To measure the clustering of a species, Condit et al. (1) used an index called the relative neighborhood density, _x1,x2. For a given species, _x1,x2 is equivalent to the average density of conspecifics in the neighborhood of individuals, normalized by the density of individuals of the species in the entire plot:

(1)

where n_x1,x2 is the number of conspecifics located between a distance x₁ and a distance x₂ from each individual, averaged over all individuals of the species; A_x1,x2 is the area of the annuli defined by the radii x₁ and x₂; and n is the number of individuals of the species in a plot of area A₀ (4). If  > 1 at distances that are short relative to the plot size, the species is considered clustered, whereas  < 1 at short distances indicates spacing or dispersion of individuals.

Self-similarity in the distribution of a single species is defined as follows. Let A₀ be a rectangular plot whose dimensions have the ratio , and let A_i = A₀/2ⁱ be the size of areas obtained from A₀ by i shape-preserving bisections (5). Given that a species is in a particular area of size A_i, let _i be the average probability that it is in at least a particular one of the two A_i + 1 contained in A_i. The distribution of the species is self-similar, or scale invariant, if _i =  is independent of i.

Self-similarity relates properties of the distribution of a species at small scales to such properties at larger scales. For example, self-similarity relates _i, here defined as the average abundance of a species in the A_i that it occupies, and n, the total abundance of the species in the plot A₀:

(2)

where  depends on the species but not on i. Eq. 2 can be used to obtain an expression for _x1,x2 in terms of the  for each species. Let _i refer to _x1,x2 in the case where x₁ is the radius of a circle of size A_i + 1 and x₂ is the radius of a circle of size A_i. For this case, the numerator in Eq. 1 can be closely approximated (6) by [1/(A_i + 1)](_i  i + 1), and hence

(3)

_i can be expressed directly in terms of the distance r_i, defined as the average of the radii of A_i and A_i + 1. Note that _i = (1/)ⁱ ₀, where ₀ = 2  (1/), and that r_i = (1/)ⁱ r₀, where r₀ = (1/2)(1 + 1/)(₀/). By defining ()^w  , we can write

(4)

where C = ₀/r₀^w.

The relative neighborhood density of self-similar distributions (Fig. 1) has characteristics in common with that of tropical forest plots described in (1). It is largest at the smallest scales and monotonically decreases with scale at a rate which is largest at small scales (7). Furthermore, since  increases with abundance (8),  at small distances will be largest for rare species. We also did a more quantitative comparison of the relative neighborhood density (as defined in Eq. 4) for 20 species chosen from (1) over a range of abundances. For each species, a linear regression was performed on the log-transformed data (9). The linear regression analysis yielded r² > 0.9 for half of the species, and r² > 0.8 for all but two (10). Fig. 2 shows three of these analyses.

The exponent w in Eq. 4 is directly related to the exponent y' that appears in the relationship between the box-counting measurement of range size R of a species and the area A of the grid cell used to measure the range, R = A₀ (A/A₀)^y'. This range-area relationship was observed by Kunin for British floral census data (11) and derived from single-species self-similarity in (2). The clustering exponent w is related to the range-area exponent y' by

(5)

Condit et al. could test Eq. 5 using their tropical forest data.

The relative neighborhood density of tropical tree species as described in (1) has characteristics in common with the relative neighborhood density expected of a self-similarly distributed species, but community-level self-similarity does not hold at the tropical forest sites studied by Condit et al. (12, 13). However, although the notion is counterintuitive, a group of species whose distributions are individually self-similar is not expected to be self-similarly distributed at the community level (2). Therefore, the lack of community-level self-similarity in the sites studied by Condit et. al is not evidence against the possibility of species-level self-similarity in those sites.

Annette Ostling
John Harte
Energy and Resources Group
University of California at Berkeley
Berkeley, CA 94720, USA
E-mail: aostling{at}socrates.berkeley.edu
Jessica Green
Department of Nuclear Engineering
University of California at Berkeley

