Mathematical analysis of an anaerobic digestion model for biogas production from solid waste

Abstract

Municipal solid waste is one of the most significant sources of gas emissions. In a context of renewable energy production and greenhouse gas emission reduction, we focus on the amount of biogas produced by the anaerobic digestion of organic matter in a controlled landfill environment. We present a mathematical model describing this process from a microbiological and biochemical point of view, including the dynamics of the methane and carbon dioxide produced. A qualitative analysis of the dynamic system shows the existence of an infinite number of non-hyperbolic equilibria inducing an attractor that has been identified. Through numerical tests, we explore how the non-connectivity of the attractor resulting from the choice of growth functions can entirely transform the performance of the process, and highlight critical initial stock values that influence the amount of biogas produced.

Introduction

The bio-degradation process of municipal waste in landfills consists of a succession of chemical reactions catalyzed by bacteria, which transform slowly the organic matter contained in the waste into mineral or gaseous elements. This transformation of organic matter can be decomposed in many steps, which differ depending on the environment being under aerobic or anaerobic conditions. The aerobic digestion lasts only a few days after the waste is deposited in the landfill, where oxygen is present in the void spaces. During this period, the biogas produced is mainly carbon dioxide. The anaerobic digestion operates within a closed environment where, in the absence of oxygen, organic matter is gradually converted into biogas, primarily composed of carbon dioxide and methane^1,2. This process unfolds in four major stages: first, hydrolysis breaks down insoluble organic polymers into soluble derivatives that become accessible to other bacteria. Next, during acidogenesis, acidogenic bacteria convert these sugars and amino acids into carbon dioxide, hydrogen, ammonia, and organic acids. In the acetogenesis stage, these organic acids are further processed by bacteria into acetic acid, along with additional ammonia, hydrogen, and carbon dioxide. Finally, methanogens convert these end products into methane and carbon dioxide, completing the biogas production process^3,4.

The mathematical modeling of the anaerobic digestion process in landfills holds significant importance for multiple reasons. It’s a fundamental tool for optimizing, and managing the generation of biogas, offering essential prospects for environmental sustainability and energy recovery.

In the literature, various models have been developed to describe the anaerobic digestion process and the associated microbial dynamics, aiming to better understand and optimize digester performance under varying microbial conditions^5–13.The Anaerobic Digestion Model No. 1 (ADM1) is a structural model that outlines the four main stages of anaerobic digestion leading to biogas production⁵. However, its complexity, due to a large number of equations, makes difficult its analysis. To overcome this, simplified models have been proposed. Rouez’s model⁶ uses a two-step process (hydrolysis/acidogenesis and methanogenesis), and was mathematically analysed later by Ouchtout et al.¹⁴ highlighting the impact of the selected bacterial growth functions. In the context of wastewater treatment, the (AM2) model by Bernard et al.⁷ is notable, incorporating Monod growth for acidogenesis and Haldane growth for methanogenesis. This choice is well-suited for control and optimization, as demonstrated by the stability analysis conducted by Benyahia et al.¹⁵ Furthermore, in the context of co-digestion, where multiple types of substrates are mixed, Berga et al.¹⁶ have provided a mathematical analysis of Rakmak et al.’s two-step model (hydrolysis/acidogenesis and methanogenesis)⁸, and highlighting the influence of the chosen bacterial growth functions. However, the Halvadakis model⁹, which captures the natural process of anaerobic digestion, is of particular interest for our study. This model provides a comprehensive description of the three main stages: hydrolysis, acidogenesis, and methanogenesis. The model considers biodegradable organic matter (easily, intermediate, and slowly), which is hydrolyzed into a dissolved organic substrate. This substrate is then transformed into acetate substrate and carbon dioxide by the acidogenic biomass. The methanogenic biomass uses the acetate substrate to produce methane and additional carbon dioxide. The substrate utilization by biomass is determined by the Monod bacterial growth function. The model also takes into account the decomposition of dead acidogenic bacteria, which is converted to acetate and carbon dioxide, and the decomposition of dead methanogenic bacteria, which is converted to methane and carbon dioxide.

In this study, we perform a mathematical analysis of the Halvadakis dynamical system, incorporating bacterial growth functions ${\mu }_a(.)$ and ${\mu }_m(.)$, which can be of either the Monod or Haldane type^17,18. Regardless of the growth function selected, our analysis confirms the existence of an infinite number of non-hyperbolic equilibria toward which the system’s trajectories converge. By employing Barbalat’s lemma and the stable manifold theorem, we identify the attractor of the dynamical system and derive an expression for the biogas produced based on initial substrate concentrations and their limiting values. Through numerical simulation, we demonstrate how the process performance is influenced by the connectivity of the attractor when the Monod growth function is chosen, and by its non-connectivity when the Haldane growth function is applied. Furthermore, we demonstrate that biogas production, as a function of the initial stock concentration, exhibits five changes in monotonicity. These changes are driven by the four regions of convergence of substrate concentrations that result from the selection of the Haldane growth function. We also analyze the impact of bacterial mortality rates on biogas production, highlighting their significant role in the overall process.

Description of the model

In this section, we introduce the mathematical model for anaerobic digestion, inspired by the Halvadakis model. Initially, the biodegradable organic substrate present in the waste, with easily, intermediate, and slowly degradable components, breakdown into dissolved organic substrate with an hydrolysis constant $k_h$. The dissolved organic substrate is then consumed by acidogenic biomass at a rate of ${\displaystyle \frac{1}{y_a}{\mu }_a(S_{\mathit{dc}})}$. A portion of this consumption supports the growth of the acidogenic biomass, which is modeled as a natality rate ${\mu }_a(S_{\mathit{dc}})$ minus a mortality rate $d_a$. The remainder of the consumed substrate, expressed as ${\displaystyle \frac{1-y_a}{y_a}{\mu }_a(S_{\mathit{dc}})}$, is partially converted to acetate substrate with a yield $y_{\mathit{ac}}$, while the rest is transformed into carbon dioxide with a yield of $(1-y_{\mathit{ac}})$. A similar process applies to the acetate substrate and methanogenic biomass. The acetate substrate is consumed by methanogenic biomass at a rate of ${\displaystyle \frac{1}{y_m}{\mu }_m(S_{\mathit{ac}})}$. Part of this consumption is used for the growth of methanogenic biomass, described by a natality rate ${\mu }_m(S_{\mathit{ac}})$ minus a mortality rate $d_m$. The remaining portion, expressed as ${\displaystyle \frac{1-y_m}{y_m}{\mu }_m(S_{\mathit{ac}})}$, leads to the production of methane with a yield $y_{C{H}_4}$, while the remainder produces carbon dioxide with a yield of $(1-y_{C{H}_4})$. The model also emphasizes the significant role of mortality rates in biogas production. These rates are included in the equations for biogas production because they account for the natural decrease in biomass. This decrease influences substrate availability, which in turn affects the rates of methane and carbon dioxide production.

