Global dynamics of a microorganism flocculation model with time delay

Songbai Guo; Wanbiao Ma

doi:10.3934/cpaa.2017091

Article Contents

2017, Volume 16, Issue 5: 1883-1891. Doi: 10.3934/cpaa.2017091

This issue Previous Article Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region Next Article Dynamics of some stochastic chemostat models with multiplicative noise

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
* Corresponding author

Received: May 31, 2016

Revised: February 28, 2017

Published: October 2017

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).

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Abstract

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.

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Mathematics Subject Classification: Primary: 92B05, 37N25; Secondary: 34K20, 34D23.

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Figure 1. Forward bifurcation.

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Figure 2. Backward bifurcation.

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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, et al., Screening of flocculant-producing microorganisms and flocculating activity, J. Environ. Sci-China, 16 (2004), 894-897.
[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.
[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, et al., Harvesting of microalgae by bio-flocculation, J Appl. Phycol., 23 (2011), 849-855.
[21]	R. W. Smith and M. Miettinen, Microorganisms in flotation and flocculation: Future technology or laboratory curiosity?, Miner. Eng., 19 (2006), 548-553.
[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, et al., 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, et al., Characterization of bioflocculants produced by bacteria isolated from crude petroleum oil, Int. J. Environ. Sci. Te., 8 (2011), 831-840.
[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.