When selecting a null model with which to test a hypothesis, it is essential to keep every feature of the randomized data as it is in the observed data, except the feature that the study aims to test (Tokeshi 1986). This study tests whether there are limitations to coexistence related to the functional characters measured. Therefore, the observed occurrences and abundances of species within points, quadrats and areas were fixed within the randomized communities at those observed, but the observed characters were randomized (Appendix S1 in Supplementary Material). That is, the actual character values measured from species within these communities were retained, not generated de novo, but the characters were assigned to species at random without replacement within the null model. By maintaining the observed community structure within all of the randomized communities, any spatial autocorrelation arising from the sampling regime cannot affect the results.
When randomizing the allocation of characters to species, several possibilities exist. Assigning to species completely at random, with no regard to the frequency of the species, could result in giving too much weight in the randomized communities to the characters associated with a very rare species, were they assigned to the occurrences of an extremely common species. To overcome this problem, the characters were randomized within two groups of species: the 50% of species most frequent across the site vs. the remainder. The selection of frequency classes to use is not simple. Too many groups will result in few species within each group, with a resulting loss of power due to the restriction upon the possible randomizations, and too few classes can potentially lead to incorrect weighting of characters. Due to the low species diversity found within this community (nine species), two groups were used as this prevents the rarest characters being allocated to one of the most abundant species while maintaining sufficient power. As the characters of a plant do not act independently of one another, but rather are part of an integrated individual (Diaz et al. 1999), in order to maintain biological realism within the randomized communities the characters associated with a species were kept together when randomly assigning the character values to a species within the randomized community, thus preserving the observed character-correlation structure.
Test statistics (see below) were calculated for samples at each of the four scales: points, mini-quadrats, quadrats and areas (Fig. 2), and averaged across the site. The frequency of species occurrences at the point and quadrat scale were used as an estimate of the abundance of species at the large-quadrat and area scales, giving both presence/absence and quantitative analyses at these two scales.
test statistics (ts)
Because of the paucity of limiting similarity studies on plant communities, it is not clear what form a non-random pattern of niches would take. Therefore, an exploratory approach was taken in this study, using a wide range of test statistics (TSs), as advocated by Fekete et al. (1976), protected by a binomial test. In each case, the test statistic was calculated through comparison of the different species co-occurring within a sample, and then compared with the test statistic calculated on the species co-occurring within the randomized community.
Characters do not act independently of one another, and species may separate from one another via a combination of characters (Précsényi et al. 1977). The communities were therefore analysed in terms of the distribution of species in multivariate functional space:
TS1, Mean NN ED: the mean nearest-neighbour Euclidean distance (Jongman et al. 1987) between species present in each sample (Fig. 3). The dissimilarity between nearest-neighbours within each sample was weighted by the product of the abundances of those two species in the samples, and the total divided by the number of comparisons. This test statistic provides an indication of how tightly packed the species are: the smaller the mean nearest-neighbour Euclidean distance the more closely packed the species.
Figure 3. The Euclidean distances between the nearest neighbours (NN ED) for (a) clumped, (b) even and (c) random distributions. Evenly spaced distributions (centre) will have a greater mean NN ED.
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TS2, Min/Max MST link: the minimum link/maximum link in the minimum spanning tree (MST) for each sample. The Euclidean distances between each species pair were used to calculate a minimum spanning tree (MST, Cormack 1971). This test statistic is a measure of the evenness of the species in functional space, a large value indicating that the species in the observed community are spaced evenly. In the extreme case, a value of 1.0 indicates that the shortest link in the MST is equal to the longest link: perfect even spacing. If the species are clumped in Euclidean/functional space, the longest link of the MST will be much greater than the shortest, giving a value closer to 0.0. As this test statistic is a ratio, it will be unaffected by the volume of trait space occupied by the species, providing an estimate of the evenness of species packing that is independent of changes in volume between observed and randomized communities.
