American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2019203

The operating diagram for a model of competition in a chemostat with an external lethal inhibitor

Bachir Bar 1, and  Tewfik Sari 2,,

1. 

Université AbouBakr Belkaid, LSDA, Tlemcen, Algérie
2. 

ITAP, Univ Montpellier, Irstea, Montpellier SupAgro, Montpellier, France

* Corresponding author

Received  January 2019 Revised  May 2019 Published  September 2019

Fund Project: The authors are supported by the France-Algeria Partnership Tassili project 15MDU949 and the Euro-Mediterranean research network TREASURE (http://www.inra.fr/treasure)

The inhibition is an important phenomenon, which promotes the stable coexistence of species, in the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external lethal inhibitor. The model is a four-dimensional system of ordinary differential equations. We give a complete analysis for the existence and local stability of all steady states. We describe the bifurcation diagram which gives the behavior of the system with respect to the operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. This diagram, is very useful to understand the model from both the mathematical and biological points of view.

Keywords: Chemostat, competition, inhibitor, stability, operating diagram.
Mathematics Subject Classification: Primary: 34C23, 34D20; Secondary: 92B05, 92D25.
Citation: Bachir Bar, Tewfik Sari. The operating diagram for a model of competition in a chemostat with an external lethal inhibitor. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019203
References:
[1]

N. Abdellatif, R. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Mathematical Biosciences and Engineering, 13 (2016), 631-652.  doi: 10.3934/mbe.2016012.  Google Scholar

[2]

M. J. De Freitas and A. G. Fredrickson, Inhibition as a factor in the maintenance of the diversity of microbial ecosystems, Journal of General Microbiology, 106 (1978), 307-320.  doi: 10.1099/00221287-106-2-307.  Google Scholar

[3]

P. de Leenheer, B. T. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248.   Google Scholar

[4]

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

[5]

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

[6]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Chemostat and bioprocesses set. Vol. 1. ISTE, London, John Wiley & Sons, Inc., Hoboken, NJ, 2017.  Google Scholar

[7]

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

[8]

S. B. Hsu, S. Hubbell and P. Waltman, A mathematical model for single nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[9]

S. B. Hsu, Y.-S. Li and P. Waltman, Competition in the presence of a lethal external inhibitor, Mathematical Biosciences, 167 (2000), 177-199.  doi: 10.1016/S0025-5564(00)00030-4.  Google Scholar

[10]

S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM Journal on Applied Mathematics, 52 (1992), 528-540.  doi: 10.1137/0152029.  Google Scholar

[11]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

[12]

W. S. Keeran, P. de Leenheer and S. S. Pilyugin, Feedback-mediated coexistence and oscillations in the chemostat, Discrete and Continuous Dynamical Systems-B, 9 (2008), 321-351.  doi: 10.3934/dcdsb.2008.9.321.  Google Scholar

[13]

R. E. Lenski and S. E. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics, Journal of Theoretical Biology, 122 (1986), 83-93.  doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar

[14]

B. T. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates, SIAM Journal on Applied Mathematics, 59 (1999), 411-422.  doi: 10.1137/S003613999631100X.  Google Scholar

[15]

J. Q. Li, Z. R. Feng, J. Zhang and J. Lou, A competition model of the chemostat with an external inhibitor, Mathematical Biosciences and Engineering, 3 (2006), 111-123.  doi: 10.3934/mbe.2006.3.111.  Google Scholar

[16]

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

[17]

S. Pavlou, Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.  doi: 10.1016/S0168-1656(99)00011-5.  Google Scholar

[18]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827.  Google Scholar

[19]

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

[20]

H. Smith and B. Tang, Competition in the gradostat: The role of the communication rate, Journal of Mathematical Biology, 27 (1989), 139-165.  doi: 10.1007/BF00276100.  Google Scholar

[21]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[22]

H. L. Smith and X.-Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete and Continuous Dynamical Systems-B, 1 (2001), 183-191.  doi: 10.3934/dcdsb.2001.1.183.  Google Scholar

[23]

