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Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain
Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics
a. | School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, China |
b. | School of Computer Science and Network Security, Dongguan University of Technology, Dongguan, 523808, Guangdong, China |
c. | College of Finance and Statistics, Hunan University, Changsha, 410079, Hunan, China |
d. | Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada |
An inhibitory uptake function is incorporated into the discrete, size-structured nonlinear chemostat model developed by Arino et al. (Journal of Mathematical Biology, 45(2002)). Different from the model with a monotonically increasing uptake function, we show that the inhibitory kinetics can induce very complex dynamics including stable equilibria, cycles and chaos (via the period-doubling cascade). In particular, when the nutrient concentration in the input feed to the chemostat $ S^0 $ is larger than the upper break-even concentration value $ \mu $, the model exhibits three types of bistability allowing a stable equilibrium to coexist with another stable equilibrium, or a stable cycle or a chaotic attractor.
References:
[1] |
J. F. Andrews,
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioeng., 10 (1968), 707-723.
|
[2] |
J. Arino, J.-L. Gouze and A. Sciandra,
A discrete, size-structured model of phytoplankton growth in the chemostat, J. Math. Biol., 45 (2002), 313-336.
doi: 10.1007/s002850200160. |
[3] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Am. Nat., 115 (1980), 151-170.
doi: 10.1086/283553. |
[4] |
L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt,
Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229.
|
[5] |
B. Boon and H. Laudeuout,
Kinetics of nitrite oxidation by Nitrobacter winogradskyi, Biochem. J., 85 (1962), 440-447.
|
[6] |
A. W. Bush and A. E. Cook,
The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theor. Biol., 63 (1976), 385-395.
|
[7] |
G. J. Butler and G. S. K. Wolkowicz,
A mathematical model of the chemostat with a general class of functions describing nutrient outake, SIAM J. Appl. Math., 45 (1985), 138-151.
doi: 10.1137/0145006. |
[8] |
E. P. Cohen and H. Eagle,
A simplified chemostat for the growth of mammalian cells: characteristics of cell growth in continuous culture, J. Exp. Med., 113 (1961), 467-474.
|
[9] |
J. M. Cushing,
A competition model for size-structured species, SIAM J. Appl. Math., 49 (1989), 838-858.
doi: 10.1137/0149049. |
[10] |
J. M. Cushing, An Introduction to Structured Population Dynamics, Reginal Conference Series in Applied Mathematics 71, SIAM, Philadelphia, PA, 1998.
doi: 10.1137/1.9781611970005. |
[11] |
D. E. Dykhuizen and A. M. Dean,
Evolution of specialists in an experimental microcosm, Genetics, 167 (2005), 2015-2026.
|
[12] |
T. B. K. Gage, F. M. Williams and J. B. Horton,
Division synchrony and the dynamics of microbial populations: A size-specific model, Theor. Pop. Bio., 26 (1984), 296-314.
doi: 10.1016/0040-5809(84)90035-2. |
[13] |
M. Golubitsky, E. B. Keeler and M. Rothschild,
Convergence of the age-structure: Applications of the projective metric, Theor. Pop. Bio., 7 (1975), 84-93.
doi: 10.1016/0040-5809(75)90007-6. |
[14] |
S. R. Hansen and S. P. Hubbell,
Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science, 207 (1980), 1491-1493.
|
[15] |
S. B. Hsu, S. P. Hubbell and P. Waltman,
A mathematical theory for single nutrient competition in countinuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.
doi: 10.1137/0132030. |
[16] |
L. Jones and S. P. Ellner,
Effects of rapid prey evolution on predator-prey cycles, J. Math. Biol., 55 (2007), 541-573.
doi: 10.1007/s00285-007-0094-6. |
[17] |
J. L. Jost, J. F. Drake, A. G. Fredrickson and H. M. Tsuchiya,
Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandu and glucose in a minimal medium, J. Bacteriol., 113 (1973), 834-841.
|
[18] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer-Verlag, New York, 1986.
doi: 10.1007/978-3-662-13159-6. |
[19] |
S. Pavlou and I. G. Kevrekidis,
Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies, Math. Biosci., 108 (1992), 1-55.
doi: 10.1016/0025-5564(92)90002-E. |
[20] |
E. Senior, A. T. Bull and J. H. Slater,
Enzyme evolution in a microbial community growing on the herbicide Dalapon, Nature, 263 (1976), 476-479.
|
[21] |
H. L. Smith,
A discrete, size-structured model of microbial growth and competition in the chemostat, J. Math. Biol., 34 (1996), 734-754.
doi: 10.1007/BF00161517. |
[22] | |
[23] |
H. L. Smith and X.-Q. Zhao,
Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 183-191.
doi: 10.3934/dcdsb.2001.1.183. |
[24] |
L. Wang and G. S. K. Wolkowicz,
A delayed chemostat model with general delayed response functions and differential removal rates, J. Math. Anal. Appl., 321 (2006), 452-468.
doi: 10.1016/j.jmaa.2005.08.014. |
[25] |
H. A. Wichman, J. Millstein and J. J. Bull,
Adaptive molecular evolution for 13,000 phage generations: A possible arms race, Genetics, 170 (2005), 19-31.
|
[26] |
L. M. Wick, H. Weilenmann and T. Egli,
The apparent clock-like evolution of Escherichia coli in glucose-limited chemostats is reproducible at large but not at small population sizes and can be explained with Monod kinetics, Microbiology (Reading, Engl.), 148 (2002), 2889-2902.
|
[27] |
G. S. K. Wolkowicz and Z. Lu,
Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.
doi: 10.1137/0152012. |
[28] |
J. Wu, H. Nie and G. S. K. Wolkowicz,
The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885.
