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A two-group age of infection epidemic model with periodic behavioral changes
The operating diagram for a model of competition in a chemostat with an external lethal inhibitor
1. | Université AbouBakr Belkaid, LSDA, Tlemcen, Algérie |
2. | ITAP, Univ Montpellier, Irstea, Montpellier SupAgro, Montpellier, France |
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.
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. |

















Meanings | Units | |
Concentrations of substrate, species and inhibitor | mass/volume | |
Input concentrations of substrate and inhibitor | mass/volume | |
Dilution rate | 1/time | |
Maximal growth rates of the competitors | 1/time | |
Half saturation constants of the competitors | mass/volume | |
Maximal growth rate of detoxification | 1/time | |
Half saturation constant of detoxification | mass/volume | |
Lethal effect of |
volume/mass | |
Growth yield coefficients | dimensionless |
Meanings | Units | |
Concentrations of substrate, species and inhibitor | mass/volume | |
Input concentrations of substrate and inhibitor | mass/volume | |
Dilution rate | 1/time | |
Maximal growth rates of the competitors | 1/time | |
Half saturation constants of the competitors | mass/volume | |
Maximal growth rate of detoxification | 1/time | |
Half saturation constant of detoxification | mass/volume | |
Lethal effect of |
volume/mass | |
Growth yield coefficients | dimensionless |
Boundary | Color | Equation in |
blue | Graph of |
|
black | Horizontal line |
|
red | Vertical line |
|
cyan | Oblique line |
|
green | Curve of equation |
Boundary | Color | Equation in |
blue | Graph of |
|
black | Horizontal line |
|
red | Vertical line |
|
cyan | Oblique line |
|
green | Curve of equation |
Regions | |||||||||
S | U | U | U | U | U | U | U | U | |
S | S | U | U | U | |||||
S | U | S | U | U | U | U | |||
S | U | S | U |
Regions | |||||||||
S | U | U | U | U | U | U | U | U | |
S | S | U | U | U | |||||
S | U | S | U | U | U | U | |||
S | U | S | U |
Case | Figures | ||||||||
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 |
Case | Figures | ||||||||
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 |
Existence | Local exponential stability | |
Always | ||
Existence | Local exponential stability | |
Always | ||
Existence | Local exponential stability | |
Always | ||
Existence | Local exponential stability | |
Always | ||
Function | |||
Function | |||
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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 |
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