doi: 10.3934/dcdsb.2020369

Mathematical analysis of a three-tiered food-web in the chemostat

a. 

University of Tunis El Manar, National Engineering School of Tunis, LAMSIN, 1002, Tunis, Tunisia

b. 

ITAP, Univ Montpellier, INRAE, Institut Agro, Montpellier, France

c. 

University of Monastir, Higher Institute of Computer Science of Mahdia, 5111, Mahdia, Tunisia

d. 

University of Manouba, National School of Computer Science, 2010, Manouba, Tunisia

* Corresponding author: Nahla Abdellatif

Received  June 2020 Published  December 2020

Fund Project: This work was supported by the Euro-Mediterranean research network TREASURE (http://www.inra.fr/treasure).

A mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web is investigated. The model is a six-dimensional system of ordinary differential equations. In our study, the phenol and the hydrogen inflowing concentrations are taken into account as well as the maintenance terms. The case of a large class of growth kinetics is considered, instead of specific kinetics. We show that the system can have up to eight types of steady states and we analytically determine the necessary and sufficient conditions for their existence according to the operating parameters. In the particular case without maintenance, the local stability conditions of all steady states are determined. The bifurcation diagram shows the behavior of the process by varying the concentration of influent chlorophenol as the bifurcating parameter. It shows that the system exhibits a bi-stability where the positive steady state can lose stability undergoing a supercritical Hopf bifurcation with the emergence of a stable limit cycle.

Citation: Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020369
References:
[1]

N. AbdellatifR. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Math. Biosci. Eng., 13 (2016), 631-652.  doi: 10.3934/mbe.2016012.  Google Scholar

[2]

M. BallykR. Staffeldt and I. Jawarneh, A nutrient-prey-predator model: Stability and bifurcations, Discrete & Continuous Dyn. Syst. - S, 13 (2020), 2975-3004.  doi: 10.3934/dcdss.2020192.  Google Scholar

[3]

B. Bar and T. Sari, The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete & Continuous Dyn. Syst. - B, 25 (2020), 2093-2120.  doi: 10.3934/dcdsb.2019203.  Google Scholar

[4]

D. J. BatstoneJ. KellerI. AngelidakiS. V. KalyhuzhnyiS. G. PavlosthathisA. RozziW. T. M. SandersH. Siegrist and V. A. Vavilin, The IWA anaerobic digestion model no 1 (ADM1), Water Sci Technol., 45 (2002), 66-73.  doi: 10.2166/wst.2002.0292.  Google Scholar

[5]

B. BenyahiaT. SariB. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, J. Proc. Control, 22 (2012), 1008-1019.  doi: 10.1016/j.jprocont.2012.04.012.  Google Scholar

[6]

O. BernardZ. Hadj-SadokD. DochainA. Genovesi and J-P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnol. Bioeng., 75 (2001), 424-438.  doi: 10.1002/bit.10036.  Google Scholar

[7]

Y. DaoudN. AbdellatifT. Sari and J. Harmand, Steady state analysis of a syntrophic model: The effect of a new input substrate concentration, Math. Model. Nat. Phenom., 13 (2018), 1-22.  doi: 10.1051/mmnp/2018037.  Google Scholar

[8]

P. De LeenheerD. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.  doi: 10.1016/j.jmaa.2006.02.036.  Google Scholar

[9]

M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete & Continuous Dyn. Syst. - B, (2020). doi: 10.3934/dcdsb.2020156.  Google Scholar

[10]

M. DellalM. Lakrib and T. Sari, The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Math. Biosci., 302 (2018), 27-45.  doi: 10.1016/j.mbs.2018.05.004.  Google Scholar

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N. Dimitrova and M. Krastanov, Nonlinear stabilizing control of an uncertain bioprocess model, Int. J. Appl. Math. Comput. Sci., 19 (2009), 441-454.  doi: 10.2478/v10006-009-0036-0.  Google Scholar

[12]

M. El HajjiF. Mazenc and J. Harmand, A mathematical study of a syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656.  doi: 10.3934/mbe.2010.7.641.  Google Scholar

[13]

M. El Hajji, N. Chorfi and M. Jleli, Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat, Electron. J. Differ. Eq., 2017, Paper No. 255, 13 pp. https://ejde.math.txstate.edu/Volumes/2017/255/elhajji.pdf.  Google Scholar

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R. Fekih-Salem, N. Abdellatif, T. Sari and J. Harmand, Analyse mathématique d'un modèle de digestion anaérobie à trois étapes, {ARIMA Rev.}, 17 (2014), 53–71. http://arima.inria.fr/017/017003.html.  Google Scholar

[15]

R. Fekih-SalemC. Lobry and T. Sari, A density-dependent model of competition for one resource in the chemostat, Math. Biosci., 286 (2017), 104-122.  doi: 10.1016/j.mbs.2017.02.007.  Google Scholar

[16]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.  doi: 10.1126/science.6767274.  Google Scholar

[17]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2017.  Google Scholar

[18]

P. A. Hoskisson and G. Hobbs, Continuous culture–making a comeback?, Microbiology, 151 (2005), 3153-3159.  doi: 10.1099/mic.0.27924-0.  Google Scholar

[19]

S.-B. HsuC. A. Klausmeier and C.-J. Lin, Analysis of a model of two parallel food chains, Discrete & Continuous Dyn. Syst. - B, 12 (2009), 337-359.  doi: 10.3934/dcdsb.2009.12.337.  Google Scholar

[20]

J. Monod, La technique de culture continue: Théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184–204 doi: 10.1016/B978-0-12-460482-7.50023-3.  Google Scholar

[21]

S. Nouaoura, N. Abdellatif, R. Fekih-Salem and T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, preprint (2020), hal-02540350. https://hal.archives-ouvertes.fr/hal-02540350. Google Scholar

[22]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete & Continuous Dyn. Syst. - B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.  Google Scholar

[23]

T. SariM. El Hajji and J. Harmand, The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat, Math. Biosci. Eng., 9 (2012), 627-645.  doi: 10.3934/mbe.2012.9.627.  Google Scholar

[24]

T. Sari and J. Harmand, A model of a syntrophic relationship between two microbial species in a chemostat including maintenance, Math. Biosci., 275 (2016), 1-9.  doi: 10.1016/j.mbs.2016.02.008.  Google Scholar

[25]

T. Sari and M. J. Wade, Generalised approach to modelling a three-tiered microbial food-web, Math. Biosci., 291 (2017), 21-37.  doi: 10.1016/j.mbs.2017.07.005.  Google Scholar

[26]

M. SbarciogM. Loccufier and E. Noldus, Determination of appropriate operating strategies for anaerobic digestion systems, Biochem. Eng. J., 51 (2010), 180-188.  doi: 10.1016/j.bej.2010.06.016.  Google Scholar

