population. Here \( \displaystyle e^{- \alpha u \tau _j}x_j (t-\tau _j)\), \(j=1,2\), represents the biomass of those microorganisms that consume nutrient \(\tau _j\) units of time prior to time t and that survive in the chemostat the \(\tau _j\) units of time necessary to complete the process of converting the nutrient to viable biomass at time t. The parameter \(\alpha \in (0,1)\) represents the proportion of bacteria that are affected by the dilution rate u. The constants \(k_1\), \(k_2\) and \(k_3\) are yield coefficients related to COD degradation, VFA production and VFA consumption respectively. For biological evidence, \(s_1^i\) and \(s_2^i\) as well as all parameters in (1) are assumed to be positive.

The functions \(\mu _1(s_1)\) and \(\mu _2(s_2)\) model the specific growth rates of the bacteria. Following [9] we impose the following assumption on \(\mu _1\) and \(\mu _2\):

Assumption A1. For each \(j=1,2\) the function \(\mu _j(s_j)\) is defined for \(s_j \in [0, +\infty )\), \(\mu _j(0)=0\), and \(\mu _j(s_j) >0\) for each \(s_j>0\); the function \(\mu _j(s_j)\) is bounded and Lipschitz continuous for all \(s_j \in [0, +\infty )\).

The Eq. (1) with \(\tau _1=\tau _2=0\) have been already investigated by the authors; thereby, global stabilizability via feedback control is proposed in [4], whereas [5] considers the case of global stabilization of the solutions using constant dilution rate u. This second approach is now extended to model (1) involving discrete delays \(\tau _j>0\), \(j=1,2\). More precisely, in this paper we define a suitable positive constant \(u_b\) and prove that for any (admissible) value of the dilution rate \(u \in (0, u_b)\) there exists an equilibrium point which is globally asymptotically stable for system (1). To our knowledge, such investigations have not been carried out for this model.

2 Global Asymptotic Stabilizability of the Model

We set \(u_b = \, \displaystyle \max \left\{ u:\ \displaystyle u\alpha e^{\alpha u \tau _1} \le \mu _1(s_1^i), \ \displaystyle u\alpha e^{\alpha u \tau _2} \le \mu _2(s_2^i)\right\} \) and make the following

Assumption A2. For each point \(\bar{u} \in (0, u_b)\) there exist points \(s_1(\bar{u}) = \bar{s}_1 \in \left( 0, s_1^i \right) \) and \(s_2(\bar{u}) = \bar{s}_2 \in \left( 0, s_2^i \right) \), such that the following equalities hold true

$$\begin{aligned} \bar{u} = \frac{ \displaystyle e^{- \alpha \bar{u} \tau _1}}{\alpha }\mu _1 (\bar{s}_1) = \frac{ \displaystyle e^{- \alpha \bar{u} \tau _2}}{\alpha }\mu _2 \left( \bar{s}_2 \right) . \end{aligned}$$

A similar assumption is called in [7] regulability of the system.

Let \(\bar{s}_1\) and \(\bar{s}_2\) be determined according to Assumption A2. Compute further

$$\begin{aligned} x_1(\bar{u}) = \bar{x}_1 = \frac{s_1^{i} - \bar{s}_1}{\alpha k_1\displaystyle e^{ \alpha \bar{u} \tau _1}},\; \;\; x_2(\bar{u}) = \bar{x}_2 = \frac{s_2^i- \bar{s}_2 + \alpha k_2 \bar{x}_1}{\alpha k_3\displaystyle e^{ \alpha \bar{u} \tau _2}}. \end{aligned}$$
(2)

Then the point \(p(\bar{u}) = \bar{p} =(\bar{s}_1, \bar{x}_1, \bar{s}_2, \bar{x}_2)\) is a nontrivial (positive) equilibrium point for system (1).

