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Dynamics of some stochastic chemostat models with multiplicative noise
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Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region
Global dynamics of a microorganism flocculation model with time delay
1. | Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China |
2. | Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada |
In this paper, we consider a microorganism flocculation model with time delay. In this model, there may exist a forward bifurcation/backward bifurcation. By constructing suitable positively invariant sets and using Lyapunov-LaSalle theorem, we study the global stability of the equilibria of the model under certain conditions. Furthermore, we also investigate the permanence of the model, and an explicit expression of the eventual lower bound of microorganism concentration is given.
References:
[1] |
T. Caraballo, X. Han and P. E. Kloeden,
Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.
doi: 10.1137/14099930X. |
[2] |
J.-P. Cheng, L.-Y. Zhang and W.-H. Wang, Screening of flocculant-producing microorganisms and flocculating activity, J. Environ. Sci-China, 16 (2004), 894-897. Google Scholar |
[3] |
Q. Dong and W. Ma,
Qualitative analysis of the chemostat model
with variable yield and a time delay, J. Math. Chem., 51 (2013), 1274-1292.
doi: 10.1007/s10910-013-0144-9. |
[4] |
Q. Dong and W. Ma,
Qualitative analysis of a chemostat model
with inhibitory exponential substrate uptake and a time delay, Int. J. Biomath., 7 (2014), 1450045, 16pp.
doi: 10.1142/S1793524514500454. |
[5] |
O. Diekmann, S. A. van Oils and S. M. Verduyn Lunel, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2.![]() ![]() |
[6] |
B. Fiedler and S.-B. Hsu,
Non-periodicity in chemostat
equations: a multi-dimensional negative Bendixson–Dulac criterion, J. Math. Biol., 59 (2009), 233-253.
doi: 10.1007/s00285-008-0229-4. |
[7] |
H. I. Freedman and S. Ruan,
Uniform persistence in functional differential equations, J. Differ. Equations, 115 (1995), 173-192.
doi: 10.1006/jdeq.1995.1011. |
[8] |
S. Guo and W. Ma,
Global behavior of delay differential equations
model of HIV infection with apoptosis, Discrete Cont. Dyn.-B, 21 (2016), 103-119.
doi: 10.3934/dcdsb.2016.21.103. |
[9] |
B. Haegeman, C. Lobry and J. Harmand, Modeling bacteria flocculation as density-dependent growth, AIChE J., 53 (2007), 535-539. Google Scholar |
[10] |
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
doi: 10.1007/978-1-4612-9892-2.![]() ![]() |
[11] |
S. B. Hsu,
Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[12] |
Y. Kuang,
Limit cycles in a chemostat-related model, SIAM J. Appl. Math., 49 (1989), 1759-1767.
doi: 10.1137/0149107. |
[13] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
![]() |
[14] |
C. Liu,
Modelling and parameter identification for a nonlinear
time-delay system in microbial batch fermentation, Appl. Math. Model., 37 (2013), 6899-6908.
doi: 10.1016/j.apm.2013.02.021. |
[15] |
Z. Li and R. Xu,
Stability analysis of a ratio-dependent chemostat
model with time delay and variable yield, Int. J. Biomath., 3 (2010), 243-253.
doi: 10.1142/S1793524510000921. |
[16] |
B. Li and Y. Kuang,
Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.
doi: 10.1006/jmaa.1999.6655. |
[17] |
X. Meng, Q. Gao and Z. Li,
The effects of delayed growth response on the dynamic behaviors of the Monod type chemostat model with impulsive input nutrient concentration, Nonlinear Anal.-Real, 11 (2010), 4476-4486.
doi: 10.1016/j.nonrwa.2010.05.030. |
[18] |
X. Meng, L. Wang and T. Zhang,
Global dynamics analysis of a nonlinear impulsive
stochastic chemostat system in a polluted environment, J. Appl. Anal. Comput., 6 (2016), 865-875.
doi: 10.11948/2016055. |
[19] |
K. Mischaikow, H. L. Smith and H. R. Thieme,
Asymptotically autonomous semiflows: Chain
recurrence and Lyapunov functions, T. Am. Math. Soc., 347 (1995), 1669-1685.
doi: 10.1090/S0002-9947-1995-1290727-7. |
[20] |
S. Salim, R. Bosma and M. H. Vermuë, Harvesting of microalgae by bio-flocculation, J Appl. Phycol., 23 (2011), 849-855. Google Scholar |
[21] |
R. W. Smith and M. Miettinen, Microorganisms in flotation and flocculation: Future technology or laboratory curiosity?, Miner. Eng., 19 (2006), 548-553. Google Scholar |
[22] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() |
[23] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, American Mathematical Society, Providence, RI, 1995.
doi: 10.1090/surv/041. |
[24] |
X. Tai, W. Ma and S. Guo,
A class of dynamic delayed model describing flocculation
of micoorganism and its theoretical analysis, Math. Pract. Theory, 45 (2015), 198-209.
|
[25] |
Z. Xu and X.-Q. Zhao,
A vector-bias malaria model with incubation period and diffusion, Discrete Cont. Dyn.-B, 17 (2012), 2615-2634.
doi: 10.3934/dcdsb.2012.17.2615. |
[26] |
S. Zaki, S. Farag and G. A. Elreesh, Characterization of bioflocculants produced by bacteria isolated from crude petroleum oil, Int. J. Environ. Sci. Te., 8 (2011), 831-840. Google Scholar |
[27] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1.![]() ![]() |
[28] |
T. Zhang, T. Zhang and X. Meng,
Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1-7.