[27] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[28]

S. SobieszekM. J. Wade and G. S. K. Wolkowicz, Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Math. Biosci. Eng., 17 (2020), 7045-7073.  doi: 10.3934/mbe.2020363.  Google Scholar

[29]

M. J. Wade, Not Just Numbers: Mathematical Modelling and Its Contribution to Anaerobic Digestion Processes, Processes, 8 (2020), 888. doi: 10.3390/pr8080888.  Google Scholar

[30]

M. J. WadeJ. HarmandB. BenyahiaT. BouchezS. ChaillouB. CloezJ.-J. GodonB. Moussa BoudjemaaA. RapaportT. SariR. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecol. Modell., 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002.  Google Scholar

[31]

M. J. WadeR. W. PattinsonN. G. Parker and J. Dolfing, Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web', J. Theor. Biol., 389 (2016), 171-186.  doi: 10.1016/j.jtbi.2015.10.032.  Google Scholar

[32]

M. Weedermann, Analysis of a model for the effects of an external toxin on anaerobic digestion, Math. Biosci. Eng., 9 (2012), 445-459.  doi: 10.3934/mbe.2012.9.445.  Google Scholar

[33]

G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.  doi: 10.1016/0025-5564(89)90025-4.  Google Scholar

[34]

A. XuJ. DolfingT. P. CurtisG. Montague and E. Martin, Maintenance affects the stability of a two-tiered microbial 'food chain'?, J. Theor. Biol., 276 (2011), 35-41.  doi: 10.1016/j.jtbi.2011.01.026.  Google Scholar

show all references

References:
[1]

N. AbdellatifR. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Math. Biosci. Eng., 13 (2016), 631-652.  doi: 10.3934/mbe.2016012.  Google Scholar

[2]

M. BallykR. Staffeldt and I. Jawarneh, A nutrient-prey-predator model: Stability and bifurcations, Discrete & Continuous Dyn. Syst. - S, 13 (2020), 2975-3004.  doi: 10.3934/dcdss.2020192.  Google Scholar

[3]

B. Bar and T. Sari, The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete & Continuous Dyn. Syst. - B, 25 (2020), 2093-2120.  doi: 10.3934/dcdsb.2019203.  Google Scholar

[4]

D. J. BatstoneJ. KellerI. AngelidakiS. V. KalyhuzhnyiS. G. PavlosthathisA. RozziW. T. M. SandersH. Siegrist and V. A. Vavilin, The IWA anaerobic digestion model no 1 (ADM1), Water Sci Technol., 45 (2002), 66-73.  doi: 10.2166/wst.2002.0292.  Google Scholar

[5]

B. BenyahiaT. SariB. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, J. Proc. Control, 22 (2012), 1008-1019.  doi: 10.1016/j.jprocont.2012.04.012.  Google Scholar

[6]

O. BernardZ. Hadj-SadokD. DochainA. Genovesi and J-P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnol. Bioeng., 75 (2001), 424-438.  doi: 10.1002/bit.10036.  Google Scholar

[7]

Y. DaoudN. AbdellatifT. Sari and J. Harmand, Steady state analysis of a syntrophic model: The effect of a new input substrate concentration, Math. Model. Nat. Phenom., 13 (2018), 1-22.  doi: 10.1051/mmnp/2018037.  Google Scholar

[8]

P. De LeenheerD. Angeli and E. D. Sontag, Crowding effects promote coexistence in the chemostat, J. Math. Anal. Appl., 319 (2006), 48-60.  doi: 10.1016/j.jmaa.2006.02.036.  Google Scholar

[9]

M. Dellal and B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete & Continuous Dyn. Syst. - B, (2020). doi: 10.3934/dcdsb.2020156.  Google Scholar

[10]

M. DellalM. Lakrib and T. Sari, The operating diagram of a model of two competitors in a chemostat with an external inhibitor, Math. Biosci., 302 (2018), 27-45.  doi: 10.1016/j.mbs.2018.05.004.  Google Scholar

[11]

N. Dimitrova and M. Krastanov, Nonlinear stabilizing control of an uncertain bioprocess model, Int. J. Appl. Math. Comput. Sci., 19 (2009), 441-454.  doi: 10.2478/v10006-009-0036-0.  Google Scholar

[12]

M. El HajjiF. Mazenc and J. Harmand, A mathematical study of a syntrophic relationship of a model of anaerobic digestion process, Math. Biosci. Eng., 7 (2010), 641-656.  doi: 10.3934/mbe.2010.7.641.  Google Scholar

[13]

M. El Hajji, N. Chorfi and M. Jleli, Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat, Electron. J. Differ. Eq., 2017, Paper No. 255, 13 pp. https://ejde.math.txstate.edu/Volumes/2017/255/elhajji.pdf.  Google Scholar

[14]

R. Fekih-Salem, N. Abdellatif, T. Sari and J. Harmand, Analyse mathématique d'un modèle de digestion anaérobie à trois étapes, {ARIMA Rev.}, 17 (2014), 53–71. http://arima.inria.fr/017/017003.html.  Google Scholar

[15]

R. Fekih-SalemC. Lobry and T. Sari, A density-dependent model of competition for one resource in the chemostat, Math. Biosci., 286 (2017), 104-122.  doi: 10.1016/j.mbs.2017.02.007.  Google Scholar

[16]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493.  doi: 10.1126/science.6767274.  Google Scholar

[17]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2017.  Google Scholar

[18]

P. A. Hoskisson and G. Hobbs, Continuous culture–making a comeback?, Microbiology, 151 (2005), 3153-3159.  doi: 10.1099/mic.0.27924-0.  Google Scholar

[19]

S.-B. HsuC. A. Klausmeier and C.-J. Lin, Analysis of a model of two parallel food chains, Discrete & Continuous Dyn. Syst. - B, 12 (2009), 337-359.  doi: 10.3934/dcdsb.2009.12.337.  Google Scholar

[20]

J. Monod, La technique de culture continue: Théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184–204 doi: 10.1016/B978-0-12-460482-7.50023-3.  Google Scholar

[21]

S. Nouaoura, N. Abdellatif, R. Fekih-Salem and T. Sari, Mathematical analysis of a three-tiered model of anaerobic digestion, preprint (2020), hal-02540350. https://hal.archives-ouvertes.fr/hal-02540350. Google Scholar

[22]

A. Rapaport and M. Veruete, A new proof of the competitive exclusion principle in the chemostat, Discrete & Continuous Dyn. Syst. - B, 24 (2019), 3755-3764.  doi: 10.3934/dcdsb.2018314.  Google Scholar

[23]

T. SariM. El Hajji and J. Harmand, The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat, Math. Biosci. Eng., 9 (2012), 627-645.  doi: 10.3934/mbe.2012.9.627.  Google Scholar

[24]

T. Sari and J. Harmand, A model of a syntrophic relationship between two microbial species in a chemostat including maintenance, Math. Biosci., 275 (2016), 1-9.  doi: 10.1016/j.mbs.2016.02.008.  Google Scholar

[25]

T. Sari and M. J. Wade, Generalised approach to modelling a three-tiered microbial food-web, Math. Biosci., 291 (2017), 21-37.  doi: 10.1016/j.mbs.2017.07.005.  Google Scholar

[26]

M. SbarciogM. Loccufier and E. Noldus, Determination of appropriate operating strategies for anaerobic digestion systems, Biochem. Eng. J., 51 (2010), 180-188.  doi: 10.1016/j.bej.2010.06.016.  Google Scholar

[27] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[28]

S. SobieszekM. J. Wade and G. S. K. Wolkowicz, Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Math. Biosci. Eng., 17 (2020), 7045-7073.  doi: 10.3934/mbe.2020363.  Google Scholar

[29]

M. J. Wade, Not Just Numbers: Mathematical Modelling and Its Contribution to Anaerobic Digestion Processes, Processes, 8 (2020), 888. doi: 10.3390/pr8080888.  Google Scholar

[30]

M. J. WadeJ. HarmandB. BenyahiaT. BouchezS. ChaillouB. CloezJ.-J. GodonB. Moussa BoudjemaaA. RapaportT. SariR. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecol. Modell., 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002.  Google Scholar

[31]

M. J. WadeR. W. PattinsonN. G. Parker and J. Dolfing, Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web', J. Theor. Biol., 389 (2016), 171-186.  doi: 10.1016/j.jtbi.2015.10.032.  Google Scholar

[32]

M. Weedermann, Analysis of a model for the effects of an external toxin on anaerobic digestion, Math. Biosci. Eng., 9 (2012), 445-459.  doi: 10.3934/mbe.2012.9.445.  Google Scholar

[33]

G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci., 93 (1989), 249-268.  doi: 10.1016/0025-5564(89)90025-4.  Google Scholar

[34]

A. XuJ. DolfingT. P. CurtisG. Montague and E. Martin, Maintenance affects the stability of a two-tiered microbial 'food chain'?, J. Theor. Biol., 276 (2011), 35-41.  doi: 10.1016/j.jtbi.2011.01.026.  Google Scholar