The mathematical model of the process described in Fig. 1 is formulated through a system of differential equations. This system is represented as follows:

$$\begin{array}{cc}& \begin{array}{c}\hfill \left\{\begin{array}{cc}& {\displaystyle \frac{d{S}_s}{\mathit{dt}}=-k_h{S}_s}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{S}_{\mathit{dc}}}{\mathit{dt}}=k_h{S}_s-\frac{1}{y_a}{\mu }_a(S_{\mathit{dc}})X_a}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{X}_a}{\mathit{dt}}=\left({\mu }_a,(S_{\mathit{dc}}),-,d_a\right)X_a}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{S}_{\mathit{ac}}}{\mathit{dt}}=y_{\mathit{ac}}\left(\frac{1-y_a}{y_a},{\mu }_a,(S_{\mathit{dc}}),+,d_a\right)X_a-\frac{1}{y_m}{\mu }_m(S_{\mathit{ac}})X_m}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{X}_m}{\mathit{dt}}=\left({\mu }_m,(S_{\mathit{ac}}),-,d_m\right)X_m.}\hfill \end{array}\right)\end{array}\hfill \end{array}$$ 1

$$\begin{array}{cc}& \begin{array}{c}\hfill \left\{\begin{array}{cc}& {\displaystyle \frac{dC{H}_4}{\mathit{dt}}=y_{C{H}_4}\left(\frac{1-y_m}{y_m},{\mu }_m,(S_{\mathit{ac}}),+,d_m\right)X_m}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{dC{O}_2}{\mathit{dt}}=\left(1,-,y_{\mathit{ac}}\right)\left(\frac{1-y_a}{y_a},{\mu }_a,(S_{\mathit{dc}}),+,d_a\right)X_a+\left(1,-,y_{C{H}_4}\right)\left(\frac{1-y_m}{y_m},{\mu }_m,(S_{\mathit{ac}}),+,d_m\right)X_m.}\hfill \end{array}\right)\end{array}\hfill \end{array}$$ 2

The bacterial growth functions denoted as ${\mu }_a$ and ${\mu }_m$, are two $C^1$-functions defined on $\mathbb{R}$ by:

• Monod law:

  $$\begin{array}{c}\hfill {\mu }_a(S_{\mathit{dc}})=\frac{{\mu }_a^{\mathit{max}}{S}_{\mathit{dc}}}{K_d+S_{\mathit{dc}}}\text{and}{\mu }_m(S_{\mathit{ac}})=\frac{{\mu }_m^{\mathit{max}}{S}_{\mathit{ac}}}{K_a+S_{\mathit{ac}}},\forall S_{\mathit{dc}},S_{\mathit{ac}}\in {\mathbb{R}}^{+},\text{or}:\end{array}$$
• Haldane law:

  $$\begin{array}{c}\hfill {\mu }_a(S_{\mathit{dc}})=\frac{{\mu }_a^{\mathit{max}}{S}_{\mathit{dc}}}{K_d+S_{\mathit{dc}}+\frac{S_{\mathit{dc}}^2}{K_{\mathit{id}}}}\text{and}{\mu }_m(S_{\mathit{ac}})=\frac{{\mu }_m^{\mathit{max}}{S}_{\mathit{ac}}}{K_a+S_{\mathit{ac}}+\frac{S_{\mathit{ac}}^2}{K_{\mathit{ia}}}},\forall S_{\mathit{dc}},S_{\mathit{ac}}\in {\mathbb{R}}^{+}.\end{array}$$

Haldane’s law includes inhibition effects, providing a more realistic representation of bacterial growth at high substrate concentrations. Monod’s growth function is appropriate for simpler systems with lower substrate concentrations and minimal inhibition effects, while the Haldane function is better suited for complex scenarios involving significant substrate inhibition, although it can affect biogas production. In the model, we have:

Fig. 1.

Schematic representation of anaerobic digestion process in landfill.

Variables:

$S_s$: Total concentration of biodegradable organic substrate (g/l),

$S_{\mathit{dc}}$: Dissolved organic substrate concentration (g/l),

$X_a$: Acidogenic biomass concentration (g/l),

$S_{\mathit{ac}}$: Acetate substrate concentration (g/l),

$X_m$: Methanogenic biomass concentration (g/l),

$C{H}_4$: Methane concentration (g/l),

$C{O}_2$: Carbon dioxide concentration (g/l).

Parameters:

$y_a$: mass conversion rate of dissolved organic substrate into acidogenic biomass,

$y_m$: mass conversion rate of acetate substrate into methanogenic biomass,

$y_{\mathit{ac}}$: acetate substrate conversion yield,

$y_{C{H}_4}$: methane conversion yield.

$K_d$: half-saturation constant of dissolved organic substrate (g/l),

$K_a$: half-saturation constant of acetate substrate (g/l),

$K_{\mathit{id}}$: inhibition constant of dissolved organic substrate (g/l),

$K_{\mathit{ia}}$: inhibition constant of acetate substrate (g/l),

$k_h$: hydrolysis constant of biodegradable organic substrate $(da{y}^{-1})$,

$d_a$: death rate of acidogenic bacteria $(da{y}^{-1})$,

$d_m$: death rate of methanogenic bacteria $(da{y}^{-1})$.

${\mu }_a^{\mathit{max}}$: maximum growth rate of acidogenic bacteria $(da{y}^{-1})$,

${\mu }_m^{\mathit{max}}$: maximum growth rate of methanogenic bacteria $(da{y}^{-1})$.

The model parameters satisfy the following conditions.

H1

. The substrate-biomass yields are strictly positives and:

$$\begin{array}{c}\hfill y_a,y_m,y_{\mathit{ac}},y_{C{H}_4}\in (0,1).\end{array}$$

H2

. The hydrolysis, half saturation and inhibition constants are strictly positives:

$$\begin{array}{c}\hfill k_h,K_d,K_a,K_{\mathit{id}},K_{\mathit{ia}}>0.\end{array}$$

H3

. The bacterial death rates are strictly positives and less than the maximum growth rates:

$$\begin{array}{c}\hfill 0<d_a<{\mu }_a^{\mathit{max}}\text{and}0<d_m<{\mu }_m^{\mathit{max}}.\end{array}$$

We define the non-empty sets:

$$\begin{array}{c}\hfill {\xi }_{\mathit{dc}}=\{s\in {\mathbb{R}}_{+};{\mu }_a(s)\le d_a\}\text{and}{\xi }_{\mathit{ac}}=\{s\in {\mathbb{R}}_{+};{\mu }_m(s)\le d_m\}.\end{array}$$

For the Monod expression, we have:

$$\begin{array}{c}\hfill {\displaystyle {\xi }_{\mathit{dc}}=[0,{\lambda }_a]\text{with}{\lambda }_a=\frac{d_a{K}_d}{{\mu }_a^{\mathit{max}}-d_a},}\end{array}$$ 3

and

$$\begin{array}{c}\hfill {\displaystyle {\xi }_{\mathit{ac}}=[0,{\lambda }_m]\text{with}{\lambda }_m=\frac{d_m{K}_a}{{\mu }_m^{\mathit{max}}-d_m}.}\end{array}$$ 4

