Mathematical Study of a Two-Stage Anaerobic Model When the Hydrolysis Is the Limiting Step
<p>The existence of the solution of <math display="inline"><semantics> <mrow> <mi>γ</mi> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p> "> Figure 2
<p>Operating diagram of System (<a href="#FD5-processes-09-02050" class="html-disp-formula">5</a>).</p> "> Figure 3
<p>Operating diagram of System (<a href="#FD9-processes-09-02050" class="html-disp-formula">9</a>).</p> "> Figure 4
<p>The operating diagram for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>2</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> mmol/L.</p> "> Figure 5
<p>The operating diagram for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> g/L.</p> "> Figure 6
<p>The operating diagram for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>0.03</mn> </mrow> </semantics></math> g/L.</p> "> Figure 7
<p>Biological interpretation for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>18</mn> </mrow> </semantics></math> g/L.</p> "> Figure 8
<p>The bifurcation diagram for the input control <span class="html-italic">D</span> for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>18</mn> </mrow> </semantics></math> g/L.</p> "> Figure 9
<p>Biological interpretation for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>14</mn> </mrow> </semantics></math> g/L.</p> "> Figure 10
<p>The bifurcation diagram of the operating diagram shown in <a href="#processes-09-02050-f004" class="html-fig">Figure 4</a> for <math display="inline"><semantics> <mrow> <msubsup> <mi>S</mi> <mn>1</mn> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mn>14</mn> </mrow> </semantics></math> g/L.</p> ">
Abstract
:1. Introduction
2. Mathematical Model
3. Analysis of the Model
3.1. The Dynamics of and
3.1.1. Study of the Steady States of System (5)
- 1.
- is LES if and only if (i.e., );
- 2.
- is LES if and only if (i.e., ), ( is stable if it exists).
3.1.2. Operating Diagram of the System (5)
3.2. The Dynamics of and
3.2.1. Study of the Steady States of System (8)
- 1.
- is LES if and only if or ;
- 2.
- is LES if and only if (stable if it exists);
- 3.
- is unstable if it exists (unstable if ).
3.2.2. Operating Diagram of the System (8)
3.3. Analysis of the Whole System (4)
3.3.1. Steady States
- , and ;
- , and ;
- , and .
- , and ;
- , and ;
- , and .
- always exists;
- exists if and only if ;
- exists if and only if ;
- exists if and only if ;
- exists if and only if and ;
- exists if and only if and .
3.3.2. Steady States Stability of the System (4)
Case | Condition | NH | S | U |
1.4 | ||||
1.5 | ||||
1.6 |
Case | Condition | NH | S | U |
2.7 | ||||
2.8 | ||||
2.9 | ||||
2.10 | , | |||
2.11 | , | |||
2.12 | ||||
2.13 | , | |||
2.14 | , | , | ||
2.15 | , |
4. Simulations
4.1. Algorithm for the Determination of the Operating Diagrams
Algorithm 1 Operating diagram |
for i varying from 1 to do; |
for j varying from 1 to ; |
determine 6 equilibria of the model |
for k varying from 1 to 6 do |
calculate the Jacobian matrix at |
calculate the eigenvalues of |
if all the conditions of existence of are fulfilled and all real parts of |
the eigenvalues of are non-positive then is stable |
else if all conditions of existence of are fulfilled and at least one |
real part of eigenvalue of is positive then is unstable |
else does not exist |
end if |
end for (k) |
end for (j) |
end for (i) |
4.2. Operating Diagrams
4.3. Practical Interpretations of the Operating Diagrams
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Steady State | Existence Condition | Stability Condition |
---|---|---|
Always exists | ||
Stable when it exists |
Region | Equ. | Equ. |
---|---|---|
S | ||
U | S |
Steady-State | Existence Condition | Stability Condition |
---|---|---|
Always exists | or | |
Stable if it exists | ||
Unstable if it exists |
Region | Equ. | Equ. | Equ. |
---|---|---|---|
S | |||
U | S | ||
S | S | U |
Equ. | Matrices and | Conditions of Stability |
---|---|---|
if , | ||
if or , | ||
and | ||
() | and if | |
and at , | ||
at | ||
and by A1 | ||
and | ||
if or | ||
() | and | |
and at , | ||
at | ||
Case | Area | Condition | ||||||
---|---|---|---|---|---|---|---|---|
1.1 | S | |||||||
1.2 | U | S | ||||||
1.3 | S | S | U |
Case | Area | Condition | ||||||
---|---|---|---|---|---|---|---|---|
2.1 | U | S | ||||||
2.2 | U | U | S | |||||
2.3 | U | S | S | U | ||||
2.4 | U | U | U | S | ||||
2.5 | U | U | S | S | U | |||
2.6 | U | U | U | S | S | U |
Parameter | Unit | Nominal Value |
---|---|---|
d | 0.5 | |
g/L | 2.1 | |
d | 1 | |
I | mmol/L | 60 |
mmol/L | 24 | |
d | 0.1 | |
d | 0.06 | |
in | 0.5 | |
g/g | 1/25 | |
g/mmol | 1/250 | |
g/mmol | 1/268 |
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Hanaki, M.; Harmand, J.; Mghazli, Z.; Rapaport, A.; Sari, T.; Ugalde, P. Mathematical Study of a Two-Stage Anaerobic Model When the Hydrolysis Is the Limiting Step. Processes 2021, 9, 2050. https://doi.org/10.3390/pr9112050
Hanaki M, Harmand J, Mghazli Z, Rapaport A, Sari T, Ugalde P. Mathematical Study of a Two-Stage Anaerobic Model When the Hydrolysis Is the Limiting Step. Processes. 2021; 9(11):2050. https://doi.org/10.3390/pr9112050
Chicago/Turabian StyleHanaki, Mohammed, Jérôme Harmand, Zoubida Mghazli, Alain Rapaport, Tewfik Sari, and Pablo Ugalde. 2021. "Mathematical Study of a Two-Stage Anaerobic Model When the Hydrolysis Is the Limiting Step" Processes 9, no. 11: 2050. https://doi.org/10.3390/pr9112050