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.
Sæther, B-E., Engen, S., Grøtan, V., Bregnballe, T., Both, C., Tryjanowski, P., Leivits, A., Wright, J., Møller, A. P., Visser, M. E., & Winkel, W. (2008). Forms of density regulation and (quasi-) stationary distributions of population sizes in birds. Oikos, 117(8), 1197-1208. https://doi.org/10.1111/j.0030-1299.2008.16420.x