For the Haldane expression, we have:

$$\begin{array}{c}\hfill {\displaystyle {\xi }_{\mathit{dc}}=[0,{\lambda }_a^{-}]\cup [{\lambda }_a^{+},+\infty )\text{with}{\lambda }_a^{\pm }=\frac{{\mu }_a^{\mathit{max}}-d_a\pm \sqrt{{\mathrm{\Delta }}_a}}{2d_a/K_{\mathit{id}}},}\end{array}$$ 5

and

$$\begin{array}{c}\hfill {\displaystyle {\xi }_{\mathit{ac}}=[0,{\lambda }_m^{-}]\cup [{\lambda }_m^{+},+\infty )\text{with}{\lambda }_m^{\pm }=\frac{{\mu }_m^{\mathit{max}}-d_m\pm \sqrt{{\mathrm{\Delta }}_m}}{2d_m/K_{\mathit{ia}}},}\end{array}$$ 6

where

$$\begin{array}{c}\hfill {\displaystyle {\mathrm{\Delta }}_a={(d_a-{\mu }_a^{\mathit{max}})}^2-4\frac{K_d}{K_{\mathit{id}}}{d}_a^2\text{and}{\mathrm{\Delta }}_m={(d_m-{\mu }_m^{\mathit{max}})}^2-4\frac{K_a}{K_{\mathit{ia}}}{d}_m^2.}\end{array}$$

Study of the asymptotic behavior

As our system (1)–(2) satisfies the conditions of Cauchy-Lipschitz theorem, the existence and uniqueness of the solution with a given initial value is guaranteed. We begin with a result on the non-negativity and boundedness of the solution.

Proposition 1

Under hypothesis (H1-H3), for any non-negative initial condition $(S_s^0,S_{\mathit{dc}}^0,X_a^0,S_{\mathit{ac}}^0,X_m^0,C{H}_4^0,C{O}_2^0)$, the solution

$(S_s(.),S_{\mathit{dc}}(.),X_a(.),S_{\mathit{ac}}(.),X_m(.),C{H}_4(.),C{O}_2(.))$ of (1)–(2) is non-negative and bounded.

Proof

We will demonstrate the non-negativity of each component of the solution.

• For the component $S_s$, we have ${\displaystyle \frac{d{S}_s}{\mathit{dt}}=-k_h{S}_s}$ with $S_s(0)=S_s^0$, which implies $S_s(t)=S_s^0{e}^{-k_{h}t}$. If $S_s^0\ge 0$, then for all $t\ge 0$, we have $S_s(t)\ge 0$.

• For the component $S_{\mathit{dc}}$, we have ${\displaystyle \frac{d{S}_{\mathit{dc}}}{\mathit{dt}}=k_h{S}_s-\frac{1}{y_a}{\mu }_a(S_{\mathit{dc}})X_a}$ with $S_{\mathit{dc}}(0)=S_{\mathit{dc}}^0$. We have ${\displaystyle \frac{d{S}_{\mathit{dc}}}{\mathit{dt}}\ge -\frac{1}{y_a}{\mu }_a(S_{\mathit{dc}})X_a}$ for all $t\ge 0$. According to the comparison theorem¹⁹, if $S_{\mathit{dc}}^0>{\underset{\_}{S}}^0$, then $S_{\mathit{dc}}(t)>\underset{\_}{S}(t)$, such that $\underset{\_}{S}(t)$ is a solution of the differential equation:

  $$\begin{array}{c}\hfill \frac{d\underset{\_}{S}}{\mathit{dt}}=-\frac{1}{y_a}{\mu }_a(\underset{\_}{S})X_a,\underset{\_}{S}(0)={\underset{\_}{S}}^0.\end{array}$$ 7

  If ${\underset{\_}{S}}^0=0$, then $\underset{\_}{S}\equiv 0$ is the only solution of (7), thus $\underset{\_}{S}(t)>0$ when ${\underset{\_}{S}}^0>0$ for all $t\ge 0$. Then, if $S_{\mathit{dc}}^0>0$, we have $S_{\mathit{dc}}(t)>0$ for all $t\ge 0$.

• For the component $X_a$, we have ${\displaystyle \frac{d{X}_a}{\mathit{dt}}=\left({\mu }_a,(S_{\mathit{dc}}),-,d_a\right)X_a}$ with $X_a(0)=X_a^0$. The term $\left({\mu }_a,(S_{\mathit{dc}}),-,d_a\right)X_a$ is linear with respect to $X_a$, thus if $X_a^0\ge 0$, we have $X_a(t)\ge 0$ for all $t\ge 0$.

• For the component $S_{\mathit{ac}}$, we have ${\displaystyle \frac{d{S}_{\mathit{ac}}}{\mathit{dt}}=y_{\mathit{ac}}\left(\frac{1-y_a}{y_a},{\mu }_a,(S_{\mathit{dc}}),+,d_a\right)X_a-\frac{1}{y_m}{\mu }_m(S_{\mathit{ac}})X_m}$ with $S_{\mathit{ac}}(0)=S_{\mathit{ac}}^0$. We have ${\displaystyle \frac{d{S}_{\mathit{ac}}}{\mathit{dt}}\ge -\frac{1}{y_m}{\mu }_m(S_{\mathit{ac}})X_m}$ for all $t\ge 0$. According to the comparison theorem, if $S_{\mathit{ac}}^0>{\overline{S}}^0$, then $S_{\mathit{ac}}(t)>\overline{S}(t)$, such that $\overline{S}(t)$ is a solution of the differential equation:

  $$\begin{array}{c}\hfill \frac{d\overline{S}}{\mathit{dt}}=-\frac{1}{y_m}{\mu }_m(\overline{S})X_m,\overline{S}(0)={\overline{S}}^0.\end{array}$$ 8

  If ${\overline{S}}^0=0$, then $\overline{S}\equiv 0$ is the only solution of (8), thus $\overline{S}(t)>0$ when ${\overline{S}}^0>0$ for all $t\ge 0$. Then, if $S_{\mathit{ac}}^0>0$, we have $S_{\mathit{ac}}(t)>0$ for all $t\ge 0$.

• For the component $X_m$, we have ${\displaystyle \frac{d{X}_m}{\mathit{dt}}=\left({\mu }_m,(S_{\mathit{ac}}),-,d_m\right)X_m}$ with $X_m(0)=X_m^0$. The term $\left({\mu }_m,(S_{\mathit{ac}}),-,d_m\right)X_m$ is linear with respect to $X_m$, thus if $X_m^0\ge 0$, we have $X_m(t)\ge 0$ for all $t\ge 0$.

• From the above, we deduce the non-negativity of the components $C{H}_4$ and $C{O}_2$.

For boundedness, it suffices to notice that our system is closed, that is:

$$\begin{array}{c}\hfill \frac{d{S}_s}{\mathit{dt}}+\frac{d{S}_{\mathit{dc}}}{\mathit{dt}}+\frac{d{X}_a}{\mathit{dt}}+\frac{d{S}_{\mathit{ac}}}{\mathit{dt}}+\frac{d{X}_m}{\mathit{dt}}+\frac{dC{H}_4}{\mathit{dt}}+\frac{dC{O}_2}{\mathit{dt}}=0,\end{array}$$

so, there exists a positive constant C such that for all $t\ge 0$, we have:

$$\begin{array}{c}\hfill S_s(t)+S_{\mathit{dc}}(t)+X_a(t)+S_{\mathit{ac}}(t)+X_m(t)+C{H}_4(t)+C{O}_2(t)=C.\end{array}$$ 9

Hence, the result is proven. $\square$

Next, we proceed to investigate the asymptotic behaviour of (1).