Assumption A3. There exist positive numbers \(\nu _1\) and \(\nu _2\) such that the following inequalities hold true

$$\begin{aligned} \mu _{1} (s_{1}^{-}) < \mu _{1} (\bar{s}_{1}) < \mu _{1} (s_{1}^{+}), \; \; \; \mu _{2} (s_{2}^{-})< \mu _{2} (\bar{s}_{2}) < \mu _{2} ( s_{2}^{+}) \end{aligned}$$

for each

$$\begin{aligned} s_1^- \in (0, \bar{s}_1), s_1^+ \in (\bar{s}_1, s_1^i+ \nu _1], s_2^- \in (0, \bar{s}_2) \text{ and } s_2^+ \in (\bar{s}_2, s_2^i + \nu _2]. \end{aligned}$$

Assumption A3 is always fulfilled when the functions \(\mu _j(\cdot )\), \(j=1,2\), are monotone increasing (like the Monod specific growth rate). If at least one function \(\mu _j(\cdot )\) is not monotone increasing (like the Haldane law) then the points \(\bar{s}_j\) have to be chosen sufficiently small in order to satisfy Assumption A3.

Denote by \(R^+\) the set of all positive real numbers and by \(C^+_{\tau }\) – the nonnegative cone of continuous functions \(\varphi :[-\tau ,0]\rightarrow R^+\), where \(\tau =\max \{\tau _1, \tau _2 \}\), and set \(C^4_{\tau }:= \{\varphi =(\varphi _{s_1},\varphi _{x_1},\varphi _{s_2},\varphi _{x_2}) \ \in C^+_{\tau } \times C^+_{\tau } \times C^+_{\tau } \times C^+_{\tau } \}\).

Let \(\bar{u} \in (0,\, u_b)\) be chosen in such a way that Assumptions A2 and A3 are satisfied. Denote by \(\varSigma \) the system obtained from (1) by substituting the parameter u by \(\bar{u}\). Using the Schauder fixed-point theorem it is easy to prove that for each \(\varphi \in C^4_{\tau }\) there exists \(\varrho >0\) and a unique solution \(\varPhi (t,\varphi )=(s_1(t, \varphi ), x_1(t, \varphi ),s_2(t, \varphi ), x_2(t, \varphi ))\) of (1) defined on \([-\tau , \varrho )\) such that \(\varPhi (t,\varphi ) = \varphi (t)\) for each \(t\in [-\tau ,0]\) (cf. Theorem 2.1 in [8]).

We shall prove below that the equilibrium point \(\bar{p}\) is globally asymptotically stable for system \(\varSigma \).

Theorem 1

Let the Assumptions A1, A2 and A3 be fulfilled and let \(\varphi _0\) be an arbitrary element of \(C^4_{\tau }\). Then the corresponding solution \(\varPhi (t, \varphi _0)\) is well defined on \([-\tau , + \infty )\) and converges asymptotically towards \(\bar{p}\).

Proof

We fix an arbitrary \(\varphi _0\in C^4_{\tau }\). Then there exists \(\varrho >0\) such that the corresponding solution \(\varPhi (t, \varphi _0)\) of \(\varSigma \) (denoted by \(\varPhi (t):= (s_1(t), x_1(t),s_2(t), x_2(t))\) for simplicity) is defined on \([-\tau ,\varrho )\). The proof uses some ideas from [13, 14]. For the reader’s convenience we subdivide the proof in five claims.

Claim 1. The components of \(\varPhi (t)\) take positive values for each \(t\in [-\tau , \varrho )\).

Proof of Claim 1. If \(s_1(t)=0\) for some \(t\in [0, \varrho )\), then \(\dot{s}_1 (t)>0\). This implies that \(s_1 (t)>0\) for each \(t\in [-\tau , \varrho )\). Analogously one can obtain that \(s_2(t)>0\) for each \(t\in [-\tau , \varrho )\). Since

$$\begin{aligned} x_j(t)= \varphi _{x_j}(0)e^{-\alpha \bar{u} t}+ \int _0^t e^{-\alpha \bar{u} (t-\sigma )}\mu _j (s_j(\sigma -\tau _j))x_j(\sigma -\tau _j)d\sigma , \ j=1,2, \end{aligned}$$

then \(x_j(t)>0\) for each \(t\in [-\tau , \varrho )\). This completes the proof of Claim 1. \(\diamondsuit \)

Claim 2. The solution \(\varPhi (t)\) of \(\varSigma \) is defined for each \(t\in [-\tau , +\infty )\) and is bounded.

