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September  2017, 16(5): 1883-1891. doi: 10.3934/cpaa.2017091

Global dynamics of a microorganism flocculation model with time delay

Songbai Guo 1,2, and  Wanbiao Ma 1,,

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

* Corresponding author

Received  June 2016 Revised  March 2017 Published  May 2017

Fund Project: This work was supported by the China Scholarship Council, the Fundamental Research Funds for the Central Universities (FRF-BY-14-036) and the National Natural Science Foundation of China (11471034)

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.

Keywords: Microorganism flocculation, Lyapunov functional, time delay, permanence, global stability.
Mathematics Subject Classification: Primary: 92B05, 37N25; Secondary: 34K20, 34D23.
Citation: Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[11]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[12]

Y. Kuang, Limit cycles in a chemostat-related model, SIAM J. Appl. Math., 49 (1989), 1759-1767.  doi: 10.1137/0149107.  Google Scholar

[13] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[11]

S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.  doi: 10.1137/0134064.  Google Scholar

[12]

Y. Kuang, Limit cycles in a chemostat-related model, SIAM J. Appl. Math., 49 (1989), 1759-1767.  doi: 10.1137/0149107.  Google Scholar

[13] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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

Figure 1.  Forward bifurcation.
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Figure 2.  Backward bifurcation.
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