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Bloch wave approach to almost periodic homogenization and approximations of effective coefficients
A competition model in the chemostat with allelopathy and substrate inhibition
a. | Ibn Khaldoun University, 14000 Tiaret, Algeria |
b. | Ecole Normale Supérieure, 27000 Mostaganem, Algeria |
c. | LDM, Djillali Liabès University, 22000 Sidi Bel Abbès, Algeria |
A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a super-critical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. 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] |
J. F. Andrews,
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.
doi: 10.1002/bit.260100602. |
[3] |
B. Bar and T. Sari,
The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete and Continuous Dynamical Systems–B, 25 (2020), 2093-2120.
doi: 10.3934/dcdsb.2019203. |
[4] |
G. J. Butler and G. S. K. Wolkowicz,
A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.
doi: 10.1137/0145006. |
[5] |
M. Dellal and B. Bar,
Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete and Continuous Dynamical Systems–B, 26 (2021), 1129-1148.
doi: 10.3934/dcdsb.2020156. |
[6] |
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. |
[7] |
R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari,
Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.
doi: 10.1016/j.jmaa.2012.07.055. |
[8] |
P. Fergola, M. Cerasuolo, A. Pollio, G. Pinto and M. Della Grecac.,
Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecological Modelling, 208 (2007), 205-214.
doi: 10.1016/j.ecolmodel.2007.05.024. |
[9] |
P. Fergola, J. Li and Z. Ma,
On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ricerche di Matematica, 60 (2011), 313-332.
doi: 10.1007/s11587-011-0108-y. |
[10] |
H. Fgaier, M. Kalmokoff, T. Ells and H. J. Eberl,
An allelopathy based model for the Listeria overgrowth phenomenon, Mathematical Biosciences, 247 (2014), 13-26.
doi: 10.1016/j.mbs.2013.10.008. |
[11] |
G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, (1934). |
[12] |
B. Haegeman and A. Rapaport,
How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, 2 (2008), 1-13.
doi: 10.1080/17513750801942537. |
[13] |
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. |
[14] |
G. Hardin,
The competitive exclusion principle, Science, 131 (1960), 1292-1297.
doi: 10.1126/science.131.3409.1292. |
[15] |
J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Vol. 1, ISTE, London, John Wiley and Sons, Inc. Hoboken, NJ, 2017. |
[16] |
J. Heßeler, J. K. Schmidt, U. Reichl and D. Flockerzi,
Coexistence in the chemostat as a result of metabolic by-products, Journal of Mathematical Biology, 53 (2006), 556-584.
doi: 10.1007/s00285-006-0012-3. |
[17] |
S. B. Hsu, S. Hubbell and P. Waltman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[18] |
S. B. Hsu, T. K. Luo and P. Waltman,
Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor, Journal of Mathematical Biology, 34 (1995), 225-238.
doi: 10.1007/BF00178774. |
[19] |
S. B. Hsu and P. Waltman,
Competition in the chemostat when one competitor produces a toxin, Japan Journal of Industrial and Applied Mathematics, 15 (1998), 471-490.
doi: 10.1007/BF03167323. |
[20] |
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 3$^rd$ edition, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[21] |
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. |
[22] |
R. E. Lenski and S. 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. |
[23] |
C. Lobry and J. Harmand,
A new hypothesis to explain the coexistence of $n$ species in the presence of a single resource, Comptes Rendus Biologies, 329 (2006), 40-46.
doi: 10.1016/j.crvi.2005.10.004. |
[24] |
I. P. Martines, H. V. Kojouharov and J. P. Grover,
A chemostat model of resource competition and allelopathy, Applied Mathematics and Computation, 215 (2009), 573-582.
doi: 10.1016/j.amc.2009.05.033. |
[25] |
S. Pavlou,
Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.
doi: 10.1016/S0168-1656(99)00011-5. |
[26] |
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. |
[27] |
M. Scheffer, S. Rinaldi, J. Huisman and F. J. Weissing,
Why plankton communities have no equilibrium: Solutions to the paradox, Hydrobiologia, 491 (2003), 9-18.
doi: 10.1023/A:1024404804748. |
[28] |
H. L. 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. |
[29] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() ![]() |
[30] |
S. Sobieszek, G. S. K. Wolkowicz and M. J. Wade,
Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Mathematical Biosciences and Engineering, 17 (2020), 7045-7073.
doi: 10.3934/mbe.2020363. |
[31] |
M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, 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. |
[32] |
M. Weedermann, G. Seo and G. 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. |
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] |
J. F. Andrews,
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnology and Bioengineering, 10 (1968), 707-723.
doi: 10.1002/bit.260100602. |
[3] |
B. Bar and T. Sari,
The operating diagram for a model of competition in a chemostat with an external lethal inhibitor, Discrete and Continuous Dynamical Systems–B, 25 (2020), 2093-2120.
doi: 10.3934/dcdsb.2019203. |
[4] |
G. J. Butler and G. S. K. Wolkowicz,
A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM Journal on Applied Mathematics, 45 (1985), 138-151.
