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Self-Similarity and Clustering in the Spatