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On a transcendental equation in the stability analysis of a population growth model

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Summary

We consider the rate equation n = rn for the density n of a single species population in a constant environment. We assume only that there is a positive constant solution n*, that the rate of increase r depends on the history of n and that r decreases for great n. The stability properties of the solution n* depend on the location of the eigenvalues of the linearized functional differential equation. These eigenvalues are the complex solutions λ of the equation λ + α∫ −1 0 exp [λa]ds(a) − 0 with α>0 and s increasing, s (−1)=0, s (0)=1. We give conditions on a and s which ensure that all eigenvalues have negative real part, or that there are eigenvalues with positive real part. In the case of the simplest smooth function s (s=id+1), we obtain a theorem which describes the distribution of all eigenvalues in the complex plane for every α>0.

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References

  1. [1]

    Dieudonne, J.: Foundations of modern analysis. New York: Academic Press 1960.

  2. [2]

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

  3. [3]

    Hutchinson, G. E.: Circular causal systems in ecology. Annals of the New York Academy of Sciences 50, 221–246 (1948).

  4. [4]

    Hale, J. K.: Functional differential equations. Berlin-Heidelberg-New York: Springer 1971.

  5. [5]

    Walther, H. O.: Asymptotic stability for some functional differential equations. Proceedings of the Royal Society of Edinburgh 74 A (1974/75).

  6. [6]

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

  7. [7]

    Zygmund, A.: Trigonometric series I, Second edition. Cambridge: University Press 1959.

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Walther, H.-. On a transcendental equation in the stability analysis of a population growth model. J. Math. Biology 3, 187–195 (1976). https://doi.org/10.1007/BF00276205

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Keywords

  • Stability Analysis
  • Growth Model
  • Complex Plane
  • Mathematical Biology
  • Matrix Theory