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.
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.
Beddington, J. R. 1974. Age distribution and the stability of simple discrete time population models.J. Theor. Biol. 47, 65–74.
Beverton, R. J. H. and S. J. Holt. 1957.On the Dynamics of Exploited Fish Populations. London: H. M. Stationary Offices, Fish. Invest.
Charlesworth, B. 1980.Evolution in Age-Structured Populations. Cambridge: Cambridge University Press.
Chaundy, T. W. and E. Phillips. 1936. The convergence of sequences defined by quadratic formulas.Quart. J. Math. 7, 74–80.
Clark, C. W. 1990.Mathematical Bioeconomics, 2nd ed. New York: Wiley.
Cook, L. M. 1965. Oscillation in the simple logistic growth model.Nature 207, 316.
Cooke, K. L. and M. Witten. 1986. One-dimensional linear and logistic harvesting models.Math. Modelling 7, 301–340.
Diekmann, O. and S. D. Mylius. 1995. About ESS, maximisation and the need to be specific about density dependence.Oikos 74, 218–224.
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.
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.
Hanski, I. 1989. Population biology of Eurasian shrews: towards a synthesis.Ann. Zool Fennici 26, 469–479.
Hassell, M. P. 1974. Density dependence in single-species populations.J. Animal Ecol. 44, 283–296.
Hassell, M. P. 1976.The Dynamics of Competition and Predation. Southampton: The Camelot Press.
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.
Krebs, C. J. 1972.Ecology: The Experimental Analysis of Distribution and Abundance. New York: Harper and Row.
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.
Li, T.-Y. and J. A. Yorke. 1975. Period three implies chaos.Amer. Math. Monthly 82, 985–992.
Macfadyen, A. 1963.Animal Ecology: Aims and Methods. London: Pitman.
May, R. M. 1973. On relationships between various types of population models.The American Naturalist 107, 46–57.
May, R. M. 1974a. Biological populations with nonoverlap** generations: stable points, stable cycles and chaos.Science 185, 645–647.
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.
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.
May, R. M. and G. F. Oster. 1976. Bifurcation and dynamic complexity in simple ecological models.The American Naturalist 110, 573–599.
Maynard Smith, J. 1968.Mathematical Ideas in Biology. Cambridge: Cambridge University Press.
Maynard Smith, J. 1974a.Models in Ecology. Cambridge: Cambridge University Press.
Maynard Smith, J. 1974b. The theory of games and the evolution of animal conflicts.J. Theor. Biol. 47, 209–221.
Maynard Smith, J. and G. R. Price. 1973. The logic of animal conflict.Nature 246, 15–18.
Metz, J. A. J. and O. Diekmannn. 1986.The Dynamics of Physiologically Structured Populations. Berlin: Springer-Verlag.
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.
Moran, P. A. P. 1950. Some remarks on animal populations dynamics.Biometrics 6, 250–258.
Murray, J. D. 1989.Mathematical Biology. New York: Springer-Verlag.
Nicholson, A. J. 1954. An outline of the dynamics of animal populations.Austral. J. Zool. 2, 9–65.
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.
Pielou, E. C. 1977.Mathematical Ecology. New York: Wiley.
Ricker, W. E. 1954. Stock and recruitment.J. Fish. Res. Board. Can. 11, 559–623.
Skellam, J. G. 1951. Random dispersal in theoretical populations.Biometrika 38, 196–218.
Stearns, S. C. 1992.The Evolution of Life Histories. London: Oxford University Press.
Usher, M. B. 1972. Developments of the Leslie matrix model. InMathematical Models in Ecology, J. N. R. Jeffers (Ed), pp. 29–60. London: Blackwell.
Utida, S. 1957. Population fluctuation, an experimental and theoretical approach.Cold Spring Harbor Symp. Quant. Biol. 22, 139–151.
Utida, S. 1967. Damped oscillation of population density at equilibrium.Res. Pop. Ecol. 9, 1–9.
Varley, G. C., G. R. Gradwell and M. P. Hassell. 1973.Insect. Population Ecology. Oxford: Blackwell.
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.
Yodzis, P. 1989.Introduction to Theoretical Ecology. New York: Harper and Row.
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02458425