doi: 10.1137/050627514. |
[29] |
H. Xia, G. S. K. Wolkowicz and L. Wang,
Transient oscillations induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489-530.
doi: 10.1007/s00285-004-0311-5. |
show all references
References:
[1] |
J. F. Andrews,
A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotech. Bioeng., 10 (1968), 707-723.
|
[2] |
J. Arino, J.-L. Gouze and A. Sciandra,
A discrete, size-structured model of phytoplankton growth in the chemostat, J. Math. Biol., 45 (2002), 313-336.
doi: 10.1007/s002850200160. |
[3] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Am. Nat., 115 (1980), 151-170.
doi: 10.1086/283553. |
[4] |
L. Becks, F. M. Hilker, H. Malchow, K. Jürgens and H. Arndt,
Experimental demonstration of chaos in a microbial food web, Nature, 435 (2005), 1226-1229.
|
[5] |
B. Boon and H. Laudeuout,
Kinetics of nitrite oxidation by Nitrobacter winogradskyi, Biochem. J., 85 (1962), 440-447.
|
[6] |
A. W. Bush and A. E. Cook,
The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theor. Biol., 63 (1976), 385-395.
|
[7] |
G. J. Butler and G. S. K. Wolkowicz,
A mathematical model of the chemostat with a general class of functions describing nutrient outake, SIAM J. Appl. Math., 45 (1985), 138-151.
doi: 10.1137/0145006. |
[8] |
E. P. Cohen and H. Eagle,
A simplified chemostat for the growth of mammalian cells: characteristics of cell growth in continuous culture, J. Exp. Med., 113 (1961), 467-474.
|
[9] |
J. M. Cushing,
A competition model for size-structured species, SIAM J. Appl. Math., 49 (1989), 838-858.
doi: 10.1137/0149049. |
[10] |
J. M. Cushing, An Introduction to Structured Population Dynamics, Reginal Conference Series in Applied Mathematics 71, SIAM, Philadelphia, PA, 1998.
doi: 10.1137/1.9781611970005. |
[11] |
D. E. Dykhuizen and A. M. Dean,
Evolution of specialists in an experimental microcosm, Genetics, 167 (2005), 2015-2026.
|
[12] |
T. B. K. Gage, F. M. Williams and J. B. Horton,
Division synchrony and the dynamics of microbial populations: A size-specific model, Theor. Pop. Bio., 26 (1984), 296-314.
doi: 10.1016/0040-5809(84)90035-2. |
[13] |
M. Golubitsky, E. B. Keeler and M. Rothschild,
Convergence of the age-structure: Applications of the projective metric, Theor. Pop. Bio., 7 (1975), 84-93.
doi: 10.1016/0040-5809(75)90007-6. |
[14] |
S. R. Hansen and S. P. Hubbell,
Single nutrient microbial competition: Agreement between experimental and theoretical forecast outcomes, Science, 207 (1980), 1491-1493.
|
[15] |
S. B. Hsu, S. P. Hubbell and P. Waltman,
A mathematical theory for single nutrient competition in countinuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.
doi: 10.1137/0132030. |
[16] |
L. Jones and S. P. Ellner,
Effects of rapid prey evolution on predator-prey cycles, J. Math. Biol., 55 (2007), 541-573.
doi: 10.1007/s00285-007-0094-6. |
[17] |
J. L. Jost, J. F. Drake, A. G. Fredrickson and H. M. Tsuchiya,
Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandu and glucose in a minimal medium, J. Bacteriol., 113 (1973), 834-841.
|
[18] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer-Verlag, New York, 1986.
doi: 10.1007/978-3-662-13159-6. |
[19] |
S. Pavlou and I. G. Kevrekidis,
Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies, Math. Biosci., 108 (1992), 1-55.
doi: 10.1016/0025-5564(92)90002-E. |
[20] |
E. Senior, A. T. Bull and J. H. Slater,
Enzyme evolution in a microbial community growing on the herbicide Dalapon, Nature, 263 (1976), 476-479.
|
[21] |
H. L. Smith,
A discrete, size-structured model of microbial growth and competition in the chemostat, J. Math. Biol., 34 (1996), 734-754.
doi: 10.1007/BF00161517. |
[22] | |
[23] |
H. L. Smith and X.-Q. Zhao,
Competitive exclusion in a discrete-time, size-structured chemostat model, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 183-191.
doi: 10.3934/dcdsb.2001.1.183. |
[24] |
L. Wang and G. S. K. Wolkowicz,
A delayed chemostat model with general delayed response functions and differential removal rates, J. Math. Anal. Appl., 321 (2006), 452-468.
doi: 10.1016/j.jmaa.2005.08.014. |
[25] |
H. A. Wichman, J. Millstein and J. J. Bull,
Adaptive molecular evolution for 13,000 phage generations: A possible arms race, Genetics, 170 (2005), 19-31.
|
[26] |
L. M. Wick, H. Weilenmann and T. Egli,
The apparent clock-like evolution of Escherichia coli in glucose-limited chemostats is reproducible at large but not at small population sizes and can be explained with Monod kinetics, Microbiology (Reading, Engl.), 148 (2002), 2889-2902.
|
[27] |
G. S. K. Wolkowicz and Z. Lu,
Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates, SIAM J. Appl. Math., 52 (1992), 222-233.
doi: 10.1137/0152012. |
[28] |
J. Wu, H. Nie and G. S. K. Wolkowicz,
The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885.
doi: 10.1137/050627514. |
[29] |
H. Xia, G. S. K. Wolkowicz and L. Wang,
Transient oscillations induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489-530.
doi: 10.1007/s00285-004-0311-5. |





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