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Hopf bifurcation of a delayed chemostat model with general monotone response functions

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Abstract

In this paper, a chemostat model with general monotone response functions and two discrete time delays is proposed to describe the dynamical behavior about predator–prey system. First, by analyzing the characteristic equation associated with the model, we obtain the conditions of the existence and stability of extinction equilibrium and positive equilibrium. Choosing delays as bifurcation parameters, the existence of Hopf bifurcations is investigated in detail. Second, by virtue of the Poincaré normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to illustrate the theoretical results and the biological significance.

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Acknowledgements

The authors would like to thank the editor and referee for making the valuable suggestions to improve this paper. This research was supported by the Natural Science Foundation of Shanxi province (2013011002-2).

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, Shanxi, People’s Republic of China

    Shulin Sun & Xing Liu

  2. School of Mathematical Science, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Cuihua Guo

Authors
  1. Shulin Sun
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  2. Cuihua Guo
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  3. Xing Liu
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Corresponding author

Correspondence to Shulin Sun.

Additional information

Communicated by Maurizio Grasselli.

This work is supported by the Natural Science Foundation of Shanxi province (2013011002-2).

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Sun, S., Guo, C. & Liu, X. Hopf bifurcation of a delayed chemostat model with general monotone response functions. Comp. Appl. Math. 37, 2714–2737 (2018). https://doi.org/10.1007/s40314-017-0476-3

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  • DOI: https://doi.org/10.1007/s40314-017-0476-3

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