N. L. Fowler, R. Deborah Overath, and Craig M. Pease. 2006. Detection of density dependence requires density manipulations and calculation of . Ecology 87:655–664.

Appendix B. Testing the correlation between density and finite rate of increase.

Test of density dependence in the unmanipulated quadrats (controls)

To test for density dependence in the 14 annual intervals of observational (non-experimental) data from the control quadrats (Data Set B; Fig. B1), we extended the method of Pollard et al. (1987). This analysis looks for a correlation between density and finite rate of increase, after accounting for the well known spurious bias in such correlations.

Pollard et al. (1987) assumed no stage structure, started with the initial observed population size, randomly sampled without replacement from the observed values of N_t+1/N_t, and then multiplied the value of N_t+1/N_t so obtained by the initial population size, to compute population size in the next year. They continued this procedure until all empirically observed values of N_t+1/N_t were used (i.e., sampling without replacement). They computed the critical value of their test statistic, the correlation of density and N_t+1/N_t, under the null model that the values of N_t+1/N_t were ordered randomly (i.e., no density dependence).

Because the effective density experienced by an individual plant is strongly affected by the sizes of the neighboring plants, we used total number of tillers, not total number of plants, as the measure of density. For each size-class-distribution vector, we estimated total number of tillers by multiplying the number of plants in each size class by the average size of plants in that size class (1.28, 3.40, 5.84, 11.15, and 38.54 tillers per plant, respectively). We obtained these average plant sizes by averaging plant size in a given size class first by quadrat, and then averaging the quadrat averages across all 14 annual intervals and all 6 quadrat-groups, using only control quadrats.

Instead of multiplying scalar values of N by values of N_t+1/N_t (equal to λ in a scalar model), as Pollard et al. did, we multiplied size-class-distribution vectors by population projection matrices randomly drawn from the empirically observed distribution of transition matrices. We computed the matrices used for this test by averaging the six (control) population projection matrices of each year, element by element, to provide an ‘average matrix’ for that year. Since there were 14 annual intervals in this data set, there were 14 ‘average’ matrices, and we calculated the principle eigenvalue (λ) of each. We obtained the initial size vector from the number of individuals in each size class in 1983, averaged over all six control quadrats. We then calculated a new size vector for each year in turn by multiplying the previous year’s size vector by one of the 14 ‘average’ matrices, selected randomly without replacement. Thus each replicate simulation represented a random permutation of the 14 ‘average’ matrices. We calculated 10,000 replicates of simulated population growth over a period of 14 annual intervals. Our test statistic is the correlation of total tiller number at the beginning of the interval with λ, the principle eigenvalue, both computed for each annual interval separately. Each of the 10,000 simulations provided one value of this test statistic.

Alternative test of density dependence in the density-manipulated quadrats

Using simulations, we determined the distribution of the correlation of initial density and λ in the density-manipulation data set (Data Set A) under the null hypothesis of density independence. This data set had 54 values of λ and 54 values of initial density (3 annual intervals × 3 treatments × 6 quadrats per treatment = 54). In each simulation, λ and the initial number of tillers in a quadrat were each randomly and separately sampled 18 times, with replacement, from the data set. After 18 pairs of values were sampled, we calculated their correlation coefficient. We repeated this procedure 10,000 times to estimate the distribution of the correlation coefficient of total tiller number and λ.

We calculated a correlation coefficient across quadrats (i.e., across space) for each year separately in the experimental data set, to avoid the auto-correlation that may exist between values from the same quadrat in different years. Not surprisingly, the expected value of our test statistic is not biased, as happens when the correlation is computed across time rather than space. Note that although these correlations of λ and density were calculated for each year separately, resulting in three correlation coefficients for Data Set A, only a single correlation coefficient of λ and density was calculated for the unmanipulated data set (Data Set B) using all years and calculating each matrix by averaging quadrats within a year as described above.

LITERATURE CITED

Pollard, E., K. H. Lakhani, and P. Rothery. 1987. The detection of density-dependence from a series of annual censuses. Ecology 68:2046–2055.

FIG. B1. Total numbers of tillers and total number of plants in each of the six control quadrats.