doi: 10.1137/0145006. |
[5] |
M. Dellal and B. Bar,
Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete and Continuous Dynamical Systems–B, 26 (2021), 1129-1148.
doi: 10.3934/dcdsb.2020156. |
[6] |
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. |
[7] |
R. Fekih-Salem, J. Harmand, C. Lobry, A. Rapaport and T. Sari,
Extensions of the chemostat model with flocculation, Journal of Mathematical Analysis and Applications, 397 (2013), 292-306.
doi: 10.1016/j.jmaa.2012.07.055. |
[8] |
P. Fergola, M. Cerasuolo, A. Pollio, G. Pinto and M. Della Grecac.,
Allelopathy and competition between Chlorella vulgaris and Pseudokirchneriella subcapitata: Experiments and mathematical model, Ecological Modelling, 208 (2007), 205-214.
doi: 10.1016/j.ecolmodel.2007.05.024. |
[9] |
P. Fergola, J. Li and Z. Ma,
On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment, Ricerche di Matematica, 60 (2011), 313-332.
doi: 10.1007/s11587-011-0108-y. |
[10] |
H. Fgaier, M. Kalmokoff, T. Ells and H. J. Eberl,
An allelopathy based model for the Listeria overgrowth phenomenon, Mathematical Biosciences, 247 (2014), 13-26.
doi: 10.1016/j.mbs.2013.10.008. |
[11] |
G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, (1934). |
[12] |
B. Haegeman and A. Rapaport,
How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, 2 (2008), 1-13.
doi: 10.1080/17513750801942537. |
[13] |
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. |
[14] |
G. Hardin,
The competitive exclusion principle, Science, 131 (1960), 1292-1297.
doi: 10.1126/science.131.3409.1292. |
[15] |
J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, Vol. 1, ISTE, London, John Wiley and Sons, Inc. Hoboken, NJ, 2017. |
[16] |
J. Heßeler, J. K. Schmidt, U. Reichl and D. Flockerzi,
Coexistence in the chemostat as a result of metabolic by-products, Journal of Mathematical Biology, 53 (2006), 556-584.
doi: 10.1007/s00285-006-0012-3. |
[17] |
S. B. Hsu, S. Hubbell and P. Waltman,
A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.
doi: 10.1137/0132030. |
[18] |
S. B. Hsu, T. K. Luo and P. Waltman,
Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor, Journal of Mathematical Biology, 34 (1995), 225-238.
doi: 10.1007/BF00178774. |
[19] |
S. B. Hsu and P. Waltman,
Competition in the chemostat when one competitor produces a toxin, Japan Journal of Industrial and Applied Mathematics, 15 (1998), 471-490.
doi: 10.1007/BF03167323. |
[20] |
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 3$^rd$ edition, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[21] |
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. |
[22] |
R. E. Lenski and S. 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. |
[23] |
C. Lobry and J. Harmand,
A new hypothesis to explain the coexistence of $n$ species in the presence of a single resource, Comptes Rendus Biologies, 329 (2006), 40-46.
doi: 10.1016/j.crvi.2005.10.004. |
[24] |
I. P. Martines, H. V. Kojouharov and J. P. Grover,
A chemostat model of resource competition and allelopathy, Applied Mathematics and Computation, 215 (2009), 573-582.
doi: 10.1016/j.amc.2009.05.033. |
[25] |
S. Pavlou,
Computing operating diagrams of bioreactors, Journal of Biotechnology, 71 (1999), 7-16.
doi: 10.1016/S0168-1656(99)00011-5. |
[26] |
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. |
[27] |
M. Scheffer, S. Rinaldi, J. Huisman and F. J. Weissing,
Why plankton communities have no equilibrium: Solutions to the paradox, Hydrobiologia, 491 (2003), 9-18.
doi: 10.1023/A:1024404804748. |
[28] |
H. L. 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. |
[29] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() ![]() |
[30] |
S. Sobieszek, G. S. K. Wolkowicz and M. J. Wade,
Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow, Mathematical Biosciences and Engineering, 17 (2020), 7045-7073.
doi: 10.3934/mbe.2020363. |
[31] |
M. J. Wade, J. Harmand, B. Benyahia, T. Bouchez, S. Chaillou, B. Cloez, 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. |
[32] |
M. Weedermann, G. Seo and G. 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. |













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Case | Figs | |||||||
1 | 1.0 | 4.0 | 1.0 | 1.0 | 0.5 | 0.2 | 0.3 | 6, 7, 8, 9, 10 |
2 | 1.5 | 2.7 | 1.0 | 1.0 | 0.08 | 0.2 | 0.3 | 11, 12, 14, 15. |
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1 | 1.0 | 4.0 | 1.0 | 1.0 | 0.5 | 0.2 | 0.3 | 6, 7, 8, 9, 10 |
2 | 1.5 | 2.7 | 1.0 | 1.0 | 0.08 | 0.2 | 0.3 | 11, 12, 14, 15. |
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