Figure 1.  Curve of the function $ \Psi(.,D) $, where $ s_2^{*1} $ and $ s_2^{*2} $ are the solutions of equation $ \Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} $
Figure 3.  (a) Magnification of saddle-node bifurcation at $ S_{\rm ch}^{\rm in} = \sigma_2 $ and the transcritical bifurcation at $ S_{\rm ch}^{\rm in} = \sigma_5 $ when $ S_{\rm ch}^{\rm in}\in\left[0.006,0.02 \right] $. (b) Magnification of the appearance and disappearance of stable limit cycles when $ S_{\rm ch}^{\rm in}\in[0.0294,0.0302] $
Figure 2.  (a) Projection of the $ \omega $-limit set in variable $ X_{\rm ch} $ as a function of $ S_{\rm ch}^{\rm in}\in[0,0.05] $ (b) A magnification of the transcritical bifurcations occurring at $ \sigma_1 $, $ \sigma_3 $ and $ \sigma_4 $ when $ S_{\rm ch}^{\rm in}\in[0,0.015] $
Figure 6.  Curve of the function $ y = F\left(S_{\rm ch}^{\rm in}\right) $ showing that $ F\left(S_{\rm ch}^{\rm in}\right)<0.0013 $, for all $ S_{\rm ch}^{\rm in}>\sigma_1 $
Figure 4.  Case $ S_{\rm ch}^{\rm in} = 0.029639\in(\sigma^*,\sigma_6) $: bi-stability of the limit cycle (in red) and SS3
Figure 5.  Case $ S_{\rm ch}^{\rm in} = 0.029639\in(\sigma^*,\sigma_6) $: trajectories of $ X_{\rm ch} $ corresponding to those in Fig. 4 showing the sustained oscillations in yellow (a) and blue (b) or the convergence to SS3 in green (c)
Figure 7.  The green line of equation $ y = YS_{\rm ch}^{\rm in} $ is above the red and blue curves of the functions $ M_0\left(D,s_2^{\ast i}\right)+M_1\left(D,s_2^{\ast i}\right) $, $ i = 1,2 $
Figure 8.  (a) Curve of the function $ \phi_4\left(S_{\rm ch}^{\rm in}\right) $ for $ S_{\rm ch}^{\rm in}>\sigma_5 $ and the solution $ \sigma_6 $ of equation $ \phi_4\left(S_{\rm ch}^{\rm in}\right) = 0 $. (b) A magnification for $ S_{\rm ch}^{\rm in}\in (\sigma_5,0.034) $
Figure 9.  Three eigenvalues of the matrix $ \mathbf{J_1} $ evaluated at SS6 as a function of $ S_{\rm ch}^{\rm in} $. Real part of the pair of eigenvalues $ \lambda_{2,3} $ for $ S_{\rm ch}^{\rm in} \in (\sigma^\star,0.05] $ where $ \sigma^\star = 0.018 $.
Table 1.  Notations, intervals and auxiliary functions
Definition
$ s_i = M_i(y,s_2) $
$ i = 0,1 $
Let $ s_2\geqslant 0 $. $ s_i = M_i(y,s_2) $ is the unique solution of $ \mu_i(s_i,s_2) = y $, for all $ 0\leqslant y<\mu_i(+\infty,s_2) $
$ s_2 = M_2(y) $$ s_2 = M_2(y) $ is the unique solution of $ \mu_2(s_2) = y $, for all $ 0\leqslant y<\mu_2(+\infty) $
$ s_2 = M_3(s_0,z) $Let $ s_0\geqslant 0 $. $ s_2 = M_3(s_0,z) $ is the unique solution of $ \mu_0(s_0,s_2) = z $, for all $ 0\leqslant z <\mu_0(s_0,+\infty) $
$ s_2^i = s_2^i(D) $
$ i = 0,1 $
$ s_2^i = s_2^i(D) $ is the unique solution of $ \mu_i\left(+\infty,s_2\right) = D+a_i $, for all $ D+a_0\!<\!\mu_0(+\infty,+\infty) $, $ \mu_1(+\infty,+\infty)\!<\!D+a_1<\mu_1(+\infty,0) $, resp.
$ I_1 $, $ I_2 $$ I_1 = \left\{D \geqslant 0: s_2^0<s_2^1\right\} $, $ I_2 = \left\{D \in I_1: s_2^0<M_2(D+a_2)<s_2^1\right\} $
$ \Psi(s_2,D) $$ \Psi\left(s_2,D\right) = (1-\omega)M_0(D+a_0,s_2)+M_1(D+a_1,s_2)+s_2 $, for all $ D\in I_1 $ and $ s_2^0<s_2<s_2^1 $
$ \phi_1(D) $$ \phi_1(D) = \inf_{s_2^0<s_2<s_2^1} \Psi(s_2,D) $, for all $ D\in I_1 $
$ \phi_2(D) $$ \phi_2(D) = \Psi\left(M_2(D+a_2),D\right) $, for all $ D\in I_2 $
$ \phi_3(D) $$ \phi_3(D) = \frac{\partial\Psi}{\partial s_2}\left(M_2(D+a_2),D\right) $, for all $ D\in I_2 $
$ J_0 $, $ J_1 $$ J_0 = \left(\max\left(0,s^{\rm in}_0-{s^{\rm in}_2}/{\omega}\right),s^{\rm in}_0\right) $, $ J_1 = \left( 0,s_1^{\rm in}\right) $
$ \psi_0(s_0) $$ \psi_0(s_0) = \mu _0\left(s_0,s_2^{\rm in}-\omega \left(s_0^{\rm in}-s_0\right) \right) $, for all $ s_0\geqslant \max\left(0,s^{\rm in}_0-{s^{\rm in}_2}/{\omega}\right) $
$ \psi_1(s_1) $$ \psi_1(s_1) = \mu _1\left(s_1,s_2^{\rm in}+s_1^{\rm in}-s_1\right) $, for all $ s_1\in \left[0,s^{\rm in}_1+{s^{\rm in}_2}\right] $
$ \varphi_i(D) $
$ i = 0,1 $
$ \varphi_i(D)=M_i\left(D+a_i,M_2(D+a_2) \right) $, resp., for all, $ D\in\left\{D\geqslant 0 : s_2^0<M_2(D+a_2)\right\} $, $ D\in\left\{D\geqslant 0 : M_2(D+a_2)<s_2^1\right\} $
Definition
$ s_i = M_i(y,s_2) $
$ i = 0,1 $
Let $ s_2\geqslant 0 $. $ s_i = M_i(y,s_2) $ is the unique solution of $ \mu_i(s_i,s_2) = y $, for all $ 0\leqslant y<\mu_i(+\infty,s_2) $
$ s_2 = M_2(y) $$ s_2 = M_2(y) $ is the unique solution of $ \mu_2(s_2) = y $, for all $ 0\leqslant y<\mu_2(+\infty) $
$ s_2 = M_3(s_0,z) $Let $ s_0\geqslant 0 $. $ s_2 = M_3(s_0,z) $ is the unique solution of $ \mu_0(s_0,s_2) = z $, for all $ 0\leqslant z <\mu_0(s_0,+\infty) $
$ s_2^i = s_2^i(D) $
$ i = 0,1 $
$ s_2^i = s_2^i(D) $ is the unique solution of $ \mu_i\left(+\infty,s_2\right) = D+a_i $, for all $ D+a_0\!<\!\mu_0(+\infty,+\infty) $, $ \mu_1(+\infty,+\infty)\!<\!D+a_1<\mu_1(+\infty,0) $, resp.
