New insights into testing for density dependence.

The reasons why tests for density dependence often differ in their results for a particular time-series were investigated using modelled time-series of 20 generations in lenght. The test of Pollard et al. (1987) is the most reliable; it had the greatest power with the three forms of density dependent data investigated (mean detection rates of 50.8-61.1%) and was least influenced by the form of the density dependence in time-series. Bulmer's first test (Bulmer 1975) had slightly lower power (mean detection rates of 27.4-56.8%) and was more affected by the form of density dependence present in the data. The mean power of the other tests was lower and detection rates were more variable. Rates were 24.6-46.2% for regression of k-value on abundance, 6.4-32.6% for regression of k-value on logarithmic abundance and 0.2-13.7% for Bulmer's second test (Bulmer 1975). Bulmer's second test is not useful because of low power. For one method, regression of k-value on abundance. density dependence was detected in 19.9% of timeseries generated using a random-walk model. For regression of k-value on logarithmically-transformed abundance the equivalent figure was 18.3% of series. These rates of spurious detection were significantly (P<0.001) greater than the generally accepted 5% level of type 1 errors and so these methods are not suitable for the analysis of time-series data for density dependence. Levels of spurious detection (from random-walk data) were around the 5% level and hence were acceptable for Bulmer's first test, Bulmer's second test, and the tests of Pollard et al. (1987), Reddinguis and den Boer (1989) and Crowley (1992). For all tests, except Bulmer's second test, the rate of detection and the amount of autocorrelation in time-series were negatively correlated. The degree of autocorrelation accounted for as much as 59.5-77.9% of the deviance in logit proportion detection for regression of k-value on abundance, Bulmer's first test, and the tests of Pollard et al. and Reddingius and den Boer. For regression of k-value on abundance this relationship accounted for less of the deviance (29.4%). Independent effects of density dependence were largely absent. It is concluded that these are tests of autocorrelation, not density dependence (or limitation). Autocorrelation was found to become positive (which is similar to values from random-walk data) as the intrinsic growth rate became either small or large. As the strength of density dependence (in the discrete exponential logistic equation) is dependent on the product of the intrinsic growth rate and the density dependent parameter α it is unclear whether this is because of variation in the strength of density dependent mortality or reproduction per se. However, small values of the intrinsic grwoth rate cause the amount of variation in the data to become small, which might hinder detection of density dependence, and large values of the intrinsic growth rate are coincident with determinstic chaos which hinders detection. The user of these tests for density dependence should be aware of their potential weakness when variation within time-series is small (which itself is difficult to judge) or if the intrinsic growth rate is large so that chaotic dynamics might result. Power and levels of variability in rates of detection using Reddingius and den Boer's test were intermediate between those of the test of Pollard et al. and Bulmer's first test. This, combined with the strong relationship between rates of detection of limitation and the value of the autocorrelation coefficient, make testing for limitation similar to testing for density dependence. Crowley's test of attraction gave the widest range of mean detection rates from density dependent data of all the tests (20.4-60.6%). The relative rates of detection for the three forms of density dependent data were opposite to those found for Bulmer's first test and the test of Pollard et al. I conclude that testing for attraction is a complementary concept to testing for density dependence. As dynamics represented in time-series generated using a stochastic form of the exponential logistic equation became chaotic, Bulmer's first test, the test of Pollard et al. and regression of k on abundance failed to detect density dependence reliably. Conversely, Crowley's test was capable of detecting attraction with a power between 96 and 100% with time-series containing both stochastically and deterministically chaotic dynamics. This difference from other tests is in agreement with the lower influence of autocorrelation.