Proposition 2

Under hypothesis (H1-H3), for any non-negative initial condition $(S_s^0,S_{\mathit{dc}}^0,X_a^0,S_{\mathit{ac}}^0,X_m^0)$, the solution of (1) verifies:

1. $\underset{t\to +\infty }{lim}{S}_s(t)=\underset{t\to +\infty }{lim}{X}_a(t)=\underset{t\to +\infty }{lim}{X}_m(t)=0.$

2. There exist two positive real numbers $S_{\mathit{dc}}^{\ast }$ and $S_{\mathit{ac}}^{\ast }$, which satisfy:

   $$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}{S}_{\mathit{dc}}(t)=S_{\mathit{dc}}^{\ast }\text{and}\underset{t\to +\infty }{lim}{S}_{\mathit{ac}}(t)=S_{\mathit{ac}}^{\ast }.\end{array}$$

Proof

We have $\underset{t\to +\infty }{lim}{S}_s(t)=\underset{t\to +\infty }{lim}{S}_s^0{e}^{-k_{h}t}=0$. To prove the convergence of $X_a$ and $S_{\mathit{dc}}$, we consider the $C^1$-function defined on $[0,+\infty )$ as follows:

$$\begin{array}{c}\hfill V_1(t)=S_s(t)+\frac{1}{y_a}{X}_a(t)+S_{\mathit{dc}}(t).\end{array}$$

We have:

$$\begin{array}{c}\hfill \frac{d{V}_1}{\mathit{dt}}=-\frac{d_a}{y_a}{X}_a.\end{array}$$

Since $X_a$ is non-negative, then $V_1$ is non-increasing, and minored by 0. So there exists a real $l_1\ge 0$ such that $\underset{t\to +\infty }{lim}{V}_1(t)=l_1(l_1<\infty )$. Moreover, one has:

$$\begin{array}{c}\hfill \frac{d^2{V}_1}{d{t}^2}=-\frac{d_a}{y_a}\left({\mu }_a,(S_{\mathit{dc}}),-,d_a\right)X_a.\end{array}$$

Since ${\displaystyle \frac{d^2{V}_1}{d{t}^2}}$ is bounded, ${\displaystyle \frac{d{V}_1}{\mathit{dt}}}$ is uniformly continuous. According to Barbalat’s lemma²⁰, we have ${\displaystyle \underset{t\to +\infty }{lim}\frac{d{V}_1}{\mathit{dt}}(t)=0}$. Then $\underset{t\to +\infty }{lim}{X}_a(t)=0$.

Additionally, since ${\displaystyle \underset{t\to +\infty }{lim}{V}_1(t)=\underset{t\to +\infty }{lim}{S}_s(t)+\frac{1}{y_a}{X}_a(t)+S_{\mathit{dc}}(t)=l_1}$, we have $\underset{t\to +\infty }{lim}{S}_{\mathit{dc}}(t)=S_{\mathit{dc}}^{\ast }=l_1$.

Similarly for $X_m$ and $S_{\mathit{ac}}$, we consider the $C^1$-function defined on $[0,+\infty )$ as follows:

$$\begin{array}{c}\hfill V_2(t)=S_{\mathit{ac}}(t)+\frac{1}{y_m}{X}_m(t)+y_{\mathit{ac}}\left(S_s,(t),+,X_a,(t),+,S_{\mathit{dc}},(t)\right),\end{array}$$

we prove that $\underset{t\to +\infty }{lim}{X}_m(t)=0$ and $\underset{t\to +\infty }{lim}{S}_{\mathit{ac}}(t)=S_{\mathit{ac}}^{\ast }$. $\square$

Proposition 3

Under hypothesis (H1–H3), for any non-negative initial condition $(S_s^0,S_{\mathit{dc}}^0,X_a^0,S_{\mathit{ac}}^0,X_m^0)$, the solution of (1) converges asymptotically to an equilibrium $(0,S_{\mathit{dc}}^{\ast },0,S_{\mathit{ac}}^{\ast },0)$, where $S_{\mathit{dc}}^{\ast }$ and $S_{\mathit{ac}}^{\ast }$ belong respectively to ${\xi }_{\mathit{dc}}$ and ${\xi }_{\mathit{ac}}$.

Proof

Let $S_{\mathit{dc}}(.)$ be a component of the solution of (1) which converges to $S_{\mathit{dc}}^{\ast }$, and suppose that $S_{\mathit{dc}}^{\ast }$ doesn’t belong to ${\xi }_{\mathit{dc}}$.

Since ${\mu }_a$ is continuous on $[0,+\infty )$, then ${\mu }_a(S_{\mathit{dc}}(t))$ tends to ${\mu }_a(S_{\mathit{dc}}^{\ast })$ when t approaches $+\infty$, leading to:

$$\begin{array}{c}\hfill \forall ϵ>0,\exists T\ge 0:\forall t>T,-ϵ<{\mu }_a(S_{\mathit{dc}}(t))-{\mu }_a(S_{\mathit{dc}}^{\ast })<ϵ.\end{array}$$

Choosing ${\displaystyle ϵ=\frac{{\mu }_a(S_{\mathit{dc}}^{\ast })-d_a}{2}>0}$, we have, for all $t>T$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mu }_a(S_{\mathit{dc}}(t))-{\mu }_a(S_{\mathit{dc}}^{\ast })>-ϵ& \Rightarrow {\mu }_a(S_{\mathit{dc}}(t))-{\mu }_a(S_{\mathit{dc}}^{\ast })+2ϵ>ϵ\hfill \\ \hfill & \Rightarrow {\mu }_a(S_{\mathit{dc}}(t))-d_a>ϵ\hfill \\ \hfill & {\displaystyle \Rightarrow \frac{d{X}_a}{\mathit{dt}}(t)>ϵX_a(t).}\hfill \end{array}\end{array}$$

Letting $s\in (T,t)$, integrating from s to t, we get $X_a(t)>X_a(s)e^{ϵ(t-s)}$, for all $t>T$. This contradicts the fact that $X_a$ is bounded. Thus, we conclude that $S_{\mathit{dc}}^{\ast }$ belongs to ${\xi }_{\mathit{dc}}$.

Similarly, we prove that $S_{\mathit{ac}}^{\ast }$ belongs to ${\xi }_{\mathit{ac}}$. $\square$

The sub-system (1) has an infinite number of equilibria that are non-hyperbolic, therefore the linearization does not allow us to conclude stability. To overcome this difficulty, we carry out a change of variables that leads to the use of a hyperbolic system. We then apply the stable manifold theorem. However, we obtain the following result.