D. V. Vayenas and S. Pavlou, Chaotic dynamics of a microbial system of coupled food chains, Ecological Modelling, 136 (2001), 285-295.  doi: 10.1016/S0304-3800(00)00437-3.  Google Scholar

[24]

M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J.-J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecological Modelling, 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002.  Google Scholar

[25]

M. J. Wade, R. W. Pattinson, N. G. Parker and J. Dolfing, Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web', Journal of Theoretical Biology, 389 (2016), 171-186.  doi: 10.1016/j.jtbi.2015.10.032.  Google Scholar

[26]

M. Weedermann, G. Seo and G. S. K. Wolkowicz, Mathematical model of anaerobic digestion in a chemostat: Effects of syntrophy and inhibition, Journal of Biological Dynamics, 7 (2013), 59-85.  doi: 10.1080/17513758.2012.755573.  Google Scholar

[27]

G. S. K. Wolkowicz and Z. Q. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.  doi: 10.1137/0152012.  Google Scholar

show all references

References:
[1]

N. Abdellatif, R. Fekih-Salem and T. Sari, Competition for a single resource and coexistence of several species in the chemostat, Mathematical Biosciences and Engineering, 13 (2016), 631-652.  doi: 10.3934/mbe.2016012.  Google Scholar

[2]

M. J. De Freitas and A. G. Fredrickson, Inhibition as a factor in the maintenance of the diversity of microbial ecosystems, Journal of General Microbiology, 106 (1978), 307-320.  doi: 10.1099/00221287-106-2-307.  Google Scholar

[3]

P. de Leenheer, B. T. Li and H. L. Smith, Competition in the chemostat: Some remarks, Canadian Applied Mathematics Quarterly, 11 (2003), 229-248.   Google Scholar

[4]

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

[5]

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

[6]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Chemostat and bioprocesses set. Vol. 1. ISTE, London, John Wiley & Sons, Inc., Hoboken, NJ, 2017.  Google Scholar

[7]

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

[8]

S. B. Hsu, S. Hubbell and P. Waltman, A mathematical model for single nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[9]

S. B. Hsu, Y.-S. Li and P. Waltman, Competition in the presence of a lethal external inhibitor, Mathematical Biosciences, 167 (2000), 177-199.  doi: 10.1016/S0025-5564(00)00030-4.  Google Scholar

[10]

S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM Journal on Applied Mathematics, 52 (1992), 528-540.  doi: 10.1137/0152029.  Google Scholar

[11]

S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Mathematical Biosciences, 187 (2004), 53-91.  doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

[12]

W. S. Keeran, P. de Leenheer and S. S. Pilyugin, Feedback-mediated coexistence and oscillations in the chemostat, Discrete and Continuous Dynamical Systems-B, 9 (2008), 321-351.  doi: 10.3934/dcdsb.2008.9.321.  Google Scholar

[13]

R. E. Lenski and S. E. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics, Journal of Theoretical Biology, 122 (1986), 83-93.  doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar

[14]

B. T. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates, SIAM Journal on Applied Mathematics, 59 (1999), 411-422.  doi: 10.1137/S003613999631100X.  Google Scholar

[15]

J. Q. Li, Z. R. Feng, J. Zhang and J. Lou, A competition model of the chemostat with an external inhibitor, Mathematical Biosciences and Engineering, 3 (2006), 111-123.  doi: 10.3934/mbe.2006.3.111.  Google Scholar

[16]

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

[17]

S. Pavlou, Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.  doi: 10.1016/S0168-1656(99)00011-5.  Google Scholar

[18]

T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Mathematical Biosciences and Engineering, 8 (2011), 827-840.  doi: 10.3934/mbe.2011.8.827.  Google Scholar

[19]

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

[20]

H. Smith and B. Tang, Competition in the gradostat: The role of the communication rate, Journal of Mathematical Biology, 27 (1989), 139-165.  doi: 10.1007/BF00276100.  Google Scholar

[21]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[22]

H. L. Smith and X.-Q. Zhao, Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete and Continuous Dynamical Systems-B, 1 (2001), 183-191.  doi: 10.3934/dcdsb.2001.1.183.  Google Scholar

