Oscillations in a plasmid turbidostat model with delayed feedback control

Previous Article
Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model
DCDS-B Home
This Issue
Next Article
Addendum

December 2011, 15(3): 893-914. doi: 10.3934/dcdsb.2011.15.893

Oscillations in a plasmid turbidostat model with delayed feedback control

Sanling Yuan ^1, , Yongli Song ^2, and Junhui Li ^3,

1.	College of Science, Shanghai University for Science and Technology, Shanghai 200093, China
2.	Department of Mathematics, Tongji University, Shanghai 200092, China
3.	R&D department, shanghai RAAS blood products Co,. Ltd, Shanghai 200245, China

Received February 2010 Revised September 2010 Published February 2011

Abstract
Related Papers

A model of competition between plasmid-bearing and plasmid-free organisms in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.

Keywords: stability, Time delay, turbidostat, Hopf bifurcation, periodic solution..

Mathematics Subject Classification: Primary: 92A15, 34K45; Secondary: 37G15, 92C4.

Citation: Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893

References:

[1]	P. De Leenheer and H. L. Smith, Feedback control for the chemostat,, J. Math. Biol., 46 (2003), 48. doi: 10.1007/s00285-002-0170-x. Google Scholar
[2]	S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor,, Math. Biosci., 187 (2004), 53. doi: 10.1016/j.mbs.2003.07.004. Google Scholar
[3]	S. B. Hsu, P. Waltman and G. S. K. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat,, J. Math. Biol., 32 (1994), 731. doi: 10.1007/BF00163024. Google Scholar
[4]	S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media,, Chem. Eng. Sci., 52 (1997), 23. doi: 10.1016/S0009-2509(96)00385-5. Google Scholar
[5]	R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics,, J. Theor. Biol., 122 (1986), 83. doi: 10.1016/S0022-5193(86)80226-0. Google Scholar
[6]	T. K. Luo and S. B. Hsu, Global Analysis of a Model of plasmid-bearing, plasmid-free Competition in a chemostat with inhibitions,, J. Math. Biol., 34 (1995), 41. doi: 10.1007/BF00180136. Google Scholar
[7]	S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor,, SIAM J. Appl. Math., 52 (1992), 528. doi: 10.1137/0152029. Google Scholar
[8]	S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin,, J. Ind. Appl. Math., 15 (1998), 471. doi: 10.1007/BF03167323. Google Scholar
[9]	J. P. Grover, "Resource Competition,'', Chapman & Hall, (1997). Google Scholar
[10]	D. Tilman, "Resource Competition and Community Structure,'', Princeton U. P., (1982). Google Scholar
[11]	J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems,, J. Theoret. Biol., 188 (1997), 121. doi: 10.1006/jtbi.1997.0458. Google Scholar
[12]	B. Li, Competition in a turbidostat for an inhibitory nutrient,, Journal of Biological Dynamics, 2 (2008), 208. doi: 10.1080/17513750802018345. Google Scholar
[13]	N. S. Panikov, "Microbial Growth Kinetics,'', Chapman & Hall, (1995). Google Scholar
