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Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
B. Li, Competition in a turbidostat for an inhibitory nutrient,, Journal of Biological Dynamics, 2 (2008), 208.
doi: 10.1080/17513750802018345. |
[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).
|
[18] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'', Cambridge University Press, (1981).
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
B. Li, Competition in a turbidostat for an inhibitory nutrient,, Journal of Biological Dynamics, 2 (2008), 208.
doi: 10.1080/17513750802018345. |
[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).
|
[18] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'', Cambridge University Press, (1981).
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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