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

Oscillations in a plasmid turbidostat model with delayed feedback control

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.
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
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show all references

References:
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J. Math. Biol., 46 (2003), 48-70. doi: 10.1007/s00285-002-0170-x.  Google Scholar

[2]

Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.  Google Scholar

[3]

J. Math. Biol., 32 (1994), 731-742. doi: 10.1007/BF00163024.  Google Scholar

[4]

Chem. Eng. Sci., 52 (1997), 23-35. doi: 10.1016/S0009-2509(96)00385-5.  Google Scholar

[5]

J. Theor. Biol., 122 (1986), 83-93. doi: 10.1016/S0022-5193(86)80226-0.  Google Scholar

[6]

J. Math. Biol., 34 (1995), 41-76. doi: 10.1007/BF00180136.  Google Scholar

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SIAM J. Appl. Math., 52 (1992), 528-540. doi: 10.1137/0152029.  Google Scholar

[8]

J. Ind. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.  Google Scholar

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Chapman & Hall, 1997. Google Scholar

[10]

Princeton U. P., Princeton, N. J., 1982. Google Scholar

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J. Theoret. Biol., 188 (1997), 121-126. doi: 10.1006/jtbi.1997.0458.  Google Scholar

[12]

Journal of Biological Dynamics, 2 (2008), 208-220. doi: 10.1080/17513750802018345.  Google Scholar

[13]

Chapman & Hall, New York, 1995. Google Scholar

[14]

Prentice Hall, Englewood Cliffs, New Jersey, 1992. Google Scholar

[15]

Boston: Kluwer Academic Publishers, 1992. Google Scholar

[16]

New York: Spring-Verlag, 1993. Google Scholar

[17]

Boston: Academic Press, 1993.  Google Scholar

[18]

Cambridge University Press, Cambridge, 1981.  Google Scholar

[19]

Biotechnology and Bioengineering, 26 (1984), 942-957. doi: 10.1002/bit.260260819.  Google Scholar

[20]

Chem. Engin. Sci., 43 (1988), 49-57. doi: 10.1016/0009-2509(88)87125-2.  Google Scholar

[21]

Discrete Continuous Dynam. Systems - B, 5 (2005), 699-718.  Google Scholar

[22]

Math. Biosci., 148 (1998), 147-159. doi: 10.1016/S0025-5564(97)10010-4.  Google Scholar

[23]

J. Math. Biol., 42 (2001), 71-94. doi: 10.1007/PL00000073.  Google Scholar

[24]

Math. Biosci., 202 (2006), 1-28. doi: 10.1016/j.mbs.2006.04.003.  Google Scholar

[25]

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[26]

Nonlinear Analysis: Real World Applications, 10 (2009), 1305-1320. doi: 10.1016/j.nonrwa.2008.01.009.  Google Scholar

[27]

Chaos, Solitons and Fractals, 32 (2007), 1419-1428. doi: 10.1016/j.chaos.2005.11.069.  Google Scholar

[28]

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[29]

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[30]

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