[23]

D. V. Vayenas and S. Pavlou, Chaotic dynamics of a microbial system of coupled food chains, Ecological Modelling, 136 (2001), 285-295.  doi: 10.1016/S0304-3800(00)00437-3.  Google Scholar

[24]

M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, J.-J. Godon, B. Moussa Boudjemaa, A. Rapaport, T. Sari, R. Arditi and C. Lobry, Perspectives in mathematical modelling for microbial ecology, Ecological Modelling, 321 (2016), 64-74.  doi: 10.1016/j.ecolmodel.2015.11.002.  Google Scholar

[25]

M. J. Wade, R. W. Pattinson, N. G. Parker and J. Dolfing, Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web', Journal of Theoretical Biology, 389 (2016), 171-186.  doi: 10.1016/j.jtbi.2015.10.032.  Google Scholar

[26]

M. Weedermann, G. Seo and G. S. K. Wolkowicz, Mathematical model of anaerobic digestion in a chemostat: Effects of syntrophy and inhibition, Journal of Biological Dynamics, 7 (2013), 59-85.  doi: 10.1080/17513758.2012.755573.  Google Scholar

[27]

G. S. K. Wolkowicz and Z. Q. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates, SIAM Journal on Applied Mathematics, 52 (1992), 222-233.  doi: 10.1137/0152012.  Google Scholar