doi: 10.1016/j.aml.2016.12.007. |
[29] |
Y.-P. Zhou and J. Zhou, A review on models of microorganism continuous fermentation and its application, Microbiol. China, 37 (2010), 269-273. Google Scholar |
show all references
References:
[1] |
T. Caraballo, X. Han and P. E. Kloeden,
Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.
doi: 10.1137/14099930X. |
[2] |
J.-P. Cheng, L.-Y. Zhang and W.-H. Wang, Screening of flocculant-producing microorganisms and flocculating activity, J. Environ. Sci-China, 16 (2004), 894-897. Google Scholar |
[3] |
Q. Dong and W. Ma,
Qualitative analysis of the chemostat model
with variable yield and a time delay, J. Math. Chem., 51 (2013), 1274-1292.
doi: 10.1007/s10910-013-0144-9. |
[4] |
Q. Dong and W. Ma,
Qualitative analysis of a chemostat model
with inhibitory exponential substrate uptake and a time delay, Int. J. Biomath., 7 (2014), 1450045, 16pp.
doi: 10.1142/S1793524514500454. |
[5] |
O. Diekmann, S. A. van Oils and S. M. Verduyn Lunel, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2.![]() ![]() |
[6] |
B. Fiedler and S.-B. Hsu,
Non-periodicity in chemostat
equations: a multi-dimensional negative Bendixson–Dulac criterion, J. Math. Biol., 59 (2009), 233-253.
doi: 10.1007/s00285-008-0229-4. |
[7] |
H. I. Freedman and S. Ruan,
Uniform persistence in functional differential equations, J. Differ. Equations, 115 (1995), 173-192.
doi: 10.1006/jdeq.1995.1011. |
[8] |
S. Guo and W. Ma,
Global behavior of delay differential equations
model of HIV infection with apoptosis, Discrete Cont. Dyn.-B, 21 (2016), 103-119.
doi: 10.3934/dcdsb.2016.21.103. |
[9] |
B. Haegeman, C. Lobry and J. Harmand, Modeling bacteria flocculation as density-dependent growth, AIChE J., 53 (2007), 535-539. Google Scholar |
[10] |
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
doi: 10.1007/978-1-4612-9892-2.![]() ![]() |
[11] |
S. B. Hsu,
Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[12] |
Y. Kuang,
Limit cycles in a chemostat-related model, SIAM J. Appl. Math., 49 (1989), 1759-1767.
doi: 10.1137/0149107. |
[13] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
![]() |
[14] |
C. Liu,
Modelling and parameter identification for a nonlinear
time-delay system in microbial batch fermentation, Appl. Math. Model., 37 (2013), 6899-6908.
doi: 10.1016/j.apm.2013.02.021. |
[15] |
Z. Li and R. Xu,
Stability analysis of a ratio-dependent chemostat
model with time delay and variable yield, Int. J. Biomath., 3 (2010), 243-253.
doi: 10.1142/S1793524510000921. |
[16] |
B. Li and Y. Kuang,
Simple food chain in a chemostat with distinct removal rates, J. Math. Anal. Appl., 242 (2000), 75-92.
doi: 10.1006/jmaa.1999.6655. |
[17] |
X. Meng, Q. Gao and Z. Li,
The effects of delayed growth response on the dynamic behaviors of the Monod type chemostat model with impulsive input nutrient concentration, Nonlinear Anal.-Real, 11 (2010), 4476-4486.
doi: 10.1016/j.nonrwa.2010.05.030. |
[18] |
X. Meng, L. Wang and T. Zhang,
Global dynamics analysis of a nonlinear impulsive
stochastic chemostat system in a polluted environment, J. Appl. Anal. Comput., 6 (2016), 865-875.
doi: 10.11948/2016055. |
[19] |
K. Mischaikow, H. L. Smith and H. R. Thieme,
Asymptotically autonomous semiflows: Chain
recurrence and Lyapunov functions, T. Am. Math. Soc., 347 (1995), 1669-1685.
doi: 10.1090/S0002-9947-1995-1290727-7. |
[20] |
S. Salim, R. Bosma and M. H. Vermuë, Harvesting of microalgae by bio-flocculation, J Appl. Phycol., 23 (2011), 849-855. Google Scholar |
[21] |
R. W. Smith and M. Miettinen, Microorganisms in flotation and flocculation: Future technology or laboratory curiosity?, Miner. Eng., 19 (2006), 548-553. Google Scholar |
[22] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043.![]() ![]() |
[23] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, American Mathematical Society, Providence, RI, 1995.
doi: 10.1090/surv/041. |
[24] |
X. Tai, W. Ma and S. Guo,
A class of dynamic delayed model describing flocculation
of micoorganism and its theoretical analysis, Math. Pract. Theory, 45 (2015), 198-209.
|
[25] |
Z. Xu and X.-Q. Zhao,
A vector-bias malaria model with incubation period and diffusion, Discrete Cont. Dyn.-B, 17 (2012), 2615-2634.
doi: 10.3934/dcdsb.2012.17.2615. |
[26] |
S. Zaki, S. Farag and G. A. Elreesh, Characterization of bioflocculants produced by bacteria isolated from crude petroleum oil, Int. J. Environ. Sci. Te., 8 (2011), 831-840. Google Scholar |
[27] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1.![]() ![]() |
[28] |
T. Zhang, T. Zhang and X. Meng,
Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1-7.
doi: 10.1016/j.aml.2016.12.007. |
[29] |
Y.-P. Zhou and J. Zhou, A review on models of microorganism continuous fermentation and its application, Microbiol. China, 37 (2010), 269-273. Google Scholar |


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