American Institute of Mathematical Sciences

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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

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), 48-70. 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), 53-91. 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), 731-742. 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), 23-35. 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), 83-93. 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), 41-76. 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), 528-540. 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), 471-490. 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., Princeton, N. J., 1982. Google Scholar

[11]

J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theoret. Biol., 188 (1997), 121-126. 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), 208-220. doi: 10.1080/17513750802018345.  Google Scholar

[13]

N. S. Panikov, "Microbial Growth Kinetics,'' Chapman & Hall, New York, 1995. Google Scholar

[14]

M. L. Shuler and F. Kargi, "Bioprocess Engineering, Basic Concepts,'' Prentice Hall, Englewood Cliffs, New Jersey, 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, Cambridge, 1981.  Google Scholar

[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.  Google Scholar

[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.  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), 699-718.  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), 147-159. 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), 71-94. 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), 1-28. 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, Volume 2009, Article ID 204632, 17 pages. 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), 1305-1320. 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, Solitons and Fractals, 32 (2007), 1419-1428. 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), 1860-1885. doi: 10.1137/050627514.  Google Scholar

[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.  Google Scholar

[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.  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), 1428-1436. 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), 48-70. 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), 53-91. 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), 731-742. 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), 23-35. 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), 83-93. 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), 41-76. 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), 528-540. 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), 471-490. 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., Princeton, N. J., 1982. Google Scholar

[11]

J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theoret. Biol., 188 (1997), 121-126. 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), 208-220. doi: 10.1080/17513750802018345.  Google Scholar

[13]

N. S. Panikov, "Microbial Growth Kinetics,'' Chapman & Hall, New York, 1995. Google Scholar

[14]

M. L. Shuler and F. Kargi, "Bioprocess Engineering, Basic Concepts,'' Prentice Hall, Englewood Cliffs, New Jersey, 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, Cambridge, 1981.  Google Scholar

[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.  Google Scholar

[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.  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), 699-718.  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), 147-159. 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), 71-94. 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), 1-28. 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, Volume 2009, Article ID 204632, 17 pages. 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), 1305-1320. 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, Solitons and Fractals, 32 (2007), 1419-1428. 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), 1860-1885. doi: 10.1137/050627514.  Google Scholar

[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.  Google Scholar

[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.  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), 1428-1436. doi: 10.1016/j.automatica.2010.06.012.  Google Scholar

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