Proposition 4

Under hypothesis (H1–H3), for each steady state $E=(0,S_{\mathit{dc}}^{\ast },0,S_{\mathit{ac}}^{\ast },0)$ with $S_{\mathit{dc}}^{\ast }\in {\xi }_{\mathit{dc}}$ and $S_{\mathit{ac}}^{\ast }\in {\xi }_{\mathit{ac}}$, there exists an invariant three-dimensional manifold $\mathbb{M}$ in ${\mathbb{R}}_{+}^5$ such that any solution of (1) with an initial condition in $\mathbb{M}$ converges asymptotically to E.

Proof

Let $(S_s,S_{\mathit{dc}},X_a,S_{\mathit{ac}},X_m)$ be a solution of (1) with an initial condition in $\mathbb{M}$. Let $Z_1$ and $Z_2$ be the two variables defined by:

$$\begin{array}{c}\hfill Z_1(t)=\frac{S_s(t)+S_{\mathit{dc}}(t)-S_{\mathit{dc}}^{\ast }}{X_a(t)}+{\beta }_1,\forall t\ge 0,\end{array}$$

and

$$\begin{array}{c}\hfill Z_2(t)=\frac{y_{\mathit{ac}}(S_s(t)+S_{\mathit{dc}}(t)+X_a(t))+S_{\mathit{ac}}(t)-y_{\mathit{ac}}S_{\mathit{dc}}^{\ast }-S_{\mathit{ac}}^{\ast }}{X_m(t)}+{\beta }_2,\forall t\ge 0,\end{array}$$

with

$$\begin{array}{c}\hfill {\beta }_1=\frac{{\mu }_a(S_{\mathit{dc}}^{\ast })}{y_a({\mu }_a(S_{\mathit{dc}}^{\ast })-d_a)}\text{and}{\beta }_2=\frac{{\mu }_m(S_{\mathit{ac}}^{\ast })}{y_m({\mu }_m(S_{\mathit{ac}}^{\ast })-d_m)}.\end{array}$$

We have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{Z}_1}{\mathit{dt}}& {\displaystyle =\frac{-\frac{1}{y_a}{\mu }_a(S_{\mathit{dc}})X_a^2-(Z_1-{\beta }_1)({\mu }_a(S_{\mathit{dc}})-d_a)X_a^2}{X_a^2}}\hfill \\ \hfill & ={\mu }_a(S_{\mathit{dc}})({\beta }_1-\frac{1}{y_a})-{\beta }_1{d}_a-({\mu }_a(S_{\mathit{dc}})-d_a)Z_1\hfill \\ \hfill & {\displaystyle =-{\gamma }_1({\mu }_a(S_{\mathit{dc}})-{\mu }_a(S_{\mathit{dc}}^{\ast }))-({\mu }_a(S_{\mathit{dc}})-d_a)Z_1,\text{with}{\gamma }_1=-\frac{d_a}{y_a({\mu }_a(S_{\mathit{dc}}^{\ast })-d_a)},}\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{Z}_2}{\mathit{dt}}& {\displaystyle =\frac{-\frac{1}{y_m}{\mu }_m(S_{\mathit{ac}})X_m^2-(Z_2-{\beta }_2)({\mu }_m(S_{\mathit{ac}})-d_m)X_m^2}{X_m^2}}\hfill \\ \hfill & ={\mu }_m(S_{\mathit{ac}})({\beta }_2-\frac{1}{y_m})-{\beta }_2{d}_m-({\mu }_m(S_{\mathit{ac}})-d_m)Z_2\hfill \\ \hfill & {\displaystyle =-{\gamma }_2({\mu }_m(S_{\mathit{ac}})-{\mu }_m(S_{\mathit{ac}}^{\ast }))-({\mu }_m(S_{\mathit{ac}})-d_m)Z_2,\text{with}{\gamma }_2=-\frac{d_m}{y_m({\mu }_m(S_{\mathit{ac}}^{\ast })-d_m)}.}\hfill \end{array}\end{array}$$

If we define $Y=(S_s,Z_1,X_a,Z_2,X_m)$, then we have:

$$\begin{array}{c}\hfill \frac{d{Z}_1}{\mathit{dt}}=-{\gamma }_1(g_a(Y)-{\mu }_a(S_{\mathit{dc}}^{\ast }))-(g_a(Y)-d_a)Z_1,\end{array}$$

and

$$\begin{array}{c}\hfill \frac{d{Z}_2}{\mathit{dt}}=-{\gamma }_2(g_m(Y)-{\mu }_m(S_{\mathit{ac}}^{\ast }))-(g_m(Y)-d_m)Z_2,\end{array}$$

with

$$\begin{array}{c}\hfill g_a(Y)={\mu }_a(S_{\mathit{dc}})={\mu }_a\left((Z_1-{\beta }_1),X_a,-,S_s,+,S_{\mathit{dc}}^{\ast }\right)\text{and}\\ \hfill g_m(Y)={\mu }_m(S_{\mathit{ac}})={\mu }_m\left((Z_2-{\beta }_2),X_m,-,y_{\mathit{ac}},(Z_1-{\beta }_1+1),X_a,+,S_{\mathit{ac}}^{\ast }\right).\end{array}$$

The equivalent system of the system (1) in the domain $\mathbb{D}=\{(S_s,Z_1,X_a,Z_2,X_m)\in {\mathbb{R}}_{+}\times \mathbb{R}\times \mathbb{R}_{+}^{\ast }\times \mathbb{R}\times \mathbb{R}_{+}^{\ast }/(Z_1-{\beta }_1)X_a-S_s+S_{\mathit{dc}}^{\ast }\ge 0;(Z_2-{\beta }_2)X_m-y_{\mathit{ac}}(Z_1-{\beta }_1+1)X_a+S_{\mathit{ac}}^{\ast }\ge 0\}$ is written:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\displaystyle \frac{d{S}_s}{\mathit{dt}}=-k_h{S}_s}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{X}_a}{\mathit{dt}}=\left(g_a,(Y),-,d_a\right)X_a}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{X}_m}{\mathit{dt}}=\left(g_m,(Y),-,d_m\right)X_m}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{Z}_1}{\mathit{dt}}=-{\gamma }_1(g_a(Y)-{\mu }_a(S_{\mathit{dc}}^{\ast }))-(g_a(Y)-d_a)Z_1}\hfill \\ \hfill \\ \hfill & {\displaystyle \frac{d{Z}_2}{\mathit{dt}}=-{\gamma }_2(g_m(Y)-{\mu }_m(S_{\mathit{ac}}^{\ast }))-(g_m(Y)-d_m)Z_2.}\hfill \end{array}\right)\end{array}$$ 10

The domain $\mathbb{D}$ is positively invariant for the system (10). Since ${\mu }_a(S_{\mathit{dc}}^{\ast })\ne d_a$ and ${\mu }_m(S_{\mathit{ac}}^{\ast })\ne d_m$, 0 is the only equilibrium of (10) on $\mathbb{D}$. The Jacobian matrix is given by:

