Skip to main content

Persistence and convergence of ecosystems: An analysis of some second order difference equations

Journal of Mathematical Biology volume 4, pages 171–182 (1977)Cite this article

Summary

Three second order difference equation models are analyzed and numerical solutions computed. It is shown that two concepts of ecosystem stability, the local property of convergence and the global property of persistence, do not coincide, and that the existence of either need not imply the other. Conditions for the existence of either form of stability are obtained and shown as parameter space diagrams. Examples of solution trajectories representative of different regions of this space are computed and discussed. A wide range of oscillatory behavior, as noted in recent papers by several authors, results. In addition, the erratic nature of regions of convergence to stable solutions is discussed.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Price includes VAT (Finland)
Tax calculation will be finalised during checkout.

References

  • Allen, J. C.: Mathematical models of species interactions in time and space. American Naturalist 109, 319–342 (1975).

    Google Scholar 

  • Beddington, J. R., Free, C. A., Lawton, J. H.: Dynamic complexity in predator-prey models framed in difference equations. Nature 255, 58–60 (1975).

    Google Scholar 

  • Cook, L. M.: Coefficients of Natural Selection. London: Hutchinson and Company 1971.

    Google Scholar 

  • Fretwell, S.: Populations in Seasonal Environments. Princeton, N. J.: Princeton University Press 1972.

    Google Scholar 

  • Hassell, M. P., Varley, G. C.: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133–1137 (1969).

    Google Scholar 

  • Léon, J. A.: Limit cycles in populations with separate generations. Journal of Theoretical Biology 49, 241–244 (1975).

    Google Scholar 

  • Leslie, P. H.: The use of matrices in certain population mathematics. Biometrika 35, 213–245 (1945).

    Google Scholar 

  • Levine, S. H.: Three Applications of Mathematical Systems Theory to Population Biology. Ph. D. Dissertation, University of Massachusetts, 1973.

  • Levine, S. H.: Discrete time modeling of ecosystems with application in environmental enrichment. Mathematical Biosciences 24, 307–317 (1975).

    Google Scholar 

  • May, R. M.: On relationships among various types of population models. American Naturalist 107, 46–57 (1973a).

    Google Scholar 

  • May, R. M.: Stability and Complexity in Model Ecosystems. Princeton, N. J.: Princeton University Press 1973b.

    Google Scholar 

  • May, R. M.: Time delay versus stability in population models with two and three trophic levels. Ecology 54, 315–325 (1973c).

    Google Scholar 

  • May, R. M.: Stable points, stable cycles and chaos. Science 186, 645–647 (1974).

    Google Scholar 

  • May, R. M.: Biological populations obeying difference equations: stable points, stable cycles and chaos. Journal of Theoretical Biology 51, 511–525 (1975).

    Google Scholar 

  • May, R. M., Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. American Naturalist 110, 573–599 (1976).

    Google Scholar 

  • Maynard Smith, J.: Mathematical Ideas in Biology. Cambridge: Cambridge University Press 1968.

    Google Scholar 

  • Maynard Smith, J.: Models in Ecology. Cambridge: Cambridge University Press 1974.

    Google Scholar 

  • Maynard Smith, J., Slatkin, M.: The stability of predatorprey systems. Ecology 54, 384–391 (1973).

    Google Scholar 

  • Nicholson, A. J., Bailey, V. A.: The balance of animal populations. Part I. Proc. Zool. Soc. London 3, 551–598 (1935).

    Google Scholar 

  • Pielou, E. C.: An Introduction to Mathematical Ecology. New York: Wiley-Interscience 1969.

    Google Scholar 

  • Pierre, D. A.: Optimization Theory with Applications. New York: Wiley 1969.

    Google Scholar 

  • Poole, R. W.: An autoregressive model of population density change in an experimental population of daphnie magna. Ocealogia (Berl.) 10, 205–221 (1972).

    Google Scholar 

  • Scudo, F. M.: Vito Volterra and theoretical ecology. Theoretical Population Biology 2, 1–23 (1971).

    Google Scholar 

  • Scudo, F. M., Levine, S. H.: Unpublished data and manuscript, Discrete time models in ecology, 1973.

  • Volterra, V.: Variazoni e fluctuazioni del numero d'individui in specie animal conviventi. Mem. Accad. Lincei 2, 31–114 (1926).

    Google Scholar 

  • Wilson, E. O., Bessert, W. H.: A Primer of Population Biology. Stamford, Connecticut: Sinauer Associates Inc. 1971.

    Google Scholar 

Download references

Author information

Affiliations

  1. Department of Engineering, Merrimack College, 01845, North Andover, MA, USA

    S. H. Levine

  2. Laboratorio di Genetica Biochimica e Evoluzionistica, Pavia, Italy

    F. M. Scudo

  3. Department of Biology, Merrimack College, 01845, North Andover, MA, USA

    D. J. Plunkett

Authors
  1. S. H. Levine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. M. Scudo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. J. Plunkett
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Levine, S.H., Scudo, F.M. & Plunkett, D.J. Persistence and convergence of ecosystems: An analysis of some second order difference equations. J. Math. Biology 4, 171–182 (1977). https://doi.org/10.1007/BF00275982

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00275982

Keywords

  • Parameter Space
  • Stochastic Process
  • Equation Model
  • Probability Theory
  • Mathematical Biology