Oscillations in a plasmid turbidostat model with delayed feedback control

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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
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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 and 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), 48-70. doi: 10.1007/s00285-002-0170-x.
[2]	S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.
[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), 731-742. doi: 10.1007/BF00163024.
[4]	S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media, Chem. Eng. Sci., 52 (1997), 23-35. doi: 10.1016/S0009-2509(96)00385-5.
[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), 83-93. doi: 10.1016/S0022-5193(86)80226-0.
[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), 41-76. doi: 10.1007/BF00180136.
[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), 528-540. doi: 10.1137/0152029.
[8]	S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, J. Ind. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.
[9]	J. P. Grover, "Resource Competition,'' Chapman & Hall, 1997.
[10]	D. Tilman, "Resource Competition and Community Structure,'' Princeton U. P., Princeton, N. J., 1982.
[11]	J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theoret. Biol., 188 (1997), 121-126. doi: 10.1006/jtbi.1997.0458.
[12]	B. Li, Competition in a turbidostat for an inhibitory nutrient, Journal of Biological Dynamics, 2 (2008), 208-220. doi: 10.1080/17513750802018345.
[13]	N. S. Panikov, "Microbial Growth Kinetics,'' Chapman & Hall, New York, 1995.
[14]	M. L. Shuler and F. Kargi, "Bioprocess Engineering, Basic Concepts,'' Prentice Hall, Englewood Cliffs, New Jersey, 1992.
[15]	K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,'' Boston: Kluwer Academic Publishers, 1992.
[16]	J. Hale and S. Lunel, "Introduction to Functional Differential Equations,'' New York: Spring-Verlag, 1993.
[17]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'' Boston: Academic Press, 1993.
[18]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'' Cambridge University Press, Cambridge, 1981.
[19]	D. F. Ryder and D. DiBiasio, An operational strategy for unstable recombinant DNA cultures, Biotechnology and Bioengineering, 26 (1984), 942-957. doi: 10.1002/bit.260260819.
[20]	G. Stephanopoulis and G. Lapidus, Chemostat dynamics of plasmid-bearing plasmid-free mixed recombinant cultures, Chem. Engin. Sci., 43 (1988), 49-57. doi: 10.1016/0009-2509(88)87125-2.
[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), 699-718.
[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), 147-159. doi: 10.1016/S0025-5564(97)10010-4.
[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), 71-94. doi: 10.1007/PL00000073.
[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), 1-28. doi: 10.1016/j.mbs.2006.04.003.
[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, Volume 2009, Article ID 204632, 17 pages. doi: 10.1155/2009/204632.
[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), 1305-1320. doi: 10.1016/j.nonrwa.2008.01.009.
[27]	Z. Xiang and X. Song, A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input, Chaos, Solitons and Fractals, 32 (2007), 1419-1428. doi: 10.1016/j.chaos.2005.11.069.
[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), 1860-1885. doi: 10.1137/050627514.
[29]	O. Tagashira and T. Hara, Delayed feedback control for a chemostat model, Math. Biosci., 201 (2006), 101-112. doi: 10.1016/j.mbs.2005.12.014.
[30]	O. Tagashira, Permanent coexistence in chemostat models with delayed feedback control, Nonlinear Analysis: Real World Applications, 10 (2009), 1443-1452. doi: 10.1016/j.nonrwa.2008.01.015.
[31]	F. Mazenc and M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements, Automatica, 46 (2010), 1428-1436. doi: 10.1016/j.automatica.2010.06.012.

show all references

References:

[1]	P. De Leenheer and H. L. Smith, Feedback control for the chemostat, J. Math. Biol., 46 (2003), 48-70. doi: 10.1007/s00285-002-0170-x.
[2]	S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.
[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), 731-742. doi: 10.1007/BF00163024.
[4]	S. B. Hsu and P. Waltman, Competition between plasmid-bearing and plasmid-free organisms in selective media, Chem. Eng. Sci., 52 (1997), 23-35. doi: 10.1016/S0009-2509(96)00385-5.
[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), 83-93. doi: 10.1016/S0022-5193(86)80226-0.
[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), 41-76. doi: 10.1007/BF00180136.
[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), 528-540. doi: 10.1137/0152029.
[8]	S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, J. Ind. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.
[9]	J. P. Grover, "Resource Competition,'' Chapman & Hall, 1997.
[10]	D. Tilman, "Resource Competition and Community Structure,'' Princeton U. P., Princeton, N. J., 1982.
[11]	J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theoret. Biol., 188 (1997), 121-126. doi: 10.1006/jtbi.1997.0458.
[12]	B. Li, Competition in a turbidostat for an inhibitory nutrient, Journal of Biological Dynamics, 2 (2008), 208-220. doi: 10.1080/17513750802018345.
[13]	N. S. Panikov, "Microbial Growth Kinetics,'' Chapman & Hall, New York, 1995.
[14]	M. L. Shuler and F. Kargi, "Bioprocess Engineering, Basic Concepts,'' Prentice Hall, Englewood Cliffs, New Jersey, 1992.
[15]	K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,'' Boston: Kluwer Academic Publishers, 1992.
[16]	J. Hale and S. Lunel, "Introduction to Functional Differential Equations,'' New York: Spring-Verlag, 1993.
[17]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'' Boston: Academic Press, 1993.
[18]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'' Cambridge University Press, Cambridge, 1981.
[19]	D. F. Ryder and D. DiBiasio, An operational strategy for unstable recombinant DNA cultures, Biotechnology and Bioengineering, 26 (1984), 942-957. doi: 10.1002/bit.260260819.
[20]	G. Stephanopoulis and G. Lapidus, Chemostat dynamics of plasmid-bearing plasmid-free mixed recombinant cultures, Chem. Engin. Sci., 43 (1988), 49-57. doi: 10.1016/0009-2509(88)87125-2.
[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), 699-718.
[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), 147-159. doi: 10.1016/S0025-5564(97)10010-4.
[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), 71-94. doi: 10.1007/PL00000073.
[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), 1-28. doi: 10.1016/j.mbs.2006.04.003.
[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, Volume 2009, Article ID 204632, 17 pages. doi: 10.1155/2009/204632.
[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), 1305-1320. doi: 10.1016/j.nonrwa.2008.01.009.
[27]	Z. Xiang and X. Song, A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input, Chaos, Solitons and Fractals, 32 (2007), 1419-1428. doi: 10.1016/j.chaos.2005.11.069.
[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), 1860-1885. doi: 10.1137/050627514.
[29]	O. Tagashira and T. Hara, Delayed feedback control for a chemostat model, Math. Biosci., 201 (2006), 101-112. doi: 10.1016/j.mbs.2005.12.014.
[30]	O. Tagashira, Permanent coexistence in chemostat models with delayed feedback control, Nonlinear Analysis: Real World Applications, 10 (2009), 1443-1452. doi: 10.1016/j.nonrwa.2008.01.015.
[31]	F. Mazenc and M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements, Automatica, 46 (2010), 1428-1436. doi: 10.1016/j.automatica.2010.06.012.

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