REFERENCES AND NOTES

1. R. Condit, et al., Science 288, 1414 (2000) [Abstract/Free Full Text] .
2. J. Harte, T. Blackburn, A. Ostling, in preparation.
3. J. Harte, A. Kinzig, J. Green, Science 284, 334 (1999) [Abstract/Free Full Text] .
4. When calculating _x1,x2 at large x₁,x₂ from actual data, some of the area covered by the annuli A_x1,x2 will fall outside of the plot area in which the location of individuals is known. In these cases, the average density of the species in the portions of the annuli inside of the plot is assumed to be representative of the average density of the species over the entire area covered by the annuli.
5. We take the set of areas A_i to all have the same shape because a self-similar distribution of a single species is expected to have shape-dependent properties in much the same way that the community-level version of self-similarity, the SAR, has shape dependence (14).
6. This expression is an approximation in the sense that the A_i and A_i + 1 relevant to _i are circles, rather than the same shape as the plot A₀. The difference between the average number of individuals in a circle and the average number of individuals in a rectangle of the same size is probably small, however, as long as the rectangle is not long and thin.
7. Some species distributions in (1) have a nonmonotonic decrease in . Any distribution, however, including one with a nonmonotonic decrease in , can be described by the probabilities _i if _i is allowed to vary with i.
8. For a given i,  increases with abundance across species that have the same _i (Eq. 2); hence,  at small scales will be largest for the rarest of such species.
9. The linear regression was performed only over distances between 5 and 200 m, where most of the annuli used in calculating  fall inside the plot.
10. For most of the 20 species examined, the value of  calculated from the expression in (15) is not equal to that determined from the linear regression analysis, an indication that the simplifying assumption in (15) is not valid for these sites.
11. W. Kunin, Science 281, 1513 (1998) [Abstract/Free Full Text] .
12. R. Condit, et al., J. Ecol. 84, 549 (1996) [CrossRef] [Web of Science].
13. J. Plotkin, et al., Proc. Natl. Acad. Sci. U.S.A. 97, 10850 (2000) [Abstract/Free Full Text] .
14. J. Harte, S. McCarthy, K. Taylor, A. Kinzig, M. L. Fisher, Oikos 86, 45 (1999) [CrossRef] [Web of Science].
15. Under the simplifying assumption that each species has no more than one individual in areas of size A₀/N₀, where N₀ is the total number of individuals across all tree species in the plot, the value of  for a species is related to its abundance (2) by: We gratefully acknowledge financial support from the Class of 1935 Distinguished Professorship Fund of the University of California at Berkeley and from the American Association of University Women.

Response: The comment by Ostling et al. is elegant and interesting--and, indeed, precisely echoes the content of several paragraphs removed from the report by Condit et al. (1) during the editing process. In the following brief discussion, I paraphrase those omitted paragraphs for the present context, and offer several other observations on the Ostling et al. comment.

If the neighborhood density function for a species declined linearly on a log-log scale, then the species' distribution would be fractal and scale-invariant because the intensity of aggregation would decay similarly at all scales. Astronomers describe the distribution of galaxies as being fractal in exactly this way. Most individual species in the forests examined by Condit et al. (1), however, did not display scale invariance across the plots. More typically for common species, neighborhood density declined at short distances more rapidly than log-linearly, and then leveled out. Other species showed more gentle declines initially, then rapid declines at greater distances.

Intriguingly, however, the aggregate behavior of the whole communities--the sum of relative neighborhood density across species--was indeed fractal, and showed very consistent patterns across forests (Fig. 1). The most abundant species had relatively gentle declines and large x-intercepts, while rare species had steep declines and smaller x-intercepts. (The x-intercept on a log-log scale is the distance at which  = 1. Because  > 1 at short distances signifies at least some degree of aggregation, the x-intercept can thus be viewed as the clump radius, or the distance at which clumping ceases to be important.) The slope of these lines reflects the fractal dimension, D, because D is equivalent to the slope plus two: D = 2 indicates spatial randomness; D = 0 would be complete clumping, with all individuals concentrated at a single point (2). D for an aggregate of all common species varied from 1.65 to 1.83 in the six plots, and for aggregated rare species varied from 0.86 to 1.41. D declined smoothly with abundance at all plots, reflecting the tendency for rare species to be more clumped. Thus, in aggregate, the forests are scale invariant, and this should reflect scale-invariance in how species composition changes through space, although Condit et al. (1) did not investigate this.

Ostling et al. have cleverly shown how their description of self-similarity corresponds with the neighborhood function. This is useful, because the method based on quadrant occupancy that they have used can be associated with geographic ranges. Perhaps they can make something of the observation in Fig. 1, that in an aggregate sense, the communities appear to be quite precisely self-similar.

Ostling et al. mention several tests that could be done using our distribution data for large forest plots; I would be happy to make data sets available if they would like to pursue the tests. And, finally, I present a challenge: Can the theories that Ostling et al. have put forth here predict ranges at much wider scales? The 50-ha plots have been excellent for testing predictions because distributions are completely known. But at larger scales, the data that I work with are far sparser--a few tens of plots, scattered over 1000 km²--and we don't know the distributions of trees at these scales. I would like to draw conclusions, based on these sparse data, to questions such as, for example, how many species are widespread and how many occur in only one area. Can self-similarity suggest a way?

Richard Condit
Center for Tropical Forest Science
Smithsonian Tropical Research Institute
Unit 0948
APO AA 34002-0948, USA
E-mail: condit{at}ctfs.stri.si.edu

REFERENCES AND NOTES

1. R. Condit, et al., Science 288, 1414 (2000) .
2. This discussion follows the proofs and illustrations in (3), in which it is shown that the cluster dimension--defined as the slope of log K versus log x, where K is Ripley's K--is equal to the fractal dimension. So K(x) = cx^D, where D is the fractal dimension and c is a constant. _x, the relative neighborhood density function as defined by Condit et al. (1), is where is the mean density across the plot and A(x) is the area of a circle with radius x. In the limit, this is equivalent to or _x = c'x^D2, so the fractal dimension is found by adding 2 to the slope of log _x versus log x.
3. H. M. Hastings, G. Sugihara, Fractals: A User's Guide for the Natural Sciences (Oxford Univ. Press, Oxford, 1993).