Figure 1.
Forward bifurcation.
-
Previous Article
Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region
- CPAA Home
- This Issue
-
Next Article
Dynamics of some stochastic chemostat models with multiplicative noise
September
2017, 16(5): 1883-1891.
doi: 10.3934/cpaa.2017091
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.
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. |
| [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. |
| [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, 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,
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. |
| [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. |
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. |
| [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, 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,
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. |
| [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. |
Figure 2.
Backward bifurcation.
| [1] |
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 |
| [2] |
Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971 |
| [3] |
Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023 |
| [4] |
Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115 |
| [5] |
Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 |
| [6] |
C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 |
| [7] |
C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603 |
| [8] |
Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689 |
| [9] |
Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727 |
| [10] |
Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401 |
| [11] |
Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 |
| [12] |
Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010 |
| [13] |
Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369 |
| [14] |
Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191 |
| [15] |
Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627 |
| [16] |
Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 |
| [17] |
Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631 |
| [18] |
Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145 |
| [19] |
Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861 |
| [20] |
Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025 |
2017 Impact Factor: 0.884
Tools
Metrics
Other articles
by authors
[Back to Top]