Population Dynamics: Introduction


Population dynamics, meaning changes in population size in time and/or space, can be understood with reference to simple mathematical models. The dynamics include fluctuation around a stable equilibrium, cycles and chaos and are driven by ecological processes such as competition and predation.

Keywords: density dependence; population regulation; mathematical models; predator–prey dynamics

Figure 1.

Two examples of population dynamics. (a) Extinction and persistence of two species of mite, one of which feeds on the other, in a laboratory experiment (details in text). (b) Fluctuations of female red grouse in Swaledale, North Yorkshire.

Figure 2.

Life cycle of the cinnabar moth in southern Britain (not to scale).

Figure 3.

Summary of a density‐independent model of cinnabar moth population dynamics. Survival and fecundity values are based on the work of Dempster and colleagues.

Figure 4.

Examples of survival and fecundity altering with population density. (a) and (b) Drosophila melanogaster. (c) and (d) Parus major.

Figure 5.

From stability to chaos with the discrete logistic equation (K=100). (a) Stable equilibrium (λ=2). (b) Two‐point limit cycles (λ=3.2). (c) Chaos (λ=3.8). All plots start with N1=10.

Figure 6.

Estimated values of λ and b (indicating the strength of density dependence) for 28 populations of insect, overlaid on regions of different dynamic behaviour. Open circles, laboratory populations; closed circles, field populations. Original source Hassell et al. (1976).

Figure 7.

(a) Geometric growth in discrete time. (b) Geometric (exponential) growth in continuous time. (c and d) Geometric growth in continuous time showing tangent to curve for (c) positive and (d) negative rates of change.

Figure 8.

Cycles of abundance of the larch bud‐moth and the larch (needle length) (original source Baltensweiler, 1993).

Figure 9.

Cycles in the number of lynx fur returns of the Hudson's Bay Company, from 1821 to 1934, grouped into five regions. Note the different scales (original source Elton and Nicholson, 1942).

Figure 10.

Dynamics of host and parasitoid in (a) laboratory compared with the Nicholson–Bailey model and (b) stable oscillations of host and parasitoid.


Further Reading

Anderson RM and May RM (1981) The population dynamics of micro‐parasites and their invertebrate hosts. Philosophical Transactions of the Royal Society B 291: 451–524.

Baltensweiler W (1993) Why the larch bud‐moth cycle collapsed in the subalpine larch‐cembran pine forests in the year 1990 for the first time since 1850. Oecologia 94: 62–66.

Dempster JP and Lakhani KH (1979) A population model for cinnabar moth and its food plant, ragwort. Journal of Animal Ecology 48: 143–163.

Ellner S and Turchin P (1995) Chaos in a noisy world: new methods and evidence from time‐series analysis. American Naturalist 245: 343–375.

Elton CS and Nicholson M (1942) The ten‐year cycle in numbers of the lynx in Canada. Journal of Animal Ecology 11: 215–244.

Hassell MP, Lawton JH and May RM (1976) Patterns of dynamical behaviour in single species populations. Journal of Animal Ecology 45: 471–486.

Krebs CJ (1994) Ecology. New York: Harper Collins.

May RM (1976) Simple mathematical models with very complicated dynamics. Nature 261: 459–467.

Nicholson AJ and Bailey VA (1935) The balance of animal populations. Proceedings of the Zoological Society, Part 1. London 3: 551–598.

Pearl R and Reed LJ (1920) On the rate of growth of the population in the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences of the USA 6: 275–288.

Varley GC (1947) The natural control of population balance in the knapweed gallfly. Journal of Animal Ecology 16: 139–187.

Volterra V (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118: 558–560.

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Gillman, Michael(Sep 2005) Population Dynamics: Introduction. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1038/npg.els.0003164]