$J(0)=\left[\begin{array}{ccccc}-k_h& 0& 0& 0& 0\\ \\ 0& {\mu }_a(S_{\mathit{dc}}^{\ast })-d_a& 0& 0& 0\\ \\ 0& 0& {\mu }_m(S_{\mathit{ac}}^{\ast })-d_m& 0& 0\\ \\ {\gamma }_1\mu _a^{_{}^{\prime }}(S_{\mathit{dc}}^{\ast })& {\gamma }_1{\beta }_1\mu _a^{_{}^{\prime }}(S_{\mathit{dc}}^{\ast })& 0& -({\mu }_a(S_{\mathit{dc}}^{\ast })-d_a)& 0\\ \\ 0& {\gamma }_2{y}_{\mathit{ac}}(1-{\beta }_2)\mu _m^{_{}^{\prime }}(S_{\mathit{ac}}^{\ast })& {\gamma }_2{\beta }_2\mu _m^{_{}^{\prime }}(S_{\mathit{ac}}^{\ast })& 0& -({\mu }_m(S_{\mathit{ac}}^{\ast })-d_m)\end{array}\right]$

We have three negative eigenvalues: $-k_h,{\mu }_a(S_{\mathit{dc}}^{\ast })-d_a\text{and}{\mu }_m(S_{\mathit{ac}}^{\ast })-d_m$, and two positive eigenvalues: $-({\mu }_a(S_{\mathit{dc}}^{\ast })-d_a)\text{and}-({\mu }_m(S_{\mathit{ac}}^{\ast })-d_m)$. By the stable manifold theorem²¹, 0 is a hyperbolic equilibrium with a three-dimensional stable manifold ${\mathbb{W}}^s$ and a two-dimensional unstable manifold ${\mathbb{W}}^u$.

We conclude that any trajectory of (10) with an initial value in the three-dimensional invariant manifold $\mathbb{M}={\mathbb{W}}^s\cap \mathbb{D}$ converges asymptotically to 0, therefore the corresponding trajectory of (1) converges asymptotically to $E=(0,S_{\mathit{dc}}^{\ast },0,S_{\mathit{ac}}^{\ast },0)$. $\square$

We can now state the corollary.

Corollary 1

Under hypothesis (H1–H3), the set:

$$\begin{array}{c}\hfill \{0\}\times {\xi }_{\mathit{dc}}\times \{0\}\times {\xi }_{\mathit{ac}}\times \{0\}\end{array}$$

is an attractor of sub-system (1).

Let’s proceed to the sub-system (2).

Proposition 5

Under hypothesis (H1–H3), for any non-negative initial condition $(S_s^0,S_{\mathit{dc}}^0,X_a^0,S_{\mathit{ac}}^0,X_m^0,C{H}_4^0,C{O}_2^0)$, the solution of (2) verifies:

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}C{H}_4(t)=C{H}_4^0+y_{C{H}_4}(\alpha -S_{\mathit{ac}}^{\ast }-y_{\mathit{ac}}S_{\mathit{dc}}^{\ast }),\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}C{O}_2(t)=C{O}_2^0+\beta -\alpha y_{C{H}_4}+(y_{\mathit{ac}}.y_{C{H}_4}-1)S_{\mathit{dc}}^{\ast }+(y_{C{H}_4}-1)S_{\mathit{ac}}^{\ast },\end{array}$$

with

$$\begin{array}{c}\hfill \alpha =y_{\mathit{ac}}(S_s^0+S_{\mathit{dc}}^0+X_a^0)+S_{\mathit{ac}}^0+X_m^0\text{and}\beta =S_s^0+S_{\mathit{dc}}^0+X_a^0+S_{\mathit{ac}}^0+X_m^0.\end{array}$$

Proof

From (2), we have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{dC{O}_2}{\mathit{dt}}& =(1-y_{C{H}_4})\left(\frac{1-y_m}{y_m},{\mu }_m,(S_{\mathit{ac}}),+,d_m\right)X_m+\left(1,-,y_{\mathit{ac}}\right)\left(\frac{1-y_a}{y_a},{\mu }_a,(S_{\mathit{dc}}),+,d_a\right)X_a\hfill \\ \hfill & {\displaystyle =\frac{1-y_{C{H}_4}}{y_{C{H}_4}}\times \frac{dC{H}_4}{\mathit{dt}}+\left(1,-,y_{\mathit{ac}}\right)\left(\frac{1-y_a}{y_a},{\mu }_a,(S_{\mathit{dc}}),+,d_a\right)X_a}\hfill \\ \hfill & {\displaystyle =\frac{1-y_{C{H}_4}}{y_{C{H}_4}}\times \frac{dC{H}_4}{\mathit{dt}}-(1-y_{\mathit{ac}})(\frac{d{S}_s}{\mathit{dt}}+\frac{d{S}_{\mathit{dc}}}{\mathit{dt}}+\frac{d{X}_a}{\mathit{dt}}).}\hfill \end{array}\end{array}$$

For all $t\ge 0$, by integrating from 0 to t, we get:

$$\begin{array}{c}\hfill C{O}_2(t)-C{O}_2^0=\frac{1-y_{C{H}_4}}{y_{C{H}_4}}\left(C,H_4,(t),-,C,H_4^0\right)-(1-y_{\mathit{ac}})(S_s(t)-S_s^0+S_{\mathit{dc}}(t)-S_{\mathit{dc}}^0+X_a(t)-X_a^0).\end{array}$$

Let’s denote $\underset{t\to +\infty }{lim}C{H}_4(t)=\widehat{C{H}_4}$ and $\underset{t\to +\infty }{lim}C{O}_2(t)=\widehat{C{O}_2}$. Upon taking limits, we obtain:

$$\begin{array}{c}\hfill {\displaystyle \widehat{C{O}_2}-\frac{1-y_{C{H}_4}}{y_{C{H}_4}}\widehat{C{H}_4}=C{O}_2^0-\frac{1-y_{C{H}_4}}{y_{C{H}_4}}C{H}_4^0-(1-y_{\mathit{ac}})(S_{\mathit{dc}}^{\ast }-S_s^0-S_{\mathit{dc}}^0-X_a^0).}\end{array}$$ 11

On the other hand, from Eq. (9), for all $t\ge 0$, we have:

$$\begin{array}{c}\hfill S_s(t)+S_{\mathit{dc}}(t)+X_a(t)+S_{\mathit{ac}}(t)+X_m(t)+C{H}_4(t)+C{O}_2(t)=S_s^0+S_{\mathit{dc}}^0+X_a^0+S_{\mathit{ac}}^0+X_m^0+C{H}_4^0+C{O}_2^0,\end{array}$$

when t tends to $+\infty$, we obtain:

$$\begin{array}{c}\hfill S_{\mathit{dc}}^{\ast }+S_{\mathit{ac}}^{\ast }+\widehat{C{H}_4}+\widehat{C{O}_2}=S_s^0+S_{\mathit{dc}}^0+X_a^0+S_{\mathit{ac}}^0+X_m^0+C{H}_4^0+C{O}_2^0.\end{array}$$ 12

From Eq. (11) and Eq. (12), we conclude that:

$\widehat{C{H}_4}=y_{C{H}_4}\left(X_m^0,+,S_{\mathit{ac}}^0,+,y_{\mathit{ac}},(S_s^0+S_{\mathit{dc}}^0+X_a^0),-,S_{\mathit{ac}}^{\ast },-,y_{\mathit{ac}},S_{\mathit{dc}}^{\ast }\right)+C{H}_4^0$, and

$\widehat{C{O}_2}=-\widehat{C{H}_4}-S_{\mathit{dc}}^{\ast }-S_{\mathit{ac}}^{\ast }+S_s^0+S_{\mathit{dc}}^0+X_a^0+S_{\mathit{ac}}^0+X_m^0+C{H}_4^0+C{O}_2^0$.