Figure 2.  Illustrative operating diagrams for $ D $ fixed: The curves $ \Gamma_i $, $ i = 0\cdots 5 $ defined in the Table 3 divide the operating plane $ (p^0,S^0) $ into at most nine regions labeled $ \mathcal J_0 $, $ \mathcal J_1 $, $ \mathcal J_2 $, $ \mathcal J_3 $, $ \mathcal J_4 $, $ \mathcal J_5^S $, $ \mathcal J_5^U $, $ \mathcal J_6^S $ and $ \mathcal J_6^U $. Some of the regions may be empty. The existence and stability of the equilibrium points in the regions of these diagrams are shown in Table 4
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Figure 3.  The biological parameters values are given in Table 5, Case 1. $ J_c = \varUpsilon_0 $: therefore $ E_c $ is LES whenever it exists. The operating diagram for $ D = 1 $ shows that $ \left(p^0 = 1,S^0 = 1\right)\in\mathcal{J}_6^S $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4
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Figure 5.  The biological parameters values are given in Table 5, Case 2. (a): The operating diagram for $ D = 1 $ shows that $ p^0 = S^0 = 1 $ belongs to $ \mathcal{J}_6^S $. (b): The operating diagram for $ D = 2.2 $. The existence and stability of the equilibrium points in the regions of these diagrams are shown in Table 4
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Figure 7.  The biological parameters values are given in Table 5, Case 3. (a): The operating diagram for $ D = 1 $. $ (b) $: A zoom showing the instability of $ E_c $ when $ p^0 = S^0 = D = 1 $. The existence and stability of the equilibrium points in the regions of these diagrams are shown in Table 4
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Figure 8.  The biological parameters values are given in Table 5, Case 3. The operating diagram for $ D = 2.2 $. (a): $ \mathcal{J}_6^U $ is unbounded; (d): a zoom near the origin showing the regions $ \mathcal{J}_1 $, $ \mathcal{J}_2 $, $ \mathcal{J}_3 $ and $ \mathcal{J}_5^S $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4
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Figure 10.  The biological parameters values are given in Table 5, Case 4. The operating diagram for (a): $ D = 0.01 $. (b): $ D = D_1 = 0.013 $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4
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Figure 12.  The biological parameters values are given in Table 5, Case 5. The operating diagram for $ D = 1.26 $. (a): The full operating diagram. (b): A zoom showing that $ \mathcal{J}_4 $ is nonempty. The existence and stability of $ E_0 $, $ E_1 $, $ E_2 $ and $ E_c $ in the regions of these diagrams are shown in Table 4
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Figure 13.  The biological parameters values are given in Table 5, Case 3. The operating diagram in the $ (S^0,D) $-plane for $ p^0 = 1 $. (a): The region $ \mathcal{J}_6^U $ is unbounded. (b): A zoom showing the instability of $ E_c $ for $ S^0 = p^0 = D = 1 $. The existence and stability of the equilibrium points are shown in Table 4
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Figure 14.  The biological parameters values are given in Table 5, Case 3. The operating diagram in the $ (p^0,D) $-plane for $ S^0 = 1 $, showing the instability of $ E_c $ for $ S^0 = p^0 = D = 1 $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4
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Figure 15.  The biological parameters values are given in Table 5, Case 3, and $ \beta_1 = \beta_2 = 100 $. The operating diagram in the $ (p^0,D) $-plane for $ S^0 = 1 $. (a): The full diagram. (b): A zoom showing the regions near the $ D $ axis showing the regions $ \mathcal{J}_1 $, $ \mathcal{J}_3 $ and $ \mathcal{J}_5^S $. The existence and stability of the equilibrium points in the regions of this diagram are shown in Table 4
