Skip to main content
SpringerLink
Log in

Number of positive periodic solutions for feedback-driven nonlinear differential equation: application to hematopoietic process

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

This paper considers a first-order nonlinear differential equation that driven by feedback dominated by time delays. Sufficient conditions which determine the number of positive periodic solutions are studied. Our findings give insight into the existence of positive periodic solutions by exploring the \(\Delta \)-points and \(\nabla \)-points that are specifically defined. Moreover, a numerical example and its simulations are provided to illustrate our main results. In this example, at least four positive periodic solutions are ensured by our results. The existence regions of these four positive periodic solutions are also clarified. Finally, we discuss the hematopoietic process (the production process of blood cells) to demonstrate the application of the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Amster, P., Balderrama, R.: On the global dynamic behaviour for a generalized haematopoiesis model with almost periodic coefficients and oscillating circulation loss rate. Math. Methods Appl. Sci. 41, 3976–3997 (2018)

    Article  MathSciNet  Google Scholar 

  2. Berezansky, L., Braverman, E., Idels, L.: Mackey-Glass model of hematopoiesis with monotone feedback revisited. Appl. Math. Comput. 219, 4892–4907 (2013)

    MathSciNet  Google Scholar 

  3. Jiang, W.H.: The existence of multiple positive periodic solutions for functional differential equations. Appl. Math. Comput. 208, 165–171 (2009)

    MathSciNet  Google Scholar 

  4. Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)

    Google Scholar 

  5. Li, J.W., Du, C.X.: Existence of positive periodic solutions for a generalized Nicholson’s blowflies model. J. Comput. Appl. Math. 221, 226–233 (2008)

    Article  MathSciNet  Google Scholar 

  6. Li, Y.K., Zhu, L.F.: Positive periodic solutions of nonlinear functional differential equations. Appl. Math. Comput. 156, 329–339 (2004)

    MathSciNet  Google Scholar 

  7. Liu, G.R., Zhao, A.M., Yan, J.R.: Existence and global attractivity of unique positive periodic solution for a Lasota–Wazewska model. Nonlinear Anal. 64, 1737–1746 (2006)

    Article  MathSciNet  Google Scholar 

  8. Liu, G.R., Yan, J.R., Zhang, F.Q.: Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis. J. Math. Anal. Appl. 334, 157–171 (2007)

    Article  MathSciNet  Google Scholar 

  9. Nicholson, A.J.: The balance of animal population. J. Anim. Ecol. 2, 132–178 (1933)

    Article  Google Scholar 

  10. Olach, R.: Positive periodic solutions of delay differential equations. Appl. Math. Lett. 26, 1141–1145 (2013)

    Article  MathSciNet  Google Scholar 

  11. Ruan, S.G.: Delay Differential Equations in Single Species Dynamics. Springer, Netherlands (2006)

    Book  Google Scholar 

  12. Shao, J.Y.: Pseudo almost periodic solutions for a Lasota–Wazewska model with an oscillating death rate. Appl. Math. Lett. 43, 90–95 (2013)

    Article  MathSciNet  Google Scholar 

  13. Tost, G.O., Vasilieva, O.: Analysis, Modelling, Optimization, and Numerical Techniques. Springer, Berlin (2013)

    Google Scholar 

  14. Wu, X.M., Li, J.W., Zhou, H.Q.: A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis. Comput. Math. Appl. 54, 840–849 (2007)

    Article  MathSciNet  Google Scholar 

  15. Yao, L.G.: Dynamics of Nicholson’s blowflies models with a nonlinear density-dependent mortality. Appl. Math. Model. 64, 185–195 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by Start-up Scientific Research Foundation of Northeast Forestry University [60201521010].

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeast Forestry University, Harbin, 150040, People’s Republic of China

    Yan Yan & Zhu Hanqi

Authors
  1. Yan Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhu Hanqi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan Yan.

Ethics declarations

Conflict of interest

The authors declare that there is no conflicts of interest.

Additional information

Communicated by Gerald Teschl.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yan, Y., Hanqi, Z. Number of positive periodic solutions for feedback-driven nonlinear differential equation: application to hematopoietic process. Monatsh Math 203, 523–542 (2024). https://doi.org/10.1007/s00605-023-01881-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-023-01881-8

Keywords

Mathematics Subject Classification

Search

Navigation