Taking $\alpha =y_{\mathit{ac}}(S_s^0+S_{\mathit{dc}}^0+X_a^0)+S_{\mathit{ac}}^0+X_m^0$ and $\beta =S_s^0+S_{\mathit{dc}}^0+X_a^0+S_{\mathit{ac}}^0+X_m^0$, we obtain the result. $\square$

Numerical results

In this section, we utilize the Matlab programming language to display numerical results for quantities of biogas produced, impacted by the final values $S_{\mathit{dc}}^{\ast }$ and $S_{\mathit{ac}}^{\ast }$ of the concentrations of the dissolved organic and acetate substrates.

The parameter values used in our numerical simulations are inspired by^10,11,14 and are presented in the following tables:

We assume that at the beginning of the process, there is no production of biogas, so that $C{H}_4^0=C{O}_2^0=0$, the other initial conditions are given in Table 1.

Table 1.

Initial conditions.

$S_s^0$	$S_{\mathit{dc}}^0$	$X_a^0$	$S_{\mathit{ac}}^0$	$X_m^0$
400	50	15	10	1, 5

Figures 2 and 3 describe the trajectories of the system (1)–(2) corresponding to the Monod and Haldane scenarios, respectively.

Fig. 2.

Trajectories of the system using the Monod growth function.

Fig. 3.

Trajectories of the system using the Haldane growth function.

In the Monod case, the concentrations of carbon dioxide and methane reach stability at approximately 200g/l and 250g/l, respectively, following a storage period of 200 days. Notably, an observation can be made that the anaerobic phase begins at around 100 days, signifying a switch wherein methane production exceeds carbon dioxide production.

In the Haldane case, the stabilization of carbon dioxide and methane production happens nearly simultaneously but after an extended storage period of 400 days. The start of the anaerobic phase is observed after approximately 300 days in this scenario.

The disparity between the Monod and Haldane cases lies in the convergence behavior of the concentrations $S_{\mathit{dc}}$ and $S_{\mathit{ac}}$. In the Monod case, the concentrations converge within connected sets, corresponding to the connected attractor. In contrast, the Haldane case shows convergence within non-connected sets, corresponding to the non-connected attractor. To validate this assertion, we fix the parameters specified in Table 2 while varying the initial values. We, then, observe the convergence of the concentrations $S_{\mathit{dc}}$ and $S_{\mathit{ac}}$.

Table 2.

Model coefficients.

${\mu }_a^{\mathit{max}}$	$K_d$	$y_a$	$d_a$	${\mu }_m^{\mathit{max}}$	$K_a$	$y_m$	$d_m$	$k_h$	$y_{\mathit{ac}}$	$y_{C{H}_4}$	$K_{\mathit{id}}$	$K_{\mathit{ia}}$
0, 5	50	0, 15	0, 05	0, 25	600	0, 06	0, 03	0, 02	0, 7	0, 8	100	50

In the Monod case, from Eq. (3) and Eq. (4), we have ${\lambda }_a=5.5$ and ${\lambda }_m=60$.

Figures 4 and 5 illustrate that for any non-negative value of $S_s^0$, $S_{\mathit{dc}}$ and $S_{\mathit{ac}}$ converge to values less than ${\lambda }_a$ and ${\lambda }_m$ respectively.

Fig. 4.

$S_{\mathit{dc}}$ trajectories for various values of $S_s^0$ in the Monod case.

Fig. 5.

$S_{\mathit{ac}}$ trajectories for various values of $S_s^0$ in the Monod case.

In the Haldane case, Eq. (5) yields ${\lambda }_a^{-}=5.59$ and ${\lambda }_a^{+}=894.4$. Figure 6 demonstrates that when:

• $S_s^0\le 3300$, $S_{\mathit{dc}}$ converges to values less than ${\lambda }_a^{-}$.

• $S_s^0>3300$, $S_{\mathit{dc}}$ converges to values higher than ${\lambda }_a^{+}$.

Therefore, we find that 3300 is the critical value of $S_s^0$ at which the limit $S_{\mathit{dc}}^{\ast }$ shifts from the interval ${\lambda }_a^{-}$ to the interval ${\lambda }_a^{+}$.

Fig. 6.

$S_{\mathit{dc}}$ trajectories for various values of $S_s^0$ in the Haldane case.

Similarly, Eq. (6) gives ${\lambda }_m^{-}=123.24$ and ${\lambda }_m^{+}=243.42$. Figure 7 shows that when:

• $0\le S_s^0\le 600$, $S_{\mathit{ac}}$ converges to values less than ${\lambda }_m^{-}$.

• $600<S_s^0<3900$, $S_{\mathit{ac}}$ converges to values higher than ${\lambda }_m^{+}$.

• $S_s^0\ge 3900$, $S_{\mathit{ac}}$ converges again to values less than ${\lambda }_m^{-}$.

Therefore, we find that 600 and 3900 are critical values of $S_s^0$ at which the limit $S_{\mathit{ac}}^{\ast }$ shifts from ${\lambda }_m^{-}$ to ${\lambda }_m^{+}$ and then back to ${\lambda }_m^{-}$.

Fig. 7.

$S_{\mathit{ac}}$ trajectories for various values of $S_s^0$ in the Haldane case.

When the Haldane growth function is considered, we observe two regions of convergence of $S_{\mathit{dc}}$ and two regions of convergence of $S_{\mathit{ac}}$. In Fig. 8, we illustrate the behavior of the substrate concentrations within the dynamic model (1) and identify the placement of these four regions. Considering initial substrate concentrations $S_s^0$ ranging from 0 to 6000 with an increment of 200, this range includes the switch points 600 and 3900 for $S_{\mathit{ac}}^{\ast }$ and 3300 for $S_{\mathit{dc}}^{\ast }$. We observe the following:

• When $S_s^0\le 600$, both $S_{\mathit{dc}}^{\ast }$ and $S_{\mathit{ac}}^{\ast }$ are below ${\lambda }_a^{-}$ and ${\lambda }_m^{-}$ respectively.

• For $600<S_s^0\le 3300$, $S_{\mathit{dc}}^{\ast }$ remains below ${\lambda }_a^{-}$, while $S_{\mathit{ac}}^{\ast }$ surpasses ${\lambda }_m^{+}$.

• Between $3300<S_s^0<3900$, both $S_{\mathit{dc}}^{\ast }$ and $S_{\mathit{ac}}^{\ast }$ exceed ${\lambda }_a^{+}$ and ${\lambda }_m^{+}$ respectively.

• When $S_s^0\ge 3900$, $S_{\mathit{dc}}^{\ast }$ rises above ${\lambda }_a^{+}$, while $S_{\mathit{ac}}^{\ast }$ falls below ${\lambda }_m^{-}$.