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Figure 1.  (a): Definitions of $ \lambda_1 = \lambda_1(D) $, $ \lambda^- = \lambda^-(D,p^0,S^0) $, $ \lambda^+ = \lambda^+(D,p^0) $ and $ \lambda_2 = \lambda_2(D) $. (b): Definition of $ p^* = p^*(D,p^0,S^{0}) $ satisfying $ W(p^*,D,p^0) = \beta_2\left(S^0-\lambda_2(D)\right) $
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Figure 4.  The biological parameters values are given in Table 1, Case 2. The regions $ \varUpsilon_0 $ and $ \varUpsilon_2 $ of $ J_c $, and the definitions of $ p_1(D) $ and $ p_2(D) $ for $ D_1<D<D_2 $: $ D_1\simeq 1.83 $, $ D_2\simeq 2.65 $, $ p_1(2.2)\simeq 0.65 $, $ p_2(2.2)\simeq 1.04 $
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Figure 6.  The biological parameters values are given in Table 1, Case 3. The regions $ \varUpsilon_0 $, $ \varUpsilon_1 $ and $ \varUpsilon_2 $ and the definitions of $ p_1(D) $, $ p_2(D) $ for $ D_1<D<D_2 $, and $ p_3(D) $, $ p_4(D) $ for $ D_3<D<D_4 $, where $ D_1\simeq 0.63 $, $ D_2\simeq 1.27 $, $ D_3\simeq2.82 $ and $ D_4\simeq 3.59 $. The figure shows the values $ p_1(2.2)\approx 0.47 $, $ p_2(2.2)\approx 4.17 $, $ p_3(2.2)\approx 0.77 $, $ p_4(2.2)\approx 2.71 $
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Figure 9.  The biological parameters values are given in Table 5, Case 4. In red curve of equation $ a_3 = 0 $, in blue, a component of the curve of equation $ \Delta = 0 $. In black, the curve $ p^0 = p_c(D) $ (a): The full regions $ \varUpsilon_0 $, $ \varUpsilon_1 $ and $ \varUpsilon_2 $. (b): a zoom showing the values $ p_1 = p_1(D) $, $ p_2 = p_2(D) $ $ p_3 = p_3(D) $ and $ p_4 = p_4(D) $ for $ D = 0.013 $. For the clarity of the figures, the points $ p_1(D) $ and $ p_2(D) $, for $ D = 0.01 $, are not depicted on the figure
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Figure 11.  The biological parameters values are given in Table 5, Case 5. In red curve of equation $ a_3 = 0 $, in blue, a component of the curve of equation $ \Delta = 0 $. In black, the curve $ p^0 = p_c(D) $ (a): The full regions $ \varUpsilon_0 $, $ \varUpsilon_1 $ and $ \varUpsilon_2 $. (b): A zoom showing the values $ D_1 $ and $ D_2 $ and $ p_1 $ and $ p_2 $ corresponding to $ D = 1.26 $
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Figure 16.  The plots of curves $ p^0 = p_c(D) $, in black, $ \Delta = 0 $, in blue, and $ a_1a_2 = a_0a_3 $ in magenta. The biological parameters values are given in Table 5, Case 2
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Figure 17.  The curve $ p^0 = p_c(D) $ is plotted in black. (a): The plots of curves $ \Delta = 0 $ ($ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $, in blue), $ a_3 = 0 $ ($ \mathcal{C}_2 $, in red). (b): The plots of curves $ a_1a_2 = a_0a_3 $ ($ \mathcal{C}_3\cup\mathcal{C}_4 $, in magenta) and $ a_2 = 0 $ ($ \mathcal{C}_5 $, in cyan). The biological parameters are given in Table 5, Case 3
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Figure 18.  The curve $ p^0 = p_c(D) $ is plotted in black. (a): The plots of curves $ \Delta = 0 $ ($ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $, in blue), $ a_3 = 0 $ ($ \mathcal{C}_2 $, in red), $ a_1a_2 = a_0a_3 $ ($ \mathcal{C}_3\cup\mathcal{C}_4 $, in magenta) and $ a_2 = 0 $ ($ \mathcal{C}_5 $, in cyan). (b): A zoom of the strip $ 0<p^0<1 $. The biological parameters are given in Table 5, Case 3