Proof of Claim 2. Denote

$$\begin{aligned} s(t) := k_2 \displaystyle e^{- \alpha \bar{u} \tau _1} s_1(t) + k_1 \displaystyle e^{- \alpha \bar{u} \tau _1} s_2(t) \;\; \text{ and } \; \; s^i= k_2 \displaystyle e^{- \alpha \bar{u} \tau _1} s_1^i + k_1 \displaystyle e^{- \alpha \bar{u} \tau _1} s_2^i. \end{aligned}$$

Then s(t) satisfies the differential equation

$$\begin{aligned} \dot{s}(t) = \bar{u} ( s^i - s(t)) -k_1k_3\displaystyle e^{- \alpha \bar{u} \tau _1} \mu _2 (s_2(t)) x_2(t). \end{aligned}$$

We set \(q_1(t):= s(t) + k_1k_3\displaystyle e^{- \alpha \bar{u} (\tau _1-\tau _2)} x_2(t+\tau _2) - s^i/\alpha \) and \(q_2(t):= s(t) + k_1k_3x_2(t+\tau _2) - s^i\). Then

$$\begin{aligned} \dot{q}_1(t)= & {} \bar{u}\left[ s^i- s(t) - \alpha k_1k_3\displaystyle e^{- \alpha \bar{u} (\tau _1-\tau _2)} x_2(t+\tau _2) \right] \\\le & {} \bar{u} \left[ s^i - \alpha \left( s(t) + k_1k_3\displaystyle e^{- \alpha \bar{u} (\tau _1-\tau _2)} x_2(t+\tau _2) \right) \right] = - \alpha \bar{u} q_1(t), \end{aligned}$$

and hence

$$\begin{aligned} q_1(t) \le q_1(0) \cdot e^{-\alpha \bar{u} t}. \end{aligned}$$
(3)

The latter inequality shows that \(q_1(t) \) is bounded. Using the fact that the values of \(s_1(t)\), \(s_2(t)\) and \(x_2(t)\) are positive, it follows that \(s_1(t)\), \(s_2(t)\) and \(x_2(t)\) are bounded as well. Analogously one can obtain that

$$\begin{aligned} q_2(t) \ge q_2(0) \cdot e^{-\bar{u} t}. \end{aligned}$$
(4)

The estimates (3), (4) and the definition of \(s(\cdot )\) imply that for each \(\varepsilon > 0\) there exists \(T_\varepsilon >0\) such that for each \(t\ge T_\varepsilon \) the following inequalities hold true

$$\begin{aligned} s^i -\varepsilon < k_2 s_1(t) + k_1 s_2(t) + k_1k_3\displaystyle e^{- \alpha \bar{u} (\tau _1-\tau _2)} x_2(t+\tau _2) < \frac{s^i}{\alpha }+\varepsilon . \end{aligned}$$
(5)

It is easy to see (in the same way as the estimates (5)) that for each \(\varepsilon >0\) there exists a finite time \(T_\varepsilon >0\) such that for all \(t \ge T_\varepsilon \) the following inequalities hold

$$\begin{aligned} s^i_1 - \varepsilon < s_1(t)+ k_1\displaystyle e^{ \alpha \bar{u} \tau _1} x_1(t+\tau _1) <\frac{s^i_1}{\alpha }+ \varepsilon . \end{aligned}$$
(6)

The inequalities (6) imply that \(x_1 (t)\) is also bounded. Thus the trajectory \(\varPhi (t)\) of \(\varSigma \) is well defined and bounded for all \(t\ge -\tau \) (cf. also Theorem 3.1 of [8]). This completes the proof of Claim 2. \(\diamondsuit \)

Claim 3. There exists \( T_0>0\) such that \(s_1(t)<s_1^i\) and \(s_2(t) < s_2^i + k_2 s_1^i/k_1\) for each \(t\ge T_0\).

