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Asymptotic methods for the Volterra-Lotka equations

Part of the Lecture Notes in Mathematics book series (LNM,volume 711)

Abstract

This contribution deals with asymptotic methods for a Volterra-Lotka system. It is shortly demonstrated how such methods can be applied in case of small and large amplitude oscillations. A more extensive analysis is given for the case where one of the equations contains a small parameter. Our analysis of such a singularly perturbed type of Volterra-Lotka system leads to an asymptotic formula of the period for oscillations with moderate and large amplitudes.

Keywords

  • Periodic Solution
  • Asymptotic Formula
  • Prey Density
  • Asymptotic Method
  • High Order Approximation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

  1. Bogoliubov, N.N., and I.A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Gordon and Beach, New York (1961).

    Google Scholar 

  2. Copson, E.T., An introduction to the theory of functions of a complex variable, Oxford University Press, London (1948).

    MATH  Google Scholar 

  3. Dutt, Ranadir, Application of Hamilton-Jacobi theory to the Lotka-Volterra oscillator, Bull. of Math. Biol. 38: 459–465 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Dutt, Ranadir and P.K. Ghosh, Nonlinear correction to Lotka-Volterra oscillation in a prey-predator system, Math. Biol. 27: 9–16 (1975).

    MATH  Google Scholar 

  5. Frame, J.S., Explicit solutions in two species Volterra systems, J. Theor. Biosci. 43: 73–81 (1974).

    CrossRef  Google Scholar 

  6. Grasman, J. and E. Veling, An asymptotic formula for the period of a Volterra-Lotka system. Math. Biosci 18: 185–189 (1973).

    CrossRef  MATH  Google Scholar 

  7. Lauwerier, H.A., A limit case of a Volterra-Lotka system, report TN 79/75, Mathematical Centre, Amsterdam (1975).

    Google Scholar 

  8. Utz, W.R. and P.E. Waltman, Periodicity and boundedness of solutions of generalized differential equations of growth, Bull. of Math. Bioph. 25: 75–93 (1963).

    CrossRef  MATH  Google Scholar 

  9. Veling, E.J.M., Een asymptotische benadering voor de periode van een Volterra-Lotka systeem, report TN 75/73, Mathematical Centre, Amsterdam (1973).

    Google Scholar 

  10. Volterra, V., Leçons sur la théorie mathématique de la lutte pour la vie, Gouthier-Villars, Paris (1931).

    MATH  Google Scholar 

  11. Waltman, P.E., The equations of growth, Bull. of Math. Bioph. 26: 39–43 (1964).

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1979 Springer-Verlag

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Grasman, J., Veling, E.J.M. (1979). Asymptotic methods for the Volterra-Lotka equations. In: Verhulst, F. (eds) Asymptotic Analysis. Lecture Notes in Mathematics, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062951

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  • DOI: https://doi.org/10.1007/BFb0062951

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09245-2

  • Online ISBN: 978-3-540-35332-4

  • eBook Packages: Springer Book Archive