$ I_1 $, $ I_2 $$ I_1 = \left\{D \geqslant 0: s_2^0<s_2^1\right\} $, $ I_2 = \left\{D \in I_1: s_2^0<M_2(D+a_2)<s_2^1\right\} $
$ \Psi(s_2,D) $$ \Psi\left(s_2,D\right) = (1-\omega)M_0(D+a_0,s_2)+M_1(D+a_1,s_2)+s_2 $, for all $ D\in I_1 $ and $ s_2^0<s_2<s_2^1 $
$ \phi_1(D) $$ \phi_1(D) = \inf_{s_2^0<s_2<s_2^1} \Psi(s_2,D) $, for all $ D\in I_1 $
$ \phi_2(D) $$ \phi_2(D) = \Psi\left(M_2(D+a_2),D\right) $, for all $ D\in I_2 $
$ \phi_3(D) $$ \phi_3(D) = \frac{\partial\Psi}{\partial s_2}\left(M_2(D+a_2),D\right) $, for all $ D\in I_2 $
$ J_0 $, $ J_1 $$ J_0 = \left(\max\left(0,s^{\rm in}_0-{s^{\rm in}_2}/{\omega}\right),s^{\rm in}_0\right) $, $ J_1 = \left( 0,s_1^{\rm in}\right) $
$ \psi_0(s_0) $$ \psi_0(s_0) = \mu _0\left(s_0,s_2^{\rm in}-\omega \left(s_0^{\rm in}-s_0\right) \right) $, for all $ s_0\geqslant \max\left(0,s^{\rm in}_0-{s^{\rm in}_2}/{\omega}\right) $
$ \psi_1(s_1) $$ \psi_1(s_1) = \mu _1\left(s_1,s_2^{\rm in}+s_1^{\rm in}-s_1\right) $, for all $ s_1\in \left[0,s^{\rm in}_1+{s^{\rm in}_2}\right] $
$ \varphi_i(D) $
$ i = 0,1 $
$ \varphi_i(D)=M_i\left(D+a_i,M_2(D+a_2) \right) $, resp., for all, $ D\in\left\{D\geqslant 0 : s_2^0<M_2(D+a_2)\right\} $, $ D\in\left\{D\geqslant 0 : M_2(D+a_2)<s_2^1\right\} $
Table 2.  Steady states of (2). All functions are defined in Table 1
$ s_0 $, $ s_1 $, $ s_2 $ and $ x_0 $, $ x_1 $, $ x_2 $ components
SS1$ s_0 = s_0^{{\rm in}} $, $ s_1 = s_1^{\rm in} $, $ s_2 = s_2^{\rm in} $ and $ x_0 = 0 $, $ x_1 = 0 $, $ x_2 = 0 $
SS2$ s_0 = s_0^{{\rm in}} $, $ s_1 = s_1^{\rm in} $, $ s_2 = M_2(D+a_2) $ and $ x_0 = 0 $, $ x_1 = 0 $, $ x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2\right) $
SS3$ s_1 = s^{\rm in}_1+s^{\rm in}_0-s_0 $ and $ s_2 = s^{\rm in}_2-\omega\left(s^{\rm in}_0-s_0\right) $, where $ s_0 $ is a solution of $ \psi_0(s_0) = D+a_0 $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right) $, $ x_1 = 0 $, $ x_2 = 0 $
SS4$ s_0 = M_0(D+a_0,s_2) $ and $ s_1 = M_1(D+a_1,s_2) $, where $ s_2 $ is a solution of $ \Psi(s_2,D) = (1-\omega)s_0^{{\rm in}}+s_1^{{\rm in}}+s_2^{{\rm in}} $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{{\rm in}}-s_0\right) $, $ x_1 = \frac{D}{D+a_1}\left(s_0^{{\rm in}}-s_0+s_1^{{\rm in}}-s_1\right) $, $ x_2 = 0 $
SS5$ s_0 = \varphi_0(D) $, $ s_1 = s_1^{\rm in} + s_0^{\rm in} - s_0 $, $ s_2 = M_2(D+a_2) $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right) $, $ x_1 = 0 $, $ x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2-\omega \left(s_0^{\rm in}-s_0 \right)\right) $
SS6$ s_0 = \varphi_0(D) $, $ s_1 = \varphi_1(D) $, $ s_2 = M_2(D+a_2) $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right) $, $ x_1 = \frac{D}{D+a_1}\left(s_0^{\rm in}-s_0+s_1^{\rm in}-s_1\right) $, $ x_2 = \frac{D}{D+a_2}\left((1-\omega)(s_0^{{\rm in}}-s_0)+s_1^{{\rm in}}-s_1+s_2^{{\rm in}}-s_2\right) $
SS7$ s_0 = s_0^{{\rm in}} $ and $ s_2 = s_2^{\rm in}+s_1^{\rm in}-s_1 $, where $ s_1 $ is a solution of $ \psi_1(s_1) = D+a_1 $ and $ x_0 = 0 $, $ x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right) $, $ x_2 = 0 $
SS8$ s_0 = s_0^{{\rm in}} $, $ s_1 = \varphi_1(D) $, $ s_2 = M_2(D+a_2) $ and $ x_0 = 0 $, $ x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right) $, $ x_2 = \frac{D}{D+a_2}\left(s_1^{\rm in}-s_1+s_2^{\rm in}-s_2\right) $
$ s_0 $, $ s_1 $, $ s_2 $ and $ x_0 $, $ x_1 $, $ x_2 $ components
SS1$ s_0 = s_0^{{\rm in}} $, $ s_1 = s_1^{\rm in} $, $ s_2 = s_2^{\rm in} $ and $ x_0 = 0 $, $ x_1 = 0 $, $ x_2 = 0 $
SS2$ s_0 = s_0^{{\rm in}} $, $ s_1 = s_1^{\rm in} $, $ s_2 = M_2(D+a_2) $ and $ x_0 = 0 $, $ x_1 = 0 $, $ x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2\right) $
SS3$ s_1 = s^{\rm in}_1+s^{\rm in}_0-s_0 $ and $ s_2 = s^{\rm in}_2-\omega\left(s^{\rm in}_0-s_0\right) $, where $ s_0 $ is a solution of $ \psi_0(s_0) = D+a_0 $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right) $, $ x_1 = 0 $, $ x_2 = 0 $
SS4$ s_0 = M_0(D+a_0,s_2) $ and $ s_1 = M_1(D+a_1,s_2) $, where $ s_2 $ is a solution of $ \Psi(s_2,D) = (1-\omega)s_0^{{\rm in}}+s_1^{{\rm in}}+s_2^{{\rm in}} $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{{\rm in}}-s_0\right) $, $ x_1 = \frac{D}{D+a_1}\left(s_0^{{\rm in}}-s_0+s_1^{{\rm in}}-s_1\right) $, $ x_2 = 0 $
SS5$ s_0 = \varphi_0(D) $, $ s_1 = s_1^{\rm in} + s_0^{\rm in} - s_0 $, $ s_2 = M_2(D+a_2) $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right) $, $ x_1 = 0 $, $ x_2 = \frac{D}{D+a_2}\left(s_2^{\rm in}-s_2-\omega \left(s_0^{\rm in}-s_0 \right)\right) $
SS6$ s_0 = \varphi_0(D) $, $ s_1 = \varphi_1(D) $, $ s_2 = M_2(D+a_2) $ and $ x_0 = \frac{D}{D+a_0}\left(s_0^{\rm in}-s_0\right) $, $ x_1 = \frac{D}{D+a_1}\left(s_0^{\rm in}-s_0+s_1^{\rm in}-s_1\right) $, $ x_2 = \frac{D}{D+a_2}\left((1-\omega)(s_0^{{\rm in}}-s_0)+s_1^{{\rm in}}-s_1+s_2^{{\rm in}}-s_2\right) $