Proof of Claim 3. First let us assume that there exists \(\bar{t}>0 \) such that \(s_1 (t) \ge s_1^i\) for all \(t\ge \bar{t}\). Then we have

$$\begin{aligned} \dot{s}_1 (t) = \bar{u} (s_1^i - s_1 (t)) - k_1 \mu _1 (s_1(t)) x_1 (t) < 0. \end{aligned}$$

Since \(s_1 (\cdot )\) and \(x_1 (\cdot )\) are bounded differentiable functions defined on \([-\tau , +\infty ) \), then \(\dot{s}_1 (\cdot )\) is an uniformly continuous function. Barbălat’s Lemma (cf. [6]) leads to

$$\begin{aligned} 0 =\lim _{t\rightarrow \infty } \dot{s}_1(t) = \lim _{t\rightarrow \infty } [\bar{u} (s_1^i - s_1 (t)) - k_1 \mu _1 (s_1(t)) x_1 (t) ]. \end{aligned}$$

Because \(s_1^i - s_1 (t)\le 0\) and \(x_1 (t)>0\), the above equalities imply that \(s_1 (t) \downarrow s_1^i\) and \(x_1 (t) \downarrow 0\) as \(t \uparrow \infty \). On the other hand, if we set (cf. Lemma 2.2 of [14])

$$\begin{aligned} z_1 (t):= x_1 (t) + \int _{t-\tau _1}^t \displaystyle e^{- \alpha \bar{u} \tau _1} \mu _1 (s_1 (\sigma ))x_1 (\sigma )d\sigma , \end{aligned}$$

we obtain according to Assumption 3 that

$$\begin{aligned} \dot{z}_1 (t) = x_1 (t) (\displaystyle e^{- \alpha \bar{u} \tau _1}\mu _1 (s_1(t)) - \alpha \bar{u}) > 0 \; \text{ for } \text{ all } \; t \ge \bar{t}, \end{aligned}$$

and so \(z_1 (t) \uparrow z_1^*>0 \) as \(t\uparrow \infty \). But this is impossible according to the definition of \(z_1(\cdot )\) and because we have already shown that \(x_1 (t) \downarrow 0\) as \(t \uparrow \infty \).

Hence, there exists a sufficiently large \(T_0>0\) with \(s_1 (T_0)\le s_1^i\). Moreover, if the equality \(s_1(\bar{t})=s^i_1\) holds true for some \(\bar{t}\ge T_0\), then we have

$$\begin{aligned} \dot{s}_1(\bar{t}) = \bar{u} (s^i_1 - s_1(\bar{t})) - k_1 \mu _1(s_1(\bar{t})) x_1 (\bar{t})= - k_1 \mu _1(s_1(\bar{t})) x_1 (\bar{t})<0. \end{aligned}$$

The last inequality shows that \(s_1(t) < s^i_1\) for each \(t>T_0\).

Further with \(s(t) = k_2 \displaystyle e^{- \alpha \bar{u} \tau _1}s_1(t) + k_1\displaystyle e^{- \alpha \bar{u} \tau _1} s_2(t)\) and \(s^i= k_2 \displaystyle e^{- \alpha \bar{u} \tau _1}s_1^i + k_1 \displaystyle e^{- \alpha \bar{u} \tau _1}s_2^i\) we obtain

$$\begin{aligned} \dot{s}(t) = \bar{u} ( s^i - s(t)) -k_1k_3 \displaystyle e^{- \alpha \bar{u} \tau _1} \mu _2 (s_2(t)) x_2(t). \end{aligned}$$

