Skip to main content
SpringerLink
Log in

Existence of a non-constant periodic solution of a non-linear autonomous functional differential equation representing the growth of a single species population

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Summary

We consider an integro-differential equation for the densityn of a single species population where the birth rate is constant and the death rate depends on the values ofn in an interval of length τ — 1 > 0. We prove the existence of a non-constant periodic solution under the conditions birth rate b > π/2 and τ- 1 small enough. The basic idea of proof (due to R. D. Nussbaum) is to employ a theorem about non-ejective fixed points for a translation operator associated with the solutions of the equation.

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

Similar content being viewed by others

References

  1. Dunkel, G.: Single species model for population growth depending on past history, in: Seminar on Differential Equations and Dynamical Systems (Lecture Notes in Mathematics Vol. 60). Berlin-Heidelberg-New York: Springer 1968.

    Google Scholar 

  2. Grafton, R. B.: A periodicity theorem for autonomous functional differential equations. Jour. Diff. Eqs.6, 87–109 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  3. Halbach, U., Burkhardt, H. J.: Sind einfache Zeitverzögerungen die Ursachen für periodische Popuiationsschwankungen? Oecologia (Berlin)9, 215–222 (1972).

    Article  Google Scholar 

  4. Horn, W. A.: Some fixed point theorems for compact maps and flows in Banach spaces. Transactions of the AMS149, 391–404 (1970).

    MathSciNet  MATH  Google Scholar 

  5. Jones, G. S.: The existence of periodic solutions off′(x)= -αf(x−1) [1 +f(x)]. Jour. Math. An. Appl.5, 435–450 (1962).

    Article  MATH  Google Scholar 

  6. Nussbaum, R. D.: Periodic solutions of some nonlinear autonomous functional differential equations (to appear).

  7. Wright, E. M.: A non-linear difference-differential equation. Jour. Reine Angewandte Math.194, 66–87 (1955).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität München, Theresienstraße 39, D-8000, München 2, Federal Republic of Germany

    Hans-Otto Walther

Authors
  1. Hans-Otto Walther
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A proof of existence was also announced by G. Dunkel in [1].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Walther, HO. Existence of a non-constant periodic solution of a non-linear autonomous functional differential equation representing the growth of a single species population. J. Math. Biology 1, 227–240 (1975). https://doi.org/10.1007/BF01273745

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01273745

Keywords

Search

Navigation