SS7$ s_0 = s_0^{{\rm in}} $ and $ s_2 = s_2^{\rm in}+s_1^{\rm in}-s_1 $, where $ s_1 $ is a solution of $ \psi_1(s_1) = D+a_1 $ and $ x_0 = 0 $, $ x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right) $, $ x_2 = 0 $
SS8$ s_0 = s_0^{{\rm in}} $, $ s_1 = \varphi_1(D) $, $ s_2 = M_2(D+a_2) $ and $ x_0 = 0 $, $ x_1 = \frac{D}{D+a_1}\left(s_1^{\rm in}-s_1\right) $, $ x_2 = \frac{D}{D+a_2}\left(s_1^{\rm in}-s_1+s_2^{\rm in}-s_2\right) $
Table 3.  Existence conditions of steady states of (2). All functions are given in Table 1
Existence conditions
SS1 Always exists
SS2$ \mu_2\left(s_2^{\rm in}\right)>D+a_2 $
SS3$ \mu_0\left(s^{\rm in}_0,s^{\rm in}_2\right)>D+a_0 $
SS4$ (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}\geqslant \phi_1(D) $, $ s_0^{\rm in} > M_0(D+a_0,s_2) $, $ s_0^{\rm in} + s_1^{\rm in} >M_0(D+a_0,s_2) + M_1(D+a_1,s_2) $ with $ s_2 $ solution of equation $ \Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} $
SS5$ s_0^{\rm in}>\varphi_0(D) $, $ s_2^{\rm in}-\omega s_0^{\rm in}>M_2(D+a_2)-\omega \varphi_0(D) $
SS6$ (1-\omega)s^{\rm in}_0+s_1^{\rm in}+s_2^{\rm in}>\phi_2(D) $, $ s_0^{\rm in}>\varphi_0(D) $, $ s_0^{\rm in}+s_1^{\rm in} >\varphi_0(D)+\varphi_1(D) $
SS7$ \mu_1\left(s^{\rm in}_1,s^{\rm in}_2\right)>D+a_1 $
SS8$ s^{\rm in}_1>\varphi_1(D) $, $ s_1^{\rm in}+s_2^{\rm in}>\varphi_1(D)+M_2(D+a_2) $
Existence conditions
SS1 Always exists
SS2$ \mu_2\left(s_2^{\rm in}\right)>D+a_2 $
SS3$ \mu_0\left(s^{\rm in}_0,s^{\rm in}_2\right)>D+a_0 $
SS4$ (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in}\geqslant \phi_1(D) $, $ s_0^{\rm in} > M_0(D+a_0,s_2) $, $ s_0^{\rm in} + s_1^{\rm in} >M_0(D+a_0,s_2) + M_1(D+a_1,s_2) $ with $ s_2 $ solution of equation $ \Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} $
SS5$ s_0^{\rm in}>\varphi_0(D) $, $ s_2^{\rm in}-\omega s_0^{\rm in}>M_2(D+a_2)-\omega \varphi_0(D) $
SS6$ (1-\omega)s^{\rm in}_0+s_1^{\rm in}+s_2^{\rm in}>\phi_2(D) $, $ s_0^{\rm in}>\varphi_0(D) $, $ s_0^{\rm in}+s_1^{\rm in} >\varphi_0(D)+\varphi_1(D) $
SS7$ \mu_1\left(s^{\rm in}_1,s^{\rm in}_2\right)>D+a_1 $
SS8$ s^{\rm in}_1>\varphi_1(D) $, $ s_1^{\rm in}+s_2^{\rm in}>\varphi_1(D)+M_2(D+a_2) $
Table 4.  Maintenance free case: the stability conditions of steady states of (2). All functions are given in Table 1 with $ a_0 = a_1 = a_2 = 0 $, while $ \phi_4 $ is defined by (4)
Stability conditions
SS1 $ \mu_0\left(s_0^{\rm in},s_2^{\rm in}\right)<D $, $ \mu_1\left(s_1^{\rm in},s_2^{\rm in}\right)<D $, $ \mu_2\left(s_2^{\rm in}\right)<D $
SS2$ s_0^{\rm in} < \varphi_0(D) $, $ s_1^{\rm in} < \varphi_1(D) $
SS3$ \mu_1\left(s_1^{\rm in}+s_0^{\rm in}-s_0,s_2^{\rm in}-\omega \left(s_0^{\rm in}-s_0\right)\right)<D $, $ s_2^{\rm in}-\omega s_0^{\rm in} < M_2(D) -\omega \varphi_0(D) $, with $ s_0 $ solution of equation $ \psi_0(s_0) = D $
SS4$ (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} < \phi_2(D) $, $ \frac{\partial\Psi}{\partial s_2} \left( s_2,D \right) >0 $, $ \phi_3(D)>0 $, with $ s_2 $ solution of equation $ \Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} $
SS5$ s_0^{\rm in}+s_1^{\rm in} < \varphi_0(D) + \varphi_1(D) $
SS6$ \phi_3(D)\geqslant 0 $, or $ \phi_3(D)<0 $ and $ \phi_4\left(D,s^{\rm in}_0,s_1^{\rm in},s_2^{\rm in}\right)>0 $
SS7$ s_1^{\rm in}+s_2^{\rm in}< M_3 \left( s_0^{\rm in},D\right) + M_1\left(D,M_3\left( s_0^{\rm in},D\right)\right) $, $ s_1^{\rm in}+s_2^{\rm in}<M_2(D)+\varphi_1(D) $
SS8$ s_0^{\rm in} < \varphi_0(D) $
Stability conditions
SS1 $ \mu_0\left(s_0^{\rm in},s_2^{\rm in}\right)<D $, $ \mu_1\left(s_1^{\rm in},s_2^{\rm in}\right)<D $, $ \mu_2\left(s_2^{\rm in}\right)<D $
SS2$ s_0^{\rm in} < \varphi_0(D) $, $ s_1^{\rm in} < \varphi_1(D) $
SS3$ \mu_1\left(s_1^{\rm in}+s_0^{\rm in}-s_0,s_2^{\rm in}-\omega \left(s_0^{\rm in}-s_0\right)\right)<D $, $ s_2^{\rm in}-\omega s_0^{\rm in} < M_2(D) -\omega \varphi_0(D) $, with $ s_0 $ solution of equation $ \psi_0(s_0) = D $
SS4$ (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} < \phi_2(D) $, $ \frac{\partial\Psi}{\partial s_2} \left( s_2,D \right) >0 $, $ \phi_3(D)>0 $, with $ s_2 $ solution of equation $ \Psi(s_2,D) = (1-\omega)s_0^{\rm in}+s_1^{\rm in}+s_2^{\rm in} $
SS5$ s_0^{\rm in}+s_1^{\rm in} < \varphi_0(D) + \varphi_1(D) $
SS6$ \phi_3(D)\geqslant 0 $, or $ \phi_3(D)<0 $ and $ \phi_4\left(D,s^{\rm in}_0,s_1^{\rm in},s_2^{\rm in}\right)>0 $
SS7$ s_1^{\rm in}+s_2^{\rm in}< M_3 \left( s_0^{\rm in},D\right) + M_1\left(D,M_3\left( s_0^{\rm in},D\right)\right) $, $ s_1^{\rm in}+s_2^{\rm in}<M_2(D)+\varphi_1(D) $
SS8$ s_0^{\rm in} < \varphi_0(D) $
Table 10.  Nominal parameter values. Units are expressed in Chemical Oxygen Demand (COD)
ParameterWade et al. [31]Unit
$ k_{m,\rm{ch}} $
$ k_{m,\rm{ph}} $