One can show in the same way as above that \(s(t)< s^i\) for each \(t\ge T_0\) (if necessary \(T_0\) can be enlarged), i. e. \(k_2 \displaystyle e^{- \alpha \bar{u} \tau _1}s_1(t) + k_1\displaystyle e^{- \alpha \bar{u} \tau _1} s_2(t) \le k_2\displaystyle e^{- \alpha \bar{u} \tau _1} s_1^i + k_1\displaystyle e^{- \alpha \bar{u} \tau _1} s_2^i\). Since \(0<s_1 (t) < s_1^i\), it follows that \(s_2(t) \le s_2^i + k_2 s_1^i/k_1\). This establishes Claim 3. \(\diamondsuit \)

Claim 4. Denote

$$\begin{aligned} \begin{array}{l} \gamma _j:= \limsup \nolimits _{t\uparrow \infty } x_j(t), \;\;\; \delta _j:= \liminf \nolimits _{t\uparrow \infty } x_j(t), \;\;\; j=1,2 \\[6pt] v_1 (t):= s_1(t) + k_1 x_1 (t+\tau _1), \;\;\; v_2 (t):= k_2s_1(t) + k_1s_2(t) + k_1k_3 x_2 (t+\tau _2), \\[6pt] \alpha _j:= \limsup \nolimits _{t\uparrow \infty } v_j(t), \;\;\; \beta _j:= \liminf \nolimits _{t\uparrow \infty } v_j(t), \;\;\; j=1,2. \end{array} \end{aligned}$$

Then the following relations hold true: \(\delta _1>0\), \(\alpha _1=\beta _1\) and \(\gamma _1=\delta _1\), \(\alpha _2=\beta _2\) and \(\gamma _2=\delta _2\).

Proof of Claim 4. Let us assume that \(\delta _1=0\). Choose an arbitrary \(\varepsilon \in (0, (s_1^i - \bar{s}_1)/(1+\displaystyle e^{ \alpha \bar{u} \tau _1}k_1))\). According to Claim 2 (see (6)) there exists \(T_\varepsilon >0\) such that for all \(t \ge T_\varepsilon \) the following inequalities hold true

$$\begin{aligned} s^i_1 - \varepsilon < s_1(t-\tau _1)+ k_1\displaystyle e^{ \alpha \bar{u} \tau _1} x_1(t) < \frac{s^i_1}{\alpha }+ \varepsilon . \end{aligned}$$
(7)

Since \(\delta _1=0\) there exists \(t_0 > \max (T_\varepsilon , T_0)\) such that \(x_1 (t_0) < \varepsilon \). We set (cf. Lemma 3.5 of [14])

$$\begin{aligned} \begin{array}{l} \sigma := \min \{x_1 (t):\ t\in [t_0-\tau _1, t_0]\} \\[8pt] \bar{t} := \sup \{t\ge t_0-\tau _1:\ x_1 (\tau ) \ge \sigma \text{ for } \text{ all } \tau \in [t_0-\tau _1, t]\}. \end{array} \end{aligned}$$

Clearly \(\sigma \in (0, \varepsilon ]\), \(\bar{t} \in [t_0-\tau _1, +\infty )\), \(x_1 (t) \ge \sigma \) for all \(t \in [t_0 - \tau _1, \bar{t}]\) and

$$\begin{aligned} x_1 (\bar{t})= \sigma , \; \; \; \dot{x}_1 (\bar{t}) \le 0. \end{aligned}$$
(8)

Taking into account (7) and the choice of \(\varepsilon \), we obtain consecutively

$$\begin{aligned} \begin{array}{l} s_1^i> s_1(\bar{t}-\tau _ 1) \ge s_1^i -k_1\displaystyle e^{ \alpha \bar{u} \tau _1} x_1 (\bar{t}) - \varepsilon \ge \\ \ge s_1^i - (1 + \displaystyle e^{ \alpha \bar{u} \tau _1} k_1) \varepsilon > \bar{s}_1, \\[8pt] \dot{x}_1 (\bar{t}) = \displaystyle e^{- \alpha \bar{u} \tau _1}\mu _1 (s_1 (\bar{t} - \tau _ 1))x_1 (\bar{t} - \tau _ 1) - \alpha \bar{u} x_1 (\bar{t}) > \alpha \bar{u} \sigma - \alpha \bar{u} \sigma = 0. \end{array} \end{aligned}$$

The last inequality contradicts (8), which means that \(\delta _1>0\).