[14]	M. L. Shuler and F. Kargi, "Bioprocess Engineering, Basic Concepts,'', Prentice Hall, (1992). Google Scholar
[15]	K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,'', Boston: Kluwer Academic Publishers, (1992). Google Scholar
[16]	J. Hale and S. Lunel, "Introduction to Functional Differential Equations,'', New York: Spring-Verlag, (1993). Google Scholar
[17]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'', Boston: Academic Press, (1993). Google Scholar
[18]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'', Cambridge University Press, (1981). Google Scholar
[19]	D. F. Ryder and D. DiBiasio, An operational strategy for unstable recombinant DNA cultures,, Biotechnology and Bioengineering, 26 (1984), 942. doi: 10.1002/bit.260260819. Google Scholar
[20]	G. Stephanopoulis and G. Lapidus, Chemostat dynamics of plasmid-bearing plasmid-free mixed recombinant cultures,, Chem. Engin. Sci., 43 (1988), 49. doi: 10.1016/0009-2509(88)87125-2. Google Scholar
[21]	S. B. Hsu and C. C. Li, A discrete-delayed model with plasmid-bearing, plalmid-free competition in a chemostat,, Discrete Continuous Dynam. Systems - B, 5 (2005), 699. Google Scholar
[22]	Z. Lu and K. P. Hadeler, Model of plasmid-bearing, plasmid-free competition in the chemostat with nutrient recycling and an inhibitor,, Math. Biosci., 148 (1998), 147. doi: 10.1016/S0025-5564(97)10010-4. Google Scholar
[23]	S. Ai, Periodic solution in a chemostat of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor,, J. Math. Biol., 42 (2001), 71. doi: 10.1007/PL00000073. Google Scholar
[24]	S. Yuan, D. Xiao and M. Han, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor,, Math. Biosci., 202 (2006), 1. doi: 10.1016/j.mbs.2006.04.003. Google Scholar
[25]	S. Yuan, Y. Zhao and A. Xiao, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with pulsed input and washout,, Mathematical Problems in Engneeing, 2009 (2046). doi: 10.1155/2009/204632. Google Scholar
[26]	S. Yuan, W. Zhang and M. Han, Global asymptotic behavior in chemostat-type competition models with delay,, Nonlinear Analysis: Real World Applications, 10 (2009), 1305. doi: 10.1016/j.nonrwa.2008.01.009. Google Scholar
[27]	Z. Xiang and X. Song, A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input,, Chaos, 32 (2007), 1419. doi: 10.1016/j.chaos.2005.11.069. Google Scholar
[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. Appl. Math., 38 (2007), 1860. doi: 10.1137/050627514. Google Scholar
[29]	O. Tagashira and T. Hara, Delayed feedback control for a chemostat model,, Math. Biosci., 201 (2006), 101. doi: 10.1016/j.mbs.2005.12.014. Google Scholar
[30]	O. Tagashira, Permanent coexistence in chemostat models with delayed feedback control,, Nonlinear Analysis: Real World Applications, 10 (2009), 1443. doi: 10.1016/j.nonrwa.2008.01.015. Google Scholar
[31]	F. Mazenc and M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements,, Automatica, 46 (2010), 1428. doi: 10.1016/j.automatica.2010.06.012. Google Scholar

