Density dependence and the response surface methodology

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Applying response surface methodology to ecological problems endeavours to disentangle the roles of endogenous and exogenous factors in population fluctuations. We explored the reliability of response surface methodology in revealing two fundamental properties of a given population series: whether the density dependence is direct or delayed, and whether it is linear or nonlinear. This was done using simulated time series with known properties as well as a large set of real population time series. We show the importance of seeking the simplest form of the response surface among several alternatives, by fitting a set of response surfaces from simple first order to more complex second order nonlinear models to each time series. Performance of the models was judged by three methods: cross-validation, adjusted coefficient of determination, and checking residual behaviour. The results show that with proper model validation, the response surface methodology is not only capable of finding the numerical relationship between population growth rate and its size or density, but can also be used to reliably reveal the delay in density dependence when it is of significant importance. However, judging nonlinearity on the basis of the response surface is generally not as evident. We reanalysed the data sets of Turchin and Taylor and show that validated analysis leads to a somewhat smaller set of dynamical alternatives being accepted. Finally, we applied the method to a long-term data set on three grouse species. The results show strong evidence for delayed density dependence. Response surface methodology and data fitting to Royama's form of feedback function show very convergent results.

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