Skip to main content
SpringerLink
Log in

Continuous versus discrete single species population models with adjustable reproductive strategies

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

Abstract

We investigate population models with both continuous and discrete elements. Birth is assumed to occur at discrete instants of time whereas death and competition for resources and space occur continuously during the season. We compare the dynamics of such discrete-continuous hybrid models with the dynamics of purely discrete models where within-season mortality and competition are modelled directly as discrete events. We show that non-monotone discrete single-species maps cannot be derived from unstructured competition processes. This result is well known in the case of fixed reproductive strategies and our results extend this to the case of adjustable reproductive strategies. It is also shown that the most commonly used non-monotone discrete maps can be derived from structured competition processes.

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

  • Allen, L. J. S., M. K. Hannigan and M. J. Strauss. 1993. Mathematical analysis of a model for a plant-herbivore system.Bull. Math. Biol. 55, 847–864.

    Article  MATH  Google Scholar 

  • Beddington, J. R. 1974. Age distribution and the stability of simple discrete time population models.J. Theor. Biol. 47, 65–74.

    Article  Google Scholar 

  • Beverton, R. J. H. and S. J. Holt. 1957.On the Dynamics of Exploited Fish Populations. London: H. M. Stationary Offices, Fish. Invest.

    Google Scholar 

  • Charlesworth, B. 1980.Evolution in Age-Structured Populations. Cambridge: Cambridge University Press.

    Google Scholar 

  • Chaundy, T. W. and E. Phillips. 1936. The convergence of sequences defined by quadratic formulas.Quart. J. Math. 7, 74–80.

    MATH  Google Scholar 

  • Clark, C. W. 1990.Mathematical Bioeconomics, 2nd ed. New York: Wiley.

    Google Scholar 

  • Cook, L. M. 1965. Oscillation in the simple logistic growth model.Nature 207, 316.

    Article  Google Scholar 

  • Cooke, K. L. and M. Witten. 1986. One-dimensional linear and logistic harvesting models.Math. Modelling 7, 301–340.

    Article  MATH  MathSciNet  Google Scholar 

  • Diekmann, O. and S. D. Mylius. 1995. About ESS, maximisation and the need to be specific about density dependence.Oikos 74, 218–224.

    Google Scholar 

  • Diekmann, O., M. Gyllenberg, J. A. J. Metz and H. R. Thieme. 1997. On the formulation and analysis of general deterministic structured population models. I. Linear theory.J. Math. Biol., to appear.

  • Fisher, R. A. 1930.The Genetical Theory of Natural Selection. Oxford: Clarendon Press.

    Google Scholar 

  • Gyllenberg, M., I. Hanski and T. Lindström. 1996. A predator-prey model with optimal suppression of reproduction in the prey.Math. Biosci. 134, 119–152.

    Article  MATH  MathSciNet  Google Scholar 

  • Hanski, I. 1989. Population biology of Eurasian shrews: towards a synthesis.Ann. Zool Fennici 26, 469–479.

    Google Scholar 

  • Hassell, M. P. 1974. Density dependence in single-species populations.J. Animal Ecol. 44, 283–296.

    Article  Google Scholar 

  • Hassell, M. P. 1976.The Dynamics of Competition and Predation. Southampton: The Camelot Press.

    Google Scholar 

  • Hassell, M. P., J. H. Lawton and R. M. May. 1976. Patterns of dynamical behavior in single species populations.J. Animal Ecol. 45, 471–486.

    Article  Google Scholar 

  • Krebs, C. J. 1972.Ecology: The Experimental Analysis of Distribution and Abundance. New York: Harper and Row.

    Google Scholar 

  • Leslie, P. H. 1957. The analysis of the data for some experiments carried out by Gause with populations of the protozoa,Paramecium aurelia andParamecium caudatum.Biometrika 44, 314–327.

    Article  MATH  Google Scholar 

  • Li, T.-Y. and J. A. Yorke. 1975. Period three implies chaos.Amer. Math. Monthly 82, 985–992.

    Article  MATH  MathSciNet  Google Scholar 

  • Macfadyen, A. 1963.Animal Ecology: Aims and Methods. London: Pitman.

    Google Scholar 

  • May, R. M. 1973. On relationships between various types of population models.The American Naturalist 107, 46–57.

    Article  Google Scholar 

  • May, R. M. 1974a. Biological populations with nonoverlapping generations: stable points, stable cycles and chaos.Science 185, 645–647.

