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, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Allen, J. C.: Mathematical models of species interactions in time and space. American Naturalist 109, 319–342 (1975).
Beddington, J. R., Free, C. A., Lawton, J. H.: Dynamic complexity in predator-prey models framed in difference equations. Nature 255, 58–60 (1975).
Cook, L. M.: Coefficients of Natural Selection. London: Hutchinson and Company 1971.
Fretwell, S.: Populations in Seasonal Environments. Princeton, N. J.: Princeton University Press 1972.
Hassell, M. P., Varley, G. C.: New inductive population model for insect parasites and its bearing on biological control. Nature 223, 1133–1137 (1969).
Léon, J. A.: Limit cycles in populations with separate generations. Journal of Theoretical Biology 49, 241–244 (1975).
Leslie, P. H.: The use of matrices in certain population mathematics. Biometrika 35, 213–245 (1945).
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).
May, R. M.: On relationships among various types of population models. American Naturalist 107, 46–57 (1973a).
May, R. M.: Stability and Complexity in Model Ecosystems. Princeton, N. J.: Princeton University Press 1973b.
May, R. M.: Time delay versus stability in population models with two and three trophic levels. Ecology 54, 315–325 (1973c).
May, R. M.: Stable points, stable cycles and chaos. Science 186, 645–647 (1974).
May, R. M.: Biological populations obeying difference equations: stable points, stable cycles and chaos. Journal of Theoretical Biology 51, 511–525 (1975).
May, R. M., Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. American Naturalist 110, 573–599 (1976).
Maynard Smith, J.: Mathematical Ideas in Biology. Cambridge: Cambridge University Press 1968.
Maynard Smith, J.: Models in Ecology. Cambridge: Cambridge University Press 1974.
Maynard Smith, J., Slatkin, M.: The stability of predatorprey systems. Ecology 54, 384–391 (1973).
Nicholson, A. J., Bailey, V. A.: The balance of animal populations. Part I. Proc. Zool. Soc. London 3, 551–598 (1935).
Pielou, E. C.: An Introduction to Mathematical Ecology. New York: Wiley-Interscience 1969.
Pierre, D. A.: Optimization Theory with Applications. New York: Wiley 1969.
Poole, R. W.: An autoregressive model of population density change in an experimental population of daphnie magna. Ocealogia (Berl.) 10, 205–221 (1972).
Scudo, F. M.: Vito Volterra and theoretical ecology. Theoretical Population Biology 2, 1–23 (1971).
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).
Wilson, E. O., Bessert, W. H.: A Primer of Population Biology. Stamford, Connecticut: Sinauer Associates Inc. 1971.
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
- Parameter Space
- Stochastic Process
- Equation Model
- Probability Theory
- Mathematical Biology