TS3, Minimum ED: the minimum Euclidean distance observed between any of the species within a sample (point, quadrat or area) provides an indication of the absolute limit to the closeness of species packing (Weiher et al. 1998).
The average Euclidean distance between all species pairs within each sample and the variance in link length in an MST were also calculated, but they duplicated results from the test statistics above, and are not presented here.
To overcome the problem that TS1 to TS3 average effects shown in several characters, some of which will be niche separators and others not, and also to pinpoint the characters responsible for effects in TS1 to TS3, the following univariate test statistics were calculated on each of the characters:
TS4, Mean NN: the mean absolute difference in character values between nearest-neighbours along the character axis of the species within each sample (Fig. 1, i.e. the univariate equivalent of Ricklefs & Travis (1980) measure of species packing). Species that are more evenly spaced than expected at random will have a greater mean NN than species from a randomized community.
TS5, Weighted deviance (WD): the mean deviance of the species in the sample from the mean over all the species in that sample, weighted by the abundance of each species (Appendix S1). Compare the ‘distance from the origin’ test statistic of James & Boecklen (1984). This is a measure of species packing that takes account of the different abundance of different species.
The maximum, minimum and range of characters observed in each sample were also calculated (cf. Weiher et al. 1998), but the results provided no further insight and are not presented here.
TS6, Variance/range: the variance in adjacent distances divided by the range of character values. By using adjacent distances (cf. Fig. 1), TS6 incorporates all the interspecies distances along the gradient, from one end to the other, into an index of species evenness. Similar TSs have been used to test for staggered flowering times (Gleeson 1981; Pleasants 1990), but as the range of characters found in the observed and randomized communities can affect the variance, the variance was divided by the range. A community with species evenly distributed throughout character space will have a lower variance/range than species within the randomized communities.
The basic theory of MacArthur & Levins is versed in terms of niche overlap (Fig. 1). Overlap indices can be calculated from the mean and standard deviation of a character for each species in the sample, given the approximation of a normal distribution (Appendix S1; Cody 1975), or for categorical data (here, root profile) using Pianka's (1973) index of niche overlap (Appendix S1).
TS7, Mean overlap: the mean niche overlap between all pairs of species co-occurring within the sample, which measures the degree of niche overlap between coexisting species.
TS8, Maximum overlap: the maximum niche overlap between any pair of species co-occurring within the sample.
TS9, Weighted AN overlap: the mean weighted niche overlap between adjacent-neighbours along the niche gradient. This is a measure of niche overlap that takes account of the different abundance of different species.
TS10, Variance in AN overlap: the variance in the degree of niche overlap between adjacent-neighbours within each sample. This is a measure of the evenness of species packing along the character axis.
The value of the test statistic expected under the null model was calculated as the average value from 10 000 randomizations, and significance (i.e. the probability of the observed result under the null model) as the proportion of randomizations in which the test statistic was equal to that observed, or more extreme, multiplied by 2.0 to effect a two-tailed test. A program to analyse each of the test statistics at each scale was written in C++ and validated with random data (Appendix S1, details of the validation are available from the first author).
Due to the categorical nature of the root profile data, this character was analysed using only the average niche overlap and the maximum niche overlap test statistics. Nitrogen and phosphorus content were not analysed using any of the niche overlap test statistics, as there were insufficient replicates to calculate the SD for a species. Test statistics that use abundance as a weighting were used on both presence/absence and abundance information, except for weighted deviance (TS5). As multiple comparisons were made on the same data, increasing the chance of a type I error, binomial tests were used to show whether the proportion of significant tests was greater than expected. A test was made separately for each character analysed and at each scale, across all of the test statistics, separately at the P = 0.025 level for each tail. Allowance for multiple testing is usually fraught with problems; here it was ecologically necessary to test over several scales and to test several characters, but the binomial tests will probably be conservative because of non-independence between the test statistics.