    Google Scholar 

  • May, R. M. 1974b. Ecosystem patterns in randomly fluctuating environments. InProgress in Theoretical Biology, R. Rosen and F. Snell (Eds), pp. 1–50. New York: Academic Press.

    Google Scholar 

  • May, R. M. 1981. Models for single populations. InTheoretical Ecology: Principles and Applications, 2nd ed., R. M. May (Ed), pp. 5–29. Oxford: Blackwell Scientific Publications.

    Google Scholar 

  • May, R. M. and G. F. Oster. 1976. Bifurcation and dynamic complexity in simple ecological models.The American Naturalist 110, 573–599.

    Article  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Maynard Smith, J. 1974b. The theory of games and the evolution of animal conflicts.J. Theor. Biol. 47, 209–221.

    Article  MathSciNet  Google Scholar 

  • Maynard Smith, J. and G. R. Price. 1973. The logic of animal conflict.Nature 246, 15–18.

    Article  Google Scholar 

  • Metz, J. A. J. and O. Diekmannn. 1986.The Dynamics of Physiologically Structured Populations. Berlin: Springer-Verlag.

    Google Scholar 

  • Milinski, M. and G. A. Parker. 1991. Competition for resources. InBehavioural Ecology, 3rd ed. J. R. Krebs and N. B. Davies (Eds), pp. 137–168. Oxford: Blackwell.

    Google Scholar 

  • Moran, P. A. P. 1950. Some remarks on animal populations dynamics.Biometrics 6, 250–258.

    Article  Google Scholar 

  • Murray, J. D. 1989.Mathematical Biology. New York: Springer-Verlag.

    Google Scholar 

  • Nicholson, A. J. 1954. An outline of the dynamics of animal populations.Austral. J. Zool. 2, 9–65.

    Article  Google Scholar 

  • Penycuick, C. J., R. M. Compton and L. Beckingham. 1968. A computer model for simulating the growth of a population, or of two interacting populations.J. Theor. Biol. 18, 316–329.

    Article  Google Scholar 

  • Pielou, E. C. 1977.Mathematical Ecology. New York: Wiley.

    Google Scholar 

  • Ricker, W. E. 1954. Stock and recruitment.J. Fish. Res. Board. Can. 11, 559–623.

    Google Scholar 

  • Skellam, J. G. 1951. Random dispersal in theoretical populations.Biometrika 38, 196–218.

    Article  MATH  MathSciNet  Google Scholar 

  • Stearns, S. C. 1992.The Evolution of Life Histories. London: Oxford University Press.

    Google Scholar 

  • Usher, M. B. 1972. Developments of the Leslie matrix model. InMathematical Models in Ecology, J. N. R. Jeffers (Ed), pp. 29–60. London: Blackwell.

    Google Scholar 

  • Utida, S. 1957. Population fluctuation, an experimental and theoretical approach.Cold Spring Harbor Symp. Quant. Biol. 22, 139–151.

    Google Scholar 

  • Utida, S. 1967. Damped oscillation of population density at equilibrium.Res. Pop. Ecol. 9, 1–9.

    Google Scholar 

  • Varley, G. C., G. R. Gradwell and M. P. Hassell. 1973.Insect. Population Ecology. Oxford: Blackwell.

    Google Scholar 

  • Williamson, M. 1974. The analysis of discrete time cycles. InEcological Stability, M. B. Usher and M. Williamson (Eds), pp. 7–33. London: Chapman and Hall.

    Google Scholar 

  • Yodzis, P. 1989.Introduction to Theoretical Ecology. New York: Harper and Row.

    Google Scholar 

Download references

Author information

Author notes

  1. Torsten Lindström

    Present address: Division of Zoology, University of Oslo, Department of Biology, Blindern, P.O. Box 1050, N-0316, Oslo, Norway

Authors and Affiliations

  1. Department of Mathematics, University of Turku, FIN-20014, Turku, Finland

    Mats Gyllenberg

  2. Department of Ecology and Systematics, University of Helsinki, FIN-00014, Helsinki, Finland

    Ilkka Hanski

  3. Department of Technology and Natural Sciences, University of Örebro, S-70182, Örebro, Sweden

    Torsten Lindström

Authors
  1. Mats Gyllenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ilkka Hanski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Torsten Lindström
    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

Gyllenberg, M., Hanski, I. & Lindström, T. Continuous versus discrete single species population models with adjustable reproductive strategies. Bltn Mathcal Biology 59, 679–705 (1997). https://doi.org/10.1007/BF02458425

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02458425

Keywords

Search

Navigation