#### General approach

The approach used (Fig. 1) involved experimental manipulation of the density of an entire community without altering initial relative abundances of species. The underlying assumption was that increasing initial density increases the potential for interactions among individuals and thus reflects potential competition intensity (Goldberg *et al*. 1995). A ‘null community’, i.e. the expected species composition if species are not differentially affected by interactions, is derived from the lowest density plots. A computer program is used to take repeated samples from this null community and generate the expected values of probability distributions of the number of species (or other diversity indices) for plots with different total densities (the solid curve in Fig. 1). These predictions are then compared with the experimentally observed values. The biological null hypothesis is that species richness at the end of an experiment will vary solely as a function of the variation in initial density, i.e. only sampling influences richness (quantity *a* in Fig. 1). This is equivalent to the statistical null hypothesis that the probability that a given individual belongs to a particular species is invariant among all plots, i.e. it is independent of density or any other parameter.

Competitive interactions could modify the effect of higher density in two distinct ways, both involving an increase in mortality at higher density (or decrease in birth rate) and therefore leading to a final density that is lower than the initial density (Fig. 1). First, to the extent that such density-dependent mortality is randomly allocated among species, richness will decrease solely due to the decrease in number of individuals sampled at the end of the experiment (quantity *b* in Fig. 1). Secondly, if this mortality falls differentially on initially rare species, richness will be further decreased relative to that expected based on sampling alone (quantity *c* in Fig. 1). If, however, competitive mortality falls differentially on initially very common species, richness might actually be increased relative to that expected based on sampling alone (not shown in Fig. 1). Thus, increased diversity at higher density, even after taking into account sampling effects, will not necessarily reflect facilitation at the individual level. The net change in final richness between plots with low and high density (quantity *d* in Fig. 1) reflects the balance of increases due to sampling and changes due to density-dependent mortality or fecundity. It is therefore impossible to quantify either kind of competitive effect or even detect that it exists from such data, unless sampling effects are also considered.

#### Experimental system

The experimental system was an annual plant community occurring on semi-stabilized sand dunes in the Negev Desert in Israel. The combination of annual life history and the sandy substrate made it possible to collect a community seed bank and to concentrate it by sieving. This concentrated seed bank was then thoroughly mixed to reduce seed aggregations and planted in an experimental garden in eight density treatments: 1/16, 1/8, 1/4, 1/2, 1, 2, 4 and 8 times the natural density of the seed bank. Because greater variation was expected in the lower density plots, the two lowest densities were replicated four times, while all higher densities were replicated twice, for a total of 20 plots. The density treatments up to and including natural density were planted in 1-m^{2} plots. However, because of the labour involved in collecting and sieving the seed bank, the higher density treatments were planted in smaller plots (0.25 m^{2}) to reduce the total amount of seed bank needed (subsequent experiments in progress use large plots regardless of density). The experimental communities were planted in January 1993, shortly after the winter rainy season had commenced, and harvested in April 1993 at the end of the growing season. Only plants within the central 80 × 80 cm area (1-m^{2} plots) or 30 × 30 cm area (0.25-m^{2} plots) were harvested, sorted to species, counted, dried and weighed. The experiment reported here is part of a larger project investigating the effects of productivity on the community-level consequences of competition and the relationship between individual and community-level responses to competition (D. E. Goldberg, R. Turkington & L. Olsvig-Whittaker, unpublished data).

We characterized the null community by two different species pools. In both cases, only plants surviving to the end of the growing season were included because it was impossible to identify all plants to species at the initial germination phase. We therefore did not have accurate measures of species richness as a function of initial density and so could not quantify effects of randomly allocated density-dependent mortality on species richness (quantity *b* in Fig. 1). However, there was very little density-dependent mortality overall in this experiment and it is therefore likely that effects of randomly allocated density-dependent mortality on richness were weak, even if present.

The first null community initially only used the surviving plants in the lowest density plots (four at 1/16 of the natural density and four at 1/8 of the natural density) as the species pool from which to draw individuals at random. This is biologically the most appropriate species pool to use because it should exhibit minimal density-dependent effects on germination or mortality. However, the total number of individuals in these plots was relatively small (420 vs. 3953 individuals summed over all densities) and thus might, by chance, have non-representative relative abundances. In addition, some relatively rare species in the high-density communities were not present at all in the low-density (no-interaction) communities. To make more plausible predicted distributions possible, we therefore added a single individual of all such species (20 out of 53 total species found in the 20 experimental plots) to this species pool.

In the second species pool, we included all surviving plants in the experiment, regardless of the initial density of the plot. This got around the problems created by the small number of individuals in the low-density plots. On the other hand, this species pool is likely to be biased by species-specific density-dependent germination or mortality at the higher densities. Thus, detection of systematically stronger deviations from expected species composition or diversity at high relative to low density should be strong evidence that, despite this bias, increasing interactions at higher density do affect community structure.

The proportion of individuals belonging to each species in the null community was used as its hypothesized ‘invariant probability’, i.e. its abundance expected under the hypothesis of no effect of interactions on species richness. These probabilities were used to generate a set of 10 000 simulated plots for each sample plot, each having the same total number of individuals as its corresponding observed sample plot. For each simulated plot, the species identity of each individual was assigned independently, using the hypothesized invariant probabilities, and these species’ identities were used to calculate species richness and the Shannon–Weiner and Simpson diversity indices. The distributions of richness and diversity values from the set of simulated plots were then compared to the corresponding observed value. For each sample plot, the fraction of simulated values greater than or equal to an observed value is referred to as its ‘realized high significance’, i.e. the probability that diversity at least as high as the observed value could be found by random sampling from the null community. Similarly, the fraction of simulated values less than or equal to each observed value is referred to as its ‘realized low significance’, i.e. the probability that diversity as low or lower than the observed value could be found by random sampling from the null community. The expected value for a given index at a particular density is estimated by the sum of all the simulated values at that density divided by the number of simulations. The deviations of observed values from these estimated expected values indicate the magnitude of any effects of density on richness and diversity, and the realized significances provide statistical bases for interpreting these magnitudes.

To make the magnitudes of deviations of observed values (O) from estimated expected values (E) more intuitively comparable, we also expressed them in units of expected value (O – E)/E. This index gives increasingly negative values as the observed value becomes increasingly smaller than the estimated expected value. This rescaling does not alter the statistical significance of an observed value.

A computer program, divden (diversity–density analysis), to carry out these analyses, available in Windows 3.1 or DOS versions, can be downloaded from the Journal of Ecology archive on the World Wide Web (for address, see cover of a recent issue).