show all references

References:

[1]	P. De Leenheer and H. L. Smith, Feedback control for the chemostat,, J. Math. Biol., 46 (2003), 48. doi: 10.1007/s00285-002-0170-x. Google Scholar
[2]	S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor,, Math. Biosci., 187 (2004), 53. doi: 10.1016/j.mbs.2003.07.004. Google Scholar
[3]	S. B. Hsu, P. Waltman and G. S. K. Wolkowicz, Global analysis of a model of plasmid-bearing, plasmid-free competition in a chemostat,, J. Math. Biol., 32 (1994), 731. doi: 10.1007/BF00163024. Google Scholar
[4]	S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media,, Chem. Eng. Sci., 52 (1997), 23. doi: 10.1016/S0009-2509(96)00385-5. Google Scholar
[5]	R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: A chemostat model based on bacteria and antibiotics,, J. Theor. Biol., 122 (1986), 83. doi: 10.1016/S0022-5193(86)80226-0. Google Scholar
[6]	T. K. Luo and S. B. Hsu, Global Analysis of a Model of plasmid-bearing, plasmid-free Competition in a chemostat with inhibitions,, J. Math. Biol., 34 (1995), 41. doi: 10.1007/BF00180136. Google Scholar
[7]	S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor,, SIAM J. Appl. Math., 52 (1992), 528. doi: 10.1137/0152029. Google Scholar
[8]	S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin,, J. Ind. Appl. Math., 15 (1998), 471. doi: 10.1007/BF03167323. Google Scholar
[9]	J. P. Grover, "Resource Competition,'', Chapman & Hall, (1997). Google Scholar
[10]	D. Tilman, "Resource Competition and Community Structure,'', Princeton U. P., (1982). Google Scholar
[11]	J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems,, J. Theoret. Biol., 188 (1997), 121. doi: 10.1006/jtbi.1997.0458. Google Scholar
[12]	B. Li, Competition in a turbidostat for an inhibitory nutrient,, Journal of Biological Dynamics, 2 (2008), 208. doi: 10.1080/17513750802018345. Google Scholar
[13]	N. S. Panikov, "Microbial Growth Kinetics,'', Chapman & Hall, (1995). Google Scholar
[14]	M. L. Shuler and F. Kargi, "Bioprocess Engineering, Basic Concepts,'', Prentice Hall, (1992). Google Scholar
[15]	K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,'', Boston: Kluwer Academic Publishers, (1992). Google Scholar
[16]	J. Hale and S. Lunel, "Introduction to Functional Differential Equations,'', New York: Spring-Verlag, (1993). Google Scholar
[17]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'', Boston: Academic Press, (1993). Google Scholar
[18]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'', Cambridge University Press, (1981). Google Scholar
[19]	D. F. Ryder and D. DiBiasio, An operational strategy for unstable recombinant DNA cultures,, Biotechnology and Bioengineering, 26 (1984), 942. doi: 10.1002/bit.260260819. Google Scholar
[20]	G. Stephanopoulis and G. Lapidus, Chemostat dynamics of plasmid-bearing plasmid-free mixed recombinant cultures,, Chem. Engin. Sci., 43 (1988), 49. doi: 10.1016/0009-2509(88)87125-2. Google Scholar
[21]	S. B. Hsu and C. C. Li, A discrete-delayed model with plasmid-bearing, plalmid-free competition in a chemostat,, Discrete Continuous Dynam. Systems - B, 5 (2005), 699. Google Scholar
[22]	Z. Lu and K. P. Hadeler, Model of plasmid-bearing, plasmid-free competition in the chemostat with nutrient recycling and an inhibitor,, Math. Biosci., 148 (1998), 147. doi: 10.1016/S0025-5564(97)10010-4. Google Scholar
[23]	S. Ai, Periodic solution in a chemostat of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor,, J. Math. Biol., 42 (2001), 71. doi: 10.1007/PL00000073. Google Scholar
[24]	S. Yuan, D. Xiao and M. Han, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor,, Math. Biosci., 202 (2006), 1. doi: 10.1016/j.mbs.2006.04.003. Google Scholar
[25]	S. Yuan, Y. Zhao and A. Xiao, Competition between plasmid-bearing and plasmid-free organisms in a chemostat with pulsed input and washout,, Mathematical Problems in Engneeing, 2009 (2046). doi: 10.1155/2009/204632. Google Scholar
[26]	S. Yuan, W. Zhang and M. Han, Global asymptotic behavior in chemostat-type competition models with delay,, Nonlinear Analysis: Real World Applications, 10 (2009), 1305. doi: 10.1016/j.nonrwa.2008.01.009. Google Scholar
[27]	Z. Xiang and X. Song, A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input,, Chaos, 32 (2007), 1419. doi: 10.1016/j.chaos.2005.11.069. Google Scholar
[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. Appl. Math., 38 (2007), 1860. doi: 10.1137/050627514. Google Scholar
[29]	O. Tagashira and T. Hara, Delayed feedback control for a chemostat model,, Math. Biosci., 201 (2006), 101. doi: 10.1016/j.mbs.2005.12.014. Google Scholar
[30]	O. Tagashira, Permanent coexistence in chemostat models with delayed feedback control,, Nonlinear Analysis: Real World Applications, 10 (2009), 1443. doi: 10.1016/j.nonrwa.2008.01.015. Google Scholar
[31]	F. Mazenc and M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements,, Automatica, 46 (2010), 1428. doi: 10.1016/j.automatica.2010.06.012. Google Scholar

[1]	Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[2]	Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[3]	Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158
[4]	Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[5]	Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[6]	Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
[7]	Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503
[8]	Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[9]	Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[10]	Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[11]	Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[12]	John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[13]	Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[14]	Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[15]	Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[16]	Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183
[17]	Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[18]	Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[19]	Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[20]	Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences