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Bifurcation analysis of a turbidostat model with distributed delay

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Abstract

In this paper, dynamic behaviors of a turbidostat model with distributed delay are concerned. Hopf bifurcations arise when the value of bifurcation parameter, the time delay of translation for the nutrient, crosses some critical values. Firstly, the type and stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Moreover, the destabilization of periodic solutions is also discussed. Finally, numerical simulation results are given to support the theoretical conclusions.

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References

  1. Li, B.: Competition in a turbidostat for an inhibitory nutrient. J. Biol. Dyn. 2, 208–220 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Li, Z., Chen, L.: Periodic solution of a turbidostat model with impulsive state feedback control. Nonlinear Dyn. 58, 525–538 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Yao, Y., Li, Z., Liu, Z.: Hopf bifurcation analysis of a turbidostat model with discrete delay. Appl. Math. Comput. 262, 267–281 (2015)

    MathSciNet  Google Scholar 

  4. Yuan, S., Li, P., Song, Y.: Delay induced oscillations in a turbidostat with feedback control. J. Math. Chem. 49, 1646–1666 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cammarota, A., Miccio, M.: Competition of two microbial species in a turbidostat. Comput. Aid. Chem. Eng. 28, 331–336 (2010)

    Article  Google Scholar 

  6. Herbert, D., Elsworth, R., Telling, R.: The continuous culture of bacteria: a theoretical and experimental study. J. Gen. Microbiol. 14, 60l–622 (1956)

    Article  Google Scholar 

  7. Smith, H., Waltman, P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  8. Bowong, S., Kurths, J.: Modeling and analysis of the transmission dynamics of tuberculosis without and with seasonality. Nonlinear Dyn. 67, 2027–2051 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  10. Sarwardi, S., Haque, M., Mandal, P.: Ratio-dependent predator–prey model of interacting population with delay effect. Nonlinear Dyn. 69, 817–836 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, S., Beretta, E.: Competitive systems with stage structure of distributed-delay type. J. Math. Anal. Appl. 323, 331–343 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Whittaker, R.: Communities and Ecosystems. Macmillan, New York (1970)

    Google Scholar 

  13. Beretta, E., Bischi, G., Solimano, F.: Stability in chemostat equations with delayed nutrient recycling. J. Math. Biol. 28, 99–111 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ruan, S.: Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling. J. Math. Biol. 31, 633–654 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ruan, S., Wolkowicz, G.: Uniform persistence in plankton models with delayed nutrient recycling. Can. Appl. Math. Quart. 3, 219–235 (1995)

    MATH  Google Scholar 

  16. Caperon, J.: Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology 50, 188–192 (1969)

    Article  Google Scholar 

  17. Ruan, S.: The effect of delays on stability and persistence in plankton models. Nonlinear Anal. 24, 575–585 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. El-Owaidy, H., Ismail, M.: Asymptotic behaviour of the chemostat model with delayed response in growth. Chaos Solitons Fractals 13, 787–795 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ellermeyer, S.: Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth. SIAM J. Appl. Math. 54, 456–465 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wolkowicz, G., Xia, H.: Global asymptotic behavior of a chemostat model with discrete delays. SIAM J. Appl. Math. 57, 1019–1043 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang, L., Wolkowicz, G.: A delayed chemostat model with general nonmonotone response functions and differential removal rates. J. Math. Anal. Appl. 321, 452–468 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Walz, N., Hintze, T., Rusche, R.: Algae and rotifer turbidostats: studies on stability of live feed cultures. Hydrobiologia 358, 127–132 (1997)

    Article  Google Scholar 

  23. Flegr, J.: Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems. J. Theor. Biol. 188, 121–126 (1997)

    Article  Google Scholar 

  24. Leenheer, P., Smith, H.: Feedback control for the chemostat. J. Math. Biol. 46, 48–70 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. MacDonald, N.: Time Lags in Biological Models. Springer, Heidelberg (1978)

    Book  MATH  Google Scholar 

  26. Beretta, E., Takeuchi, Y.: Qualitative properties of chemostat equations with time delays: boundedness, local and global asymptotic stability. Differ. Equ. Dyn. Syst. 2, 19–40 (1994)

    MATH  MathSciNet  Google Scholar 

  27. He, X., Ruan, S., Xia, H.: Global stability in chemostat-type equations with distributed delays. SIAM J. Math. Anal. 29, 681–96 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Yuan, S., Han, M.: Bifurcation analysis of a chemostat model with two distributed delays. Chaos Solitons Fractals 20, 995–1004 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ruan, S., Wolkowicz, G.: Bifurcation analysis of a chemostat model with a distributed delay. J. Math. Anal. Appl. 204, 786–812 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. MacDonald, N.: Time lag in simple chemostat models. Biotechnol. Bioeng. 18, 805–812 (1976)

    Article  Google Scholar 

  31. Wolkowicz, G., Xia, H., Ruan, S.: Competition in the chemostat: a distributed delay model and its global asymptotic behavior. SIAM J. Appl. Math. 57, 1281–1310 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Li, B., Gail, S., Wolkowicz, K., Kuang, Y.: Global asymptotic behavior of a chemostat model with two perfectly complementary resources and distributed delay. SIAM J. Appl. Math. 60, 2058–2086 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wang, C., Guo, S., Xu, Y., Ma, J., Tang, J., Alzahrani, F., Hobiny, A.: Formation of autapse connected to neuron and its biological function. Complexity 2017, 5436737 (2017)

    MATH  MathSciNet  Google Scholar 

  34. Xu, Y., Ying, H., Jia, Y., Ma, J., Hayat, T.: Autaptic regulation of electrical activities in neuron under electromagnetic induction. Sci. Rep.-UK 7, 43452 (2017)

  35. Ma, J., Qin, H., Song, X., Chu, R.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B 29, 1450239 (2015)

    Article  Google Scholar 

  36. Song, Q., Yan, H., Zhao, Z., Liu, Y.: Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays. Neural Netw. 81, 1–10 (2016)

    Article  Google Scholar 

  37. Cushing, J.M.: Integrodifferential Equations and Delay Models in Population Dynamics. Springer, Berlin, Heidelberg (1977)

    Book  MATH  Google Scholar 

  38. Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University press, Cambridge (1981)

    MATH  Google Scholar 

  39. Diekmann, O., van Gils, S., Verduyn Lunel, S., Walther, H.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  40. Smith, H.: An Introduction to Delay Differential Equations with Sciences Applications to the Life. Springer, New York (2010)

    Google Scholar 

  41. Wei, J., Wang, H., Jiang, W.: Bifurcation Theory and Application of Delay Differential Equations. Science Press, Beijing (2012)

    Google Scholar 

  42. Morita, Y.: Destabilization of periodic solutions arising in delay-diffusion system in several space dimensions. Jpn. J. Appl. 1, 39–65 (1984)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

We are very grateful to the anonymous referees and the editor for their careful reading of the original manuscript and their kind comments and valuable suggestions that lead to truly significant improvement of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 11561022) and China Postdoctoral Science Foundation (Grant No. 2014M562008).

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Authors and Affiliations

  1. Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, Hubei, People’s Republic of China

    Yu Mu, Zuxiong Li, Huili Xiang & Hailing Wang

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Zuxiong Li, Huili Xiang & Hailing Wang

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  1. Yu Mu
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  2. Zuxiong Li
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  3. Huili Xiang
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  4. Hailing Wang
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Correspondence to Zuxiong Li.

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Mu, Y., Li, Z., Xiang, H. et al. Bifurcation analysis of a turbidostat model with distributed delay. Nonlinear Dyn 90, 1315–1334 (2017). https://doi.org/10.1007/s11071-017-3728-2

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  • DOI: https://doi.org/10.1007/s11071-017-3728-2

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