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Table 6.  Meanings and units of the variables and parameters of (1)
Meanings Units
$ S $, $ x $, $ y $, $ p $ Concentrations of substrate, species and inhibitor mass/volume
$ S^0 $, $ p^0 $ Input concentrations of substrate and inhibitor mass/volume
$ D $ Dilution rate 1/time
$ m_1 $, $ m_2 $ Maximal growth rates of the competitors 1/time
$ K_1 $, $ K_2 $ Half saturation constants of the competitors mass/volume
$ \delta $ Maximal growth rate of detoxification 1/time
$ K $ Half saturation constant of detoxification mass/volume
$ \gamma $ Lethal effect of $ p $ on $ x $ volume/mass
$ \beta_1 $, $ \beta_2 $ Growth yield coefficients dimensionless
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Meanings Units
$ S $, $ x $, $ y $, $ p $ Concentrations of substrate, species and inhibitor mass/volume
$ S^0 $, $ p^0 $ Input concentrations of substrate and inhibitor mass/volume
$ D $ Dilution rate 1/time
$ m_1 $, $ m_2 $ Maximal growth rates of the competitors 1/time
$ K_1 $, $ K_2 $ Half saturation constants of the competitors mass/volume
$ \delta $ Maximal growth rate of detoxification 1/time
$ K $ Half saturation constant of detoxification mass/volume
$ \gamma $ Lethal effect of $ p $ on $ x $ volume/mass
$ \beta_1 $, $ \beta_2 $ Growth yield coefficients dimensionless
Table 3.  Boundaries of the regions in the operating diagram. The color code is used in Figs. 2, 3, 5, 7, 8, 10, 12, 13, 14, 15
Boundary Color Equation in $ \left(p^0, S^0\right) $-plane, with $ D\in I_c $ fixed
$ \Gamma_1 $ blue Graph of $ S^0 =f_1^{-1}(D+\gamma p^0) $
$ \Gamma_2 $ black Horizontal line $ S^0 = \lambda_2(D) $
$ \Gamma_3 $ red Vertical line $ p^0 = p_c(D) $ and $ S^0> \lambda_2(D) $
$ \Gamma_4 $ cyan Oblique line $ S^0=\frac{D\left(p^0-p_c(D)\right)}{\beta_2g(p_c(D))}+\lambda_2(D) $ and $ p^0>p_c(D) $
$ \Gamma_5 $ green Curve of equation $ F_3(D,p^0,S^0)=0 $
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Boundary Color Equation in $ \left(p^0, S^0\right) $-plane, with $ D\in I_c $ fixed
$ \Gamma_1 $ blue Graph of $ S^0 =f_1^{-1}(D+\gamma p^0) $
$ \Gamma_2 $ black Horizontal line $ S^0 = \lambda_2(D) $
$ \Gamma_3 $ red Vertical line $ p^0 = p_c(D) $ and $ S^0> \lambda_2(D) $
$ \Gamma_4 $ cyan Oblique line $ S^0=\frac{D\left(p^0-p_c(D)\right)}{\beta_2g(p_c(D))}+\lambda_2(D) $ and $ p^0>p_c(D) $
$ \Gamma_5 $ green Curve of equation $ F_3(D,p^0,S^0)=0 $
Table 4.  Existence and stability of equilibrium points in the regions of the operating diagram, shown in Figs. 2, 3, 5, 7, 8, 10, 12, 13, 14, 15
Regions $ \mathcal J_0 $ $ \mathcal J_1 $ $ \mathcal J_2 $ $ \mathcal J_3 $ $ \mathcal J_4 $ $ \mathcal J_{5}^S $ $ \mathcal J_{5}^U $ $ \mathcal J_{6}^S $ $ \mathcal J_{6}^U $
$ E_0 $ S U U U U U U U U
$ E_1 $ S S U U U
$ E_2 $ S U S U U U U
$ E_{c} $ S U S U
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Regions $ \mathcal J_0 $ $ \mathcal J_1 $ $ \mathcal J_2 $ $ \mathcal J_3 $ $ \mathcal J_4 $ $ \mathcal J_{5}^S $ $ \mathcal J_{5}^U $ $ \mathcal J_{6}^S $ $ \mathcal J_{6}^U $
$ E_0 $ S U U U U U U U U
$ E_1 $ S S U U U
$ E_2 $ S U S U U U U
$ E_{c} $ S U S U
Table 5.  Biological parameters values used in the numerical computations shown in the figures. The yields are $ \beta_1 = \beta_2 = 1 $, excepted for Fig. 15 in which $ \beta_1 = \beta_2 = 100 $. The last column of the table shows the value of $ \overline{D} $ such that $ E_c $ exists for $ D\in(0,\overline{D}) $
Case $ m_1 $ $ m_2 $ $ K_1 $ $ K_2 $ $ \delta $ $ K $ $ \gamma $ Figures $ \overline{D} $
1 4.0 5.0 0.3 1.0 3.0 0.3 4.0 24 3.54