The proof of the equalities \(\alpha _j=\beta _j\) and \(\gamma _j=\delta _j\), \(j=1,2\), is based on similar ideas used in the proofs of Lemma 4.3 of [14] and Theorem 3.1 of [13], so we omit it here due to the limited paper length. \(\diamondsuit \)

Claim 5. The equilibrium point \(\bar{p}\) is locally asymptotically stable for all values of the delays \(\tau _1 \ge 0\) and \(\tau _2 \ge 0\).

Proof of Claim 5. Denote for simplicity \(a = k_1 \mu _1^\prime (\bar{s}_1)\bar{x}_1\) and \(b=k_3 \mu _2^\prime (\bar{s}_2)\bar{x}_2\). It follows from Assumption A3 that \(a>0\) and \(b>0\) hold true. It is straightforward to see that the characteristic equation of \(\varSigma \) corresponding to the equilibrium point \(\bar{p}\) has the form

$$\begin{aligned} 0=P(\lambda ; \tau _1,\tau _2)= P_1(\lambda ; \tau _1) \times P_2(\lambda ; \tau _2), \end{aligned}$$

where \(\lambda \) is a complex number and

$$\begin{aligned} P_1(\lambda ; \tau _1)= & {} \lambda ^2 + (\bar{u} +a + \alpha \bar{u})\lambda + \alpha \bar{u} (\bar{u} +a) - \alpha \bar{u} (\bar{u} + \lambda ) e^{-\lambda \tau _1}, \\ P_2(\lambda ; \tau _2)= & {} \lambda ^2 + (\bar{u} +b + \alpha \bar{u})\lambda + \alpha \bar{u} (\bar{u} +b) - \alpha \bar{u} (\bar{u} + \lambda ) e^{-\lambda \tau _2}. \end{aligned}$$

First it is straightforward to see that if \(\tau _1=\tau _2=0\) then there exist no roots \(\lambda \) of \(P(\lambda ; \tau _1,\tau _2)=0\) with \(Re(\lambda )\ge 0\). Let \(\tau _1>0\) and \(\tau _2>0\). We are looking for purely imaginary roots \(\lambda = i \omega \) of \(P_j(\lambda ; \tau _j)=0\) with \(\omega >0\), \(j=1,2\). For \(P_1(i\omega ; \tau _1)=0\) we obtain

$$\begin{aligned}&-\omega ^2 + (\bar{u} +a+\alpha \bar{u})i\omega + \alpha \bar{u} (\bar{u} +a) - \alpha \bar{u}(\bar{u} + i \omega ) e^{-i\omega \tau _1} =0, \\&-\omega ^2 + (\bar{u} +a +\alpha \bar{u})i \omega + \alpha \bar{u} (\bar{u} +a) - \alpha \bar{u} (\bar{u} + i \omega )(\cos (\tau _1 \omega ) - i \sin (\tau _1 \omega ))=0. \end{aligned}$$

Separating the real and the imaginary parts of the last equation implies

$$\begin{aligned} \begin{array}{l} -\omega ^2 + \alpha \bar{u} (\bar{u} +a) = \alpha \bar{u}^2 \cos (\tau _1 \omega ) + \alpha \bar{u} \omega \sin (\omega \tau _1),\\[8pt] (\bar{u} +a + \alpha \bar{u}) \omega = -\alpha \bar{u}^2 \sin (\tau _1 \omega ) + \alpha \bar{u} \omega \cos (\omega \tau _1). \end{array} \end{aligned}$$
(9)

Squaring both sides of the Eq. (9) and adding leads to

$$\begin{aligned} \omega ^4 + (\bar{u} +a)^2 \omega ^2 + \alpha ^2 \bar{u}^2 a (2 \bar{u} +a) =0. \end{aligned}$$

Obviously, the latter equation does not possess positive real roots since \(a>0\). The same conclusion holds true for \(P_2(i\omega ; \tau _2)=0\). Therefore, \(P(\lambda ; \tau _1, \tau _2)=0\) does not have purely imaginary roots for any \(\tau _1>0\) and \(\tau _2>0\). Applying Lemma 2 from [10] (see also [11, 12] for similar results) to the exponential polynomial \(P(\lambda ; \tau _1, \tau _2)\) we obtain that the characteristic equation does not have roots with nonnegative real parts. This means that for any \(\tau _1 \ge 0\) and \(\tau _2 \ge 0\) the equilibrium \(\bar{p}\) is locally asymptotically stable. \(\diamondsuit \)

The local asymptotic stability of the equilibrium \(\bar{p}\) together with the convergence of the solution \(\varPhi (t)\) and the attractivity of \(\bar{p}\), proved above throughout Claims 1 to 4, imply that \(\bar{p}\) is globally asymptotically stable.

The proof of Theorem 1 is completed. \(\blacklozenge \)

3 Computer Simulation

Consider the following specific growth rate functions in the model (1), taken from [1–3]:

$$\begin{aligned} \mu _1(s_1) = \dfrac{m_1 s_1}{k_{s_1} + s_1} \text{(Monod } \text{ law) }, \;\; \mu _2(s_2) = \dfrac{m_2 s_2}{k_{s_2} + s_2 +(s_2/k_I)^2} \text{(Haldane } \text{ law) }. \end{aligned}$$

In the simulation process we shall use the following numerical values for the model coefficients, which are obtained by real experiments and given in [1]:

$$\begin{aligned} \begin{array}{lllllll} k_1=10.53 &{} k_2=28.6 &{} k_3=1074 &{} s_1^i = 7.5 &{} s_2^i=75 &{} \alpha = 0.5 \\ m_1=1.2 &{} k_{s_1}=7.1 &{} m_2 =0.74 &{} k_{s_2}=9.28 &{} k_I=16 \end{array} \end{aligned}$$

As an example let us take \(\tau _1=2\) and \(\tau _2=7\). Within the above coefficient values we compute the admissible upper bound \(u_b=0.646\) for u, thus \(u \in (0, 0.646)\).

Consider \(\bar{u} = 0.25\). Then the corresponding internal equilibrium is \(\bar{p} = (1.096, 0.9472, 6.432, 0.06674)\). Using the initial conditions \(\varphi _{s_1}(t) =2\), \(\varphi _{x_1}(t)=0.1\) for \(t \in [-\tau _1, 0]\), and \( \varphi _{s_2}(t)= 10\), \(\varphi _{x_2}(t)=0.05\) for \(t \in [-\tau _2, 0]\), the numerical outputs are visualized in Fig. 1.

Fig. 1.
figure 1

Time evolution of \(s_1(t)\), \(s_2(t)\) (left) and \(x_1(t)\), \(x_2(t)\) (right)

Full size image

4 Conclusion

In this paper we investigate a bioreactor model for wastewater treatment by anaerobic digestion. The model Eq. (1) involve discrete delays, describing the time delay in nutrient conversion to viable biomass. Using a properly chosen admissible value for the dilution rate \(\bar{u}\) we prove the global convergence of the solutions towards an equilibrium point, corresponding to \(\bar{u}\). To authors’ knowledge, such kind of investigations have not been yet fulfilled for this delay bioreactor model. Numerical simulation is included to confirm the theoretical results.