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Hopf bifurcation analysis for the model of the chemostat with one species of organism. (English) Zbl 1272.92040

Summary: We consider the dynamics of the chemostat model with time delay. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. By using normal form theory and the center manifold method, we derive the explicit formulas determining the stability and direction of bifurcating periodic solutions. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

MSC:

92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
37N25 Dynamical systems in biology
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[1] Li, X.; Pan, J.; Huang, Q., Hopf bifurcation analysis of some modified chemostat models, Northeastern Mathematical Journal, 14, 4, 392-400 (1998) · Zbl 0922.92037
[2] Li, X.-y.; Qian, M.-h.; Yang, J.-p.; Huang, Q.-C., Hopf bifurcations of a chemostat system with bi-parameters, Northeastern Mathematical Journal, 20, 2, 167-174 (2004) · Zbl 1060.34057
[3] Freedman, H. I.; So, J. W.-H.; Waltman, P., Coexistence in a model of competition in the chemostat incorporating discrete delays, SIAM Journal on Applied Mathematics, 49, 3, 859-870 (1989) · Zbl 0676.92013 · doi:10.1137/0149050
[4] Hale, J., Theory of Functional Differential Equations (1977), Springer · Zbl 0352.34001
[5] Hale, J. K.; Lunel, S. M. V., Introduction to Functional Differential Equations (1995), Springer
[6] Mircea, G.; Neamtu, M.; Opris, D., Dynamical Systems from Economy, Mechanic and Biology Described by Differential Equations with Time Delay (2003), Mirton
[7] Wei, J.; Yu, C., Hopf bifurcation analysis in a model of oscillatory gene expression with delay, Proceedings of the Royal Society of Edinburgh A, 139, 4, 879-895 (2009) · Zbl 1185.34124 · doi:10.1017/S0308210507000091
[8] Monk, N. A. M., Oscillatory expression of Hes1, p53, and NF-\(κB\) driven by transcriptional time delays, Current Biology, 13, 16, 1409-1413 (2003) · doi:10.1016/S0960-9822(03)00494-9
[9] Song, Y.; Wei, J., Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos, Chaos, Solitons & Fractals, 22, 1, 75-91 (2004) · Zbl 1112.37303 · doi:10.1016/j.chaos.2003.12.075
[10] Song, Y.; Wei, J.; Han, M., Local and global Hopf bifurcation in a delayed hematopoiesis model, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 14, 11, 3909-3919 (2004) · Zbl 1090.37547 · doi:10.1142/S0218127404011697
[11] Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete & Impulsive Systems A, 10, 6, 863-874 (2003) · Zbl 1068.34072
[12] Wei, J., Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20, 11, 2483-2498 (2007) · Zbl 1141.34045 · doi:10.1088/0951-7715/20/11/002
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