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Forms of density regulation and (quasi-) stationary distributions of population sizes in birds


  • Bernt-Erik Sæther,

  • Steinar Engen,

  • Vidar Grøtan,

  • Thomas Bregnballe,

  • Christiaan Both,

  • Piotr Tryjanowski,

  • Agu Leivits,

  • Jonathan Wright,

  • Anders Pape Møller,

  • Marcel E. Visser,

  • Wolfgang Winkel

B.-E. Sæther (, V. Grøtan and J. Wright, Centre for Conservation Biology, Dept of Biology, Norwegian Univ. of Science and Technology, NO–7491 Trondheim, Norway. – S. Engen, Centre for Conservation Biology, Dept of Mathematical Sciences, Norwegian Univ. of Science and Technology, NO–7491 Trondheim, Norway. – T. Bregnballe, Dept of Wildlife Ecology and Biodiversity, National Environmental Res. Inst., Univ. of Aarhus, Kalø, Grenåvej 14, DK–8410 Rønde, Denmark. – C. Both, Animal Ecology Group, Centre for Ecological and Evolutionary Studies, Univ. of Groningen, PO Box 14, NL–9750 AA Haren, the Netherlands. – P. Tryjanowski, Dept of Behavioural Ecology, Adam Mickiewicz Univ., Umultowska 89, PL–61614 Poznan, Poland. – A. Leivits, Nigula Nature Reserve Administration, Vana-Järve, EE–86301 Tali Pr'mumaa, Estonia. – A. P. Møller, Laboratorie de Parasitologie Evolutive, CNRS UMR 7103, Univ. Pierre et Marie Curie, Bâtiment A, 7ème étage, 7 Quai St. Bernard, Case 237, FR–75252 Paris Cedex 05, France. – M. E. Visser, Netherlands Inst. of Ecology, PO Box 40, NL–6666 ZG Heteren, the Netherlands. – W. Winkel, Inst. of Avian Research ‘Vogelwarte Helgoland’, An der Vogelwarte 21, DE–26386 Wilhemshaven, Germany.


The theta-logistic model of density regulation is an especially flexible class of density regulation models where different forms of non-linear density regulation can be expressed by only one parameter, θ. Estimating the parameters of the theta-logistic model is, however, challenging. This is mainly due to the need for information concerning population growth at low densities as well as data on fluctuations around the carrying capacity K in order to estimate the strength of density regulation. Here we estimate parameters of the theta-logistic model for 28 populations of three species of birds that have grown from very small population sizes followed by a period of fluctuations around K. We then use these parameters to estimate the quasi-stationary distribution of population size. There were often large uncertainties in these parameters specifying the form of density regulation that were generally independent of the duration of the study period. In contrast, precision in the estimates of environmental variance increased with the length of the time series. In most of the populations, a large proportion of the probability density of the (quasi-) stationary distribution of population sizes was located at intermediate population sizes relative to K. Thus, we suggest that the (quasi-) stationary distribution of population sizes represents a useful summary statistic that in many cases provides a more robust characterisation of basic population dynamics (e.g. range of variation in population fluctuations or proportion of time spent close to K) than can be obtained from analyses of single model parameters.