Fig. 8.

$S_s$, $S_{\mathit{dc}}$, and $S_{\mathit{ac}}$ trajectories behavior for various values of $S_s^0$ in the Haldane case.

This figure reveals the impact of inhibition phenomena on the process. Specifically, inhibition can prevent the bacteria from efficiently consuming the substrate, resulting in high concentrations substrate remaining in the process. As a result, the accumulation of unconsumed substrate affects negatively biogas production, as will be further illustrated in the next figure. In contrast, when the Monod growth function is chosen, only one region of convergence is observed, and the remaining substrate does not reach a high concentration.

In the following, we will focus on simulations in the Haldane case.

Figure 9 shows the total biogas (methane + carbon dioxide) produced as a function of the initial substrate concentration $S_s^0$. According to Proposition 5, the quantity of biogas produced is influenced by the limits $S_{\mathit{dc}}^{\ast }$ and $S_{\mathit{ac}}^{\ast }$, which shift from lower to higher values as $S_s^0$ varies. In this figure, we observe that for $0\le S_s^0\le 3300$, $S_{\mathit{dc}}$ converges below ${\lambda }_a^{-}$. This indicates that the dissolved organic substrate is effectively consumed by acidogenic bacteria, resulting in a significant increase in biogas production, which reaches a peak concentration of 1050g/l. However, when $S_s^0>3300$, $S_{\mathit{dc}}$ converges above ${\lambda }_a^{+}$, leading to less effective consumption of the dissolved organic substrate by acidogenic bacteria. Consequently, biogas production declines sharply, with a noticeable fall between 3300 and 3400 in $S_s^0$, where biogas output falls to 400g/l. Beyond this point, biogas concentration continues to decrease. Within this range, for $S_s^0>600$, $S_{\mathit{dc}}$ converges above ${\lambda }_m^{+}$, resulting in poor consumption of acetate substrate by methanogenic bacteria, which leads to a decrease in biogas concentration over a small interval, followed by a gradual increase. Similarly, for $S_s^0>3900$, $S_{\mathit{dc}}$ converges below ${\lambda }_m^{-}$, leading to effective consumption of acetate substrate by methanogenic bacteria and a temporary increase in biogas concentration, which is then followed by a continuous decrease.

Fig. 9.

Biogas production as function of initial stock concentration $S_s^0$.

These observations highlight that the choice of growth functions significantly affects biogas production. When both bacterial growth functions are Haldane, biogas production exhibits five distinct switches in monotonicity. In contrast, when one Monod function and one Haldane function are used, there is a single switch value of $S_s^0$ at which biogas production exhibits one change in monotonicity, shifting from increasing to decreasing. When both functions are Monod, biogas production increases linearly with $S_s^0$ without any decrease. This occurs because there is no inhibition phenomenon present, allowing the bacteria to continuously consume the substrate.

Let’s examine the impact of bacterial mortality rates on biogas production by considering a small value of $S_s^0$ yielding Biogas- and a large value of $S_s^0$ yielding Biogas+. It can be seen in Fig. 10 that when the initial stock concentration $S_s^0$ is less than 3300, the mortality rate of acidogenic bacteria has almost no influence on the biogas production. In the other hand, Fig. 11 shows that the amount of biogas is first constant and then decreases as function of the mortality rate of methanogenic bacteria.

Fig. 10.

Biogas production as function of mortality rate $d_a$.

Fig. 11.

Biogas production as function of mortality rate $d_m$.

When $S_s^0$ is higher than 3400, both Figs. 10 and 11 illustrate that the biogas production remains constant when the mortality rates of acidogenic and methanogenic bacteria are below 0.025. However, it starts decreasing once the mortality rates surpass this value. Notably, in this scenario, biogas production decreases sharply with mortality rate of acidogenic bacteria.

Conclusion

Our study has provided a detailed analysis of biogas production through anaerobic digestion of solid waste, modeled by a system of differential equations. Through our mathematical analysis, we identified the attractor of the system. Our numerical simulations provide important insights into system behavior. When both bacterial growth functions are Monod, the attractor is connected, and biogas production increases linearly with the initial stock concentration $S_s^0$. When one growth function is Monod and the other is Haldane, the attractor becomes non-connected with two regions of convergence, and biogas production as function of $S_s^0$ exhibits a single change in monotonicity. When both functions are Haldane, the attractor is non-connected with four regions of convergence, and biogas production as function of $S_s^0$ exhibits five changes in monotonicity. Additionally, biogas production decreases with higher bacterial mortality rates.

In our future work, we will focus on managing the initial stock concentration to avoid exceeding the switch value of $S_s^0$ that results in optimal biogas production and will explore the impact of spatial diffusion effects on biogas production.

List of symbols

$S_s$

Total concentration of biodegradable organic substrate (g/l)

$S_{\mathit{dc}}$

Dissolved organic substrate concentration (g/l)

$X_a$

Acidogenic biomass concentration (g/l)

$S_{\mathit{ac}}$

Acetate substrate concentration (g/l)

$X_m$

Methanogenic biomass concentration (g/l)

$C{H}_4$

Methane concentration (g/l)

$C{O}_2$

Carbon dioxide concentration (g/l)

Y

Vector of variables

$y_a$

Mass conversion rate of dissolved organic substrate into acidogenic biomass

$y_m$

Mass conversion rate of acetate substrate into methanogenic biomass

$y_{\mathit{ac}}$

Acetate substrate conversion yield $y_{C{H}_4}$ Methane conversion yield

$K_d$

Half-saturation constant of dissolved organic substrate (g/l)

$K_a$

Half-saturation constant of acetate substrate (g/l)

$K_{\mathit{id}}$

Inhibition constant of dissolved organic substrate (g/l)

$K_{\mathit{ia}}$

Inhibition constant of acetate substrate (g/l)

$k_h$

Hydrolysis constant of biodegradable organic substrate $(da{y}^{-1})$

$d_a$

Death rate of acidogenic bacteria $(da{y}^{-1})$

$d_m$

Death rate of methanogenic bacteria $(da{y}^{-1})$

${\mu }_a^{\mathit{max}}$

Maximum growth rate of acidogenic bacteria $(da{y}^{-1})$

${\mu }_m^{\mathit{max}}$

Maximum growth rate of methanogenic bacteria $(da{y}^{-1})$

t

Time (day)

$S_{\mathit{dc}}^{\ast }$

Limit concentration of dissolved organic substrate (g/l)

$S_{\mathit{ac}}^{\ast }$

Limit concentration of acetate substrate (g/l)

$\widehat{C{H}_4}$

Limit concentration of methane (g/l)

$\widehat{C{O}_2}$

Limit concentration of carbon dioxide (g/l)

${\mu }_a(.)$

Growth function of acidogenic bacteria

${\mu }_m(.)$

Growth function of methanogenic bacteria

Author contributions

I.A. analyzed the outcomes. A.A. and I.A. conducted the numerical simulations. N.E.K. and A.A. identified the problem and provided supervision. All authors reviewed the manuscript.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.