American Institute of Mathematical Sciences

Advanced
June  2020, 25(6): 2093-2120. 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  June 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (6) : 2093-2120. 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.

[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.

[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. 

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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. 

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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) $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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 $
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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 $
Table Options

Download as excel

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 $
Table Options

Download as excel

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) $
Table Options

Download as excel

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) $
[1]

Jianquan Li, Zuren Feng, Juan Zhang, Jie Lou. A competition model of the chemostat with an external inhibitor. Mathematical Biosciences & Engineering, 2006, 3 (1) : 111-123. doi: 10.3934/mbe.2006.3.111
[2]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
[3]

Georg Hetzer, Wenxian Shen. Two species competition with an inhibitor involved. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 39-57. doi: 10.3934/dcds.2005.12.39
[4]

Jifa Jiang, Fensidi Tang. The complete classification on a model of two species competition with an inhibitor. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 659-672. doi: 10.3934/dcds.2008.20.659
[5]

Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1279-1297. doi: 10.3934/cpaa.2013.12.1279
[6]

Nahla Abdellatif, Radhouane Fekih-Salem, Tewfik Sari. Competition for a single resource and coexistence of several species in the chemostat. Mathematical Biosciences & Engineering, 2016, 13 (4) : 631-652. doi: 10.3934/mbe.2016012
[7]

Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621
[8]

Hua Nie, Feng-Bin Wang. Competition for one nutrient with recycling and allelopathy in an unstirred chemostat. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2129-2155. doi: 10.3934/dcdsb.2015.20.2129
[9]

Mohamed Dellal, Bachir Bar, Mustapha Lakrib. A competition model in the chemostat with allelopathy and substrate inhibition. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2025-2050. doi: 10.3934/dcdsb.2021120
[10]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319
[11]

Hua Nie, Sze-bi Hsu, Jianhua Wu. A competition model with dynamically allocated toxin production in the unstirred chemostat. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1373-1404. doi: 10.3934/cpaa.2017066
[12]

Xiaoqing He, Sze-Bi Hsu, Feng-Bin Wang. A periodic-parabolic Droop model for two species competition in an unstirred chemostat. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4427-4451. doi: 10.3934/dcds.2020185
[13]

Sze-Bi Hsu, Cheng-Che Li. A discrete-delayed model with plasmid-bearing, plasmid-free competition in a chemostat. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 699-718. doi: 10.3934/dcdsb.2005.5.699
[14]

Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098
[15]

Zhiqi Lu. Global stability for a chemostat-type model with delayed nutrient recycling. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 663-670. doi: 10.3934/dcdsb.2004.4.663
[16]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323
[17]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[18]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002
[19]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[20]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (285)
  • HTML views (403)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy