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Bifurcation analysis of a chemostat model of plasmid-bearing and plasmid-free competition with pulsed input. (English) Zbl 1442.92201

Summary: A chemostat model of plasmid-bearing and plasmid-free competition with pulsed input is proposed. The invasion threshold of the plasmid-bearing and plasmid-free organisms is obtained according to the stability of the boundary periodic solution. By use of standard techniques of bifurcation theory, the periodic oscillations in substrate, plasmid-bearing, and plasmid-free organisms are shown when some conditions are satisfied. Our results can be applied to control bioreactor aimed at producing commercial producers through genetically altered organisms.

MSC:

92D40 Ecology
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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[1] Mazenc, F.; Malisoff, M.; De Leenheer, P., On the stability of periodic solutions in the perturbed chemostat, Mathematical Biosciences and Engineering, 4, 2, 319-338 (2007) · Zbl 1126.93424
[2] Mazenc, F.; Malisoff, M.; Harmand, J., Stabilization and robustness analysis for a chemostat model with two species and monod growth rates via a Lyapunov approach, Proceedings of the 46th IEEE Conference on Decision and Control
[3] Mazenc, F.; Malisoff, M.; Harmand, J., Further results on stabilization of periodic trajectories for a chemostat with two species, IEEE Transactions on Automatic Control, 53, 66-74 (2008) · Zbl 1366.92075
[4] Nie, H.; Wu, J. H., Positive solutions of a competition model for two resources in the unstirred chemostat, Journal of Mathematical Analysis and Applications, 355, 1, 231-242 (2009) · Zbl 1184.35159
[5] Yuan, S. L.; Zhang, W. G.; Han, M. A., Global asymptotic behavior in chemostat-type competition models with delay, Nonlinear Analysis. Real World Applications, 10, 3, 1305-1320 (2009) · Zbl 1162.34335
[6] Tagashira, O., Permanent coexistence in chemostat models with delayed feedback control, Nonlinear Analysis. Real World Applications, 10, 3, 1443-1452 (2009) · Zbl 1162.34334
[7] Nelson, M. I.; Sidhu, H. S., Analysis of a chemostat model with variable yield coefficient, Journal of Mathematical Chemistry, 38, 4, 605-615 (2005) · Zbl 1096.92049
[8] Nelson, M. I.; Sidhu, H. S., Analysis of a chemostat model with variable yield coefficient: tessier kinetics, Journal of Mathematical Chemistry, 46, 2, 303-321 (2009) · Zbl 1196.92044
[9] Smith, H. L.; Waltman, P., The Theory of the Chemostat. The Theory of the Chemostat, Cambridge Studies in Mathematical Biology, 13 (1995), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1139.92029
[10] Xiang, Z. Y.; Song, X. Y., A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input, Chaos, Solitons & Fractals, 32, 4, 1419-1428 (2007) · Zbl 1126.92061
[11] Shi, X.; Song, X.; Zhou, X., Analysis of a model of plasmid-bearing, plasmid-free competition in a pulsed chemostat, Advances in Complex Systems, 9, 3, 263-276 (2006) · Zbl 1108.92045
[12] Pal, R.; Basu, D.; Banerjee, M., Modelling of phytoplankton allelopathy with Monod-Haldane-type functional response—a mathematical study, BioSystems, 95, 3, 243-253 (2009)
[13] Cushing, J. M., Periodic time-dependent predator-prey systems, SIAM Journal on Applied Mathematics, 32, 1, 82-95 (1977) · Zbl 0348.34031
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