$ k_{m,\rm{H_{2}}} $
29
26
35
$ \rm{kgCOD_{S}/kgCOD_{X}/d} $
$ K_{S,\rm{ch}} $
$ K_{S,\rm{H_{2},c}} $
$ K_{S,\rm{ph}} $
$ K_{I,\rm{H_{2}}} $
$ K_{S,\rm{H_{2}}} $
0.053
$ 10^{-6} $
0.302
3.5$ \!\times\!10^{-6} $
2.5$ \!\times\!10^{-5} $
$ \rm{kgCOD/m^{3}} $
$ Y_{\rm{ch}} $
$ Y_{\rm{ph}} $
$ Y_{\rm{H_{2}}} $
0.019
0.04
0.06
$ \rm{kgCOD_X/kgCOD_S} $
ParameterWade et al. [31]Unit
$ k_{m,\rm{ch}} $
$ k_{m,\rm{ph}} $
$ k_{m,\rm{H_{2}}} $
29
26
35
$ \rm{kgCOD_{S}/kgCOD_{X}/d} $
$ K_{S,\rm{ch}} $
$ K_{S,\rm{H_{2},c}} $
$ K_{S,\rm{ph}} $
$ K_{I,\rm{H_{2}}} $
$ K_{S,\rm{H_{2}}} $
0.053
$ 10^{-6} $
0.302
3.5$ \!\times\!10^{-6} $
2.5$ \!\times\!10^{-5} $
$ \rm{kgCOD/m^{3}} $
$ Y_{\rm{ch}} $
$ Y_{\rm{ph}} $
$ Y_{\rm{H_{2}}} $
0.019
0.04
0.06
$ \rm{kgCOD_X/kgCOD_S} $
Table 11.  Auxiliary functions in the case of growth functions given by (9)
Auxiliary function Definition domain
$ M_0(y,s_2)=\frac{yK_0(L_0+s_2)}{m_0s_2-y(L_0+s_2)} $ $ 0\leqslant y<\frac{m_0s_2}{L_0+s_2} $
$ M_1(y,s_2)=\frac{yK_1(K_I+s_2)}{m_1K_I-y(K_I+s_2)} $ $ 0\leqslant y<\frac{m_1K_I}{K_I+s_2} $
$ M_2(y)=\frac{yK_2}{m_2-y} $ $ 0\leqslant y<m_2 $
$ M_3(s_0,z)=\frac{zL_0(K_0+s_0)}{m_0s_0-z(K_0+s_0)} $ $ 0\leqslant z<\frac{m_0s_0}{K_0+s_0} $
$ s_2^0(D)=\frac{L_0(D+a_0)}{m_0-D-a_0} $ $ D+a_0<m_0 $
$ s_2^1(D)=\frac{K_I(m_1-D-a_1)}{D+a_1} $ $ D+a_1<m_1 $
$ \Psi(s_2,D) $ =$ (1-\omega)\frac{(D+a_0)K_0(L_0+s_2)}{m_0s_2-(D+a_0)(L_0+s_2)} $ + $ \frac{(D+a_1)K_1(K_I+s_2)}{m_1K_I-(D+a_1)(K_I+s_2)}+s_2 $ $ \left\{D\in I_1 : s_2^0<s_2<s_2^1 \right\} $
$ \psi_0(s_0)=\frac{m_0s_0\left(s_2^{\rm in}-\omega\left(s_0^{\rm in}-s_0\right)\right)}{(K_0+s_0)\left(L_0+s_2^{\rm in}-\omega\left(s_0^{\rm in}-s_0\right)\right)} $$ s_0\in\left[\max\left(0,s^{\rm in}_0\!-\!{s^{\rm in}_2}/{\omega}\right),+\infty\right) $
$ \psi_1(s_1)= \frac{m_1s_1K_I}{\left(K_1+s_1\right)\left(K_I+s_2^{\rm in}+s_1^{\rm in}-s_1\right)} $$ s_1\in \left[0,s^{\rm in}_1+{s^{\rm in}_2}\right] $
Auxiliary function Definition domain
$ M_0(y,s_2)=\frac{yK_0(L_0+s_2)}{m_0s_2-y(L_0+s_2)} $ $ 0\leqslant y<\frac{m_0s_2}{L_0+s_2} $
$ M_1(y,s_2)=\frac{yK_1(K_I+s_2)}{m_1K_I-y(K_I+s_2)} $ $ 0\leqslant y<\frac{m_1K_I}{K_I+s_2} $
$ M_2(y)=\frac{yK_2}{m_2-y} $ $ 0\leqslant y<m_2 $
$ M_3(s_0,z)=\frac{zL_0(K_0+s_0)}{m_0s_0-z(K_0+s_0)} $ $ 0\leqslant z<\frac{m_0s_0}{K_0+s_0} $
$ s_2^0(D)=\frac{L_0(D+a_0)}{m_0-D-a_0} $ $ D+a_0<m_0 $
$ s_2^1(D)=\frac{K_I(m_1-D-a_1)}{D+a_1} $ $ D+a_1<m_1 $
$ \Psi(s_2,D) $ =$ (1-\omega)\frac{(D+a_0)K_0(L_0+s_2)}{m_0s_2-(D+a_0)(L_0+s_2)} $ + $ \frac{(D+a_1)K_1(K_I+s_2)}{m_1K_I-(D+a_1)(K_I+s_2)}+s_2 $ $ \left\{D\in I_1 : s_2^0<s_2<s_2^1 \right\} $
$ \psi_0(s_0)=\frac{m_0s_0\left(s_2^{\rm in}-\omega\left(s_0^{\rm in}-s_0\right)\right)}{(K_0+s_0)\left(L_0+s_2^{\rm in}-\omega\left(s_0^{\rm in}-s_0\right)\right)} $$ s_0\in\left[\max\left(0,s^{\rm in}_0\!-\!{s^{\rm in}_2}/{\omega}\right),+\infty\right) $
$ \psi_1(s_1)= \frac{m_1s_1K_I}{\left(K_1+s_1\right)\left(K_I+s_2^{\rm in}+s_1^{\rm in}-s_1\right)} $$ s_1\in \left[0,s^{\rm in}_1+{s^{\rm in}_2}\right] $
Table 5.  Definitions of the critical values of $ \sigma_i $, $ i = 1,\ldots,6 $
Definition Value
$ \sigma_1=M_0\left(D,S_{\rm H_2}^{\rm in} \right)/Y $ $ 0.001017 $
$ \sigma_2=(\phi_1(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y) $ $ 0.009159 $
$ \sigma_3=\varphi_0(D)/Y $ $ 0.010846 $
$ \sigma_4=(S_{\rm H_2}^{\rm in}-M_2(D)+\omega\varphi_0(D))/(\omega Y) $ $ 0.011191 $
$ \sigma_5=(\phi_2(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y) $ $ 0.016575 $
$ \sigma_6 $ is the solution of equation $ \phi_4(S_{\rm ch}^{\rm in})=0 $ $ 0.029877 $
Definition Value
$ \sigma_1=M_0\left(D,S_{\rm H_2}^{\rm in} \right)/Y $ $ 0.001017 $
$ \sigma_2=(\phi_1(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y) $ $ 0.009159 $
$ \sigma_3=\varphi_0(D)/Y $ $ 0.010846 $
$ \sigma_4=(S_{\rm H_2}^{\rm in}-M_2(D)+\omega\varphi_0(D))/(\omega Y) $ $ 0.011191 $
$ \sigma_5=(\phi_2(D)-S_{\rm H_2}^{\rm in})/((1-\omega)Y) $ $ 0.016575 $
$ \sigma_6 $ is the solution of equation $ \phi_4(S_{\rm ch}^{\rm in})=0 $ $ 0.029877 $
Table 6.  Existence and stability of steady states, with respect to $ S_{\rm ch}^{\rm in} $. The letter S (resp. U) means that the corresponding steady state is LES (resp. unstable). No letter means that the steady state does not exist
Interval of $ S_{\rm ch}^{\rm in} $ SS1 SS2 SS3 $ \rm SS4^1 $ $ \rm SS4^2 $ SS5 SS6
$ (0,\sigma_1) $ U S
$ (\sigma_1,\sigma_2) $ U S U
$ (\sigma_2,\sigma_3) $ U S U U U
$ (\sigma_3,\sigma_4) $ U U U U U S
$ (\sigma_4,\sigma_5) $ U U S U U
$ (\sigma_5,\sigma_6) $ U U S U U U
$ (\sigma_6,+\infty) $ U U S U U S
Interval of $ S_{\rm ch}^{\rm in} $ SS1 SS2 SS3 $ \rm SS4^1 $ $ \rm SS4^2 $ SS5 SS6
$ (0,\sigma_1) $ U S
$ (\sigma_1,\sigma_2) $ U S U
$ (\sigma_2,\sigma_3) $ U S U U U