2 4.0 5.0 0.06 1.0 5.0 1.3 4.0 25, 5, 16 3.94
3 4.0 5.0 0.03 1.0 5.0 1.3 4.0 26, 7, 8, 13, 14, 15, 17, 18 3.97
4 1.7 2 0.4 0.9 15 0.03 0.025 9, 10 1.46
5 4.0 5.0 0.03 1.0 0.5 1.3 4.0 11, 12 3.97
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Case $ m_1 $ $ m_2 $ $ K_1 $ $ K_2 $ $ \delta $ $ K $ $ \gamma $ Figures $ \overline{D} $
1 4.0 5.0 0.3 1.0 3.0 0.3 4.0 24 3.54
2 4.0 5.0 0.06 1.0 5.0 1.3 4.0 25, 5, 16 3.94
3 4.0 5.0 0.03 1.0 5.0 1.3 4.0 26, 7, 8, 13, 14, 15, 17, 18 3.97
4 1.7 2 0.4 0.9 15 0.03 0.025 9, 10 1.46
5 4.0 5.0 0.03 1.0 0.5 1.3 4.0 11, 12 3.97
Table 1.  Existence and stability of equilibrium points $ E_0 $, $ E_1 $, $ E_2 $ and $ E_c $ of (3), given in Prop. 1. Here, $ \lambda_2 $ and $ \lambda^+ $ are given by (5), $ \lambda^- $ is given by (7) and $ A_1 $, $ A_2 $, $ A_3 $ and $ A_4 $ are given by (17)
Existence Local exponential stability
$ E_0 $ Always $ \min(\lambda^+,\lambda_2)>S^0 $
$ E_1 $ $ \lambda^+<S^0 $ $ \lambda^+<\lambda_2 $
$ E_2 $ $ \lambda_2<S^0 $ $ \lambda_2<\lambda^- $
$ E_{c} $ $ \lambda^-<\lambda_2<\min(\lambda^+,S^0) $ $ A_3(A_1A_2-A_3)> A_1^2A_4 $
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Existence Local exponential stability
$ E_0 $ Always $ \min(\lambda^+,\lambda_2)>S^0 $
$ E_1 $ $ \lambda^+<S^0 $ $ \lambda^+<\lambda_2 $
$ E_2 $ $ \lambda_2<S^0 $ $ \lambda_2<\lambda^- $
$ E_{c} $ $ \lambda^-<\lambda_2<\min(\lambda^+,S^0) $ $ A_3(A_1A_2-A_3)> A_1^2A_4 $
Table 2.  Existence and stability of equilibrium points of (3), with respect to the operating parameters $ D $, $ S^0 $ and $ p^0 $. The functions $ F_1 $, $ F_2 $, $ F_3 $ are defined by (19), (20), (21), respectively
Existence Local exponential stability
$ E_0 $ Always $ D>\max(f_1(S^0)-\gamma p^0,f_2(S^0)) $
$ E_1 $ $ D<f_1(S^0)-\gamma p^0 $ $ D<F_1(D,p^0) $
$ E_2 $ $ D<f_2(S^0) $ $ S^0<F_2(D,p^0) $
$ E_{c} $ $ D>F_1(D,p^0) $ & $ S^0>F_2(D,p^0) $ $ F_3(D,p^0,S^0)>0 $
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Existence Local exponential stability
$ E_0 $ Always $ D>\max(f_1(S^0)-\gamma p^0,f_2(S^0)) $
$ E_1 $ $ D<f_1(S^0)-\gamma p^0 $ $ D<F_1(D,p^0) $
$ E_2 $ $ D<f_2(S^0) $ $ S^0<F_2(D,p^0) $
$ E_{c} $ $ D>F_1(D,p^0) $ & $ S^0>F_2(D,p^0) $ $ F_3(D,p^0,S^0)>0 $
Table 7.  The signs of functions $ a_2 $, $ a_3 $, $ a_1a_2-a_0a_3 $ and $ \Delta $. Here $ \mathcal{A} = R_1\cup\mathcal{A}_1\cup R_2\cup\mathcal{A}_2 $
Function $<0 $ $ =0 $ $>0 $
$ \Delta $ $ Ext\left(\mathcal{C}_1\right)\setminus \mathcal{A} $ $ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $ $ Int\left(\mathcal{C}_1\right)\cup R_1\cup R_2 $
$ a_3 $ $ Int\left(\mathcal{C}_2\right) $ $ \mathcal{C}_2 $ $ Ext\left(\mathcal{C}_2\right) $
$ a_1a_2-a_0a_3 $ $ Int\left(\mathcal{C}_3\right)\cap Ext\left(\mathcal{C}_4\right) $ $ \mathcal{C}_3\cup\mathcal{C}_4 $ $ Ext\left(\mathcal{C}_3\right)\cup Int\left(\mathcal{C}_4\right) $
$ a_2 $ $ Int\left(\mathcal{C}_5\right) $ $ \mathcal{C}_5 $ $ Ext\left(\mathcal{C}_5\right) $
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Function $<0 $ $ =0 $ $>0 $
$ \Delta $ $ Ext\left(\mathcal{C}_1\right)\setminus \mathcal{A} $ $ \mathcal{C}_1\cup\mathcal{A}_1\cup\mathcal{A}_2 $ $ Int\left(\mathcal{C}_1\right)\cup R_1\cup R_2 $
$ a_3 $ $ Int\left(\mathcal{C}_2\right) $ $ \mathcal{C}_2 $ $ Ext\left(\mathcal{C}_2\right) $
$ a_1a_2-a_0a_3 $ $ Int\left(\mathcal{C}_3\right)\cap Ext\left(\mathcal{C}_4\right) $ $ \mathcal{C}_3\cup\mathcal{C}_4 $ $ Ext\left(\mathcal{C}_3\right)\cup Int\left(\mathcal{C}_4\right) $
$ a_2 $ $ Int\left(\mathcal{C}_5\right) $ $ \mathcal{C}_5 $ $ Ext\left(\mathcal{C}_5\right) $
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