$ (\sigma_3,\sigma_4) $ U U U U U S
$ (\sigma_4,\sigma_5) $ U U S U U
$ (\sigma_5,\sigma_6) $ U U S U U U
$ (\sigma_6,+\infty) $ U U S U U S
Table 7.  Nature of the bifurcations corresponding to the critical values of $ \sigma_i $, $ i = 1,\ldots,6 $, defined in Table 5. There exists also a critical value $ \sigma^*\simeq 0.029638 $ corresponding to the value of $ S_{\rm ch}^{\rm in} $ where the stable limit cycle disappears when $ S_{\rm ch}^{\rm in} $ is decreasing
Type of the bifurcation
$ \sigma_1 $ Transcritical bifurcation of SS1 and SS3
$ \sigma_2 $ Saddle-node bifurcation of $ \rm SS4^1 $ and $ \rm SS4^2 $
$ \sigma_3 $ Transcritical bifurcation of SS2 and SS5
$ \sigma_4 $ Transcritical bifurcation of SS3 and SS5
$ \sigma_5 $ Transcritical bifurcation of $ \rm SS4^1 $ and SS6
$ \sigma^* $ Disappearance of the stable limit cycle
$ \sigma_6 $ Supercritical Hopf bifurcation
Type of the bifurcation
$ \sigma_1 $ Transcritical bifurcation of SS1 and SS3
$ \sigma_2 $ Saddle-node bifurcation of $ \rm SS4^1 $ and $ \rm SS4^2 $
$ \sigma_3 $ Transcritical bifurcation of SS2 and SS5
$ \sigma_4 $ Transcritical bifurcation of SS3 and SS5
$ \sigma_5 $ Transcritical bifurcation of $ \rm SS4^1 $ and SS6
$ \sigma^* $ Disappearance of the stable limit cycle
$ \sigma_6 $ Supercritical Hopf bifurcation
Table 8.  Colors used in Figs. 2 and 3. The solid (resp. dashed) lines are used for LES (resp. unstable) steady states
SS1 SS2 SS3 $ \rm SS4^1 $ $ \rm SS4^2 $ SS5 SS6
Red Blue Purple Dark Green Magenta Green Cyan
SS1 SS2 SS3 $ \rm SS4^1 $ $ \rm SS4^2 $ SS5 SS6
Red Blue Purple Dark Green Magenta Green Cyan
Table 9.  Existence and local stability conditions of steady states of (1), when $ S_{\rm ph}^{\rm in} = 0 $ and $ k_{\rm dec, ch} = k_{\rm dec, ph} = k_{\rm dec, H_2} = 0 $. All functions are given in Tables 1 and 11, while $ \phi_4 $ and $ \mu_i $ are given by (4) and (9)
Existence conditions Stability conditions
SS1 Always exists $ \mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)<D $, $ \mu_2\left(S_{\rm H_2}^{\rm in}\right)<D $
SS2$ \mu_2\left(S_{\rm H_2}^{\rm in}\right)>D $$ YS_{\rm ch}^{\rm in} < \varphi_0(D) $
SS3$ \mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)>D $$ \mu_1\left(YS_{\rm ch}^{\rm in}-s_0,S_{\rm H_2}^{\rm in}-\omega \left(YS_{\rm ch}^{\rm in}-s_0\right)\right)<D $
$ S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in}<M_2(D)-\omega \varphi_0(D) $
with $ s_0 $ solution of $ \psi_0(s_0) = D $
SS4$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}\geqslant \phi_1(D) $, $ YS_{\rm ch}^{\rm in} > M_0(D,s_2)+ M_1(D,s_2) $ with $ s_2 $ solution of $ \Psi(s_2,D) = (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in} $$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}< \phi_2(D) $, $ \frac{\partial\Psi}{\partial s_2} \left(s_2,D \right) >0 $, $ \phi_3(D) >0 $
SS5$ YS_{\rm ch}^{\rm in}>\varphi_0(D) $, $ S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in} \! > \! M_2(D)-\omega \varphi_0(D) $$ YS_{\rm ch}^{\rm in}<\varphi_0(D)+\varphi_1(D) $
SS6$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}>\phi_2(D) $, $ YS_{\rm ch}^{\rm in}>\varphi_0(D)+\varphi_1(D) $$ \phi_3(D)\!\geqslant\! 0 $ or $ \phi_3(D)\!<\!0 $ and $ \phi_4(D,S_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in})\!>\!0 $
Existence conditions Stability conditions
SS1 Always exists $ \mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)<D $, $ \mu_2\left(S_{\rm H_2}^{\rm in}\right)<D $
SS2$ \mu_2\left(S_{\rm H_2}^{\rm in}\right)>D $$ YS_{\rm ch}^{\rm in} < \varphi_0(D) $
SS3$ \mu_0\left(YS_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in}\right)>D $$ \mu_1\left(YS_{\rm ch}^{\rm in}-s_0,S_{\rm H_2}^{\rm in}-\omega \left(YS_{\rm ch}^{\rm in}-s_0\right)\right)<D $
$ S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in}<M_2(D)-\omega \varphi_0(D) $
with $ s_0 $ solution of $ \psi_0(s_0) = D $
SS4$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}\geqslant \phi_1(D) $, $ YS_{\rm ch}^{\rm in} > M_0(D,s_2)+ M_1(D,s_2) $ with $ s_2 $ solution of $ \Psi(s_2,D) = (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in} $$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}< \phi_2(D) $, $ \frac{\partial\Psi}{\partial s_2} \left(s_2,D \right) >0 $, $ \phi_3(D) >0 $
SS5$ YS_{\rm ch}^{\rm in}>\varphi_0(D) $, $ S_{\rm H_2}^{\rm in}-\omega YS_{\rm ch}^{\rm in} \! > \! M_2(D)-\omega \varphi_0(D) $$ YS_{\rm ch}^{\rm in}<\varphi_0(D)+\varphi_1(D) $
SS6$ (1-\omega)YS_{\rm ch}^{\rm in}+S_{\rm H_2}^{\rm in}>\phi_2(D) $, $ YS_{\rm ch}^{\rm in}>\varphi_0(D)+\varphi_1(D) $$ \phi_3(D)\!\geqslant\! 0 $ or $ \phi_3(D)\!<\!0 $ and $ \phi_4(D,S_{\rm ch}^{\rm in},S_{\rm H_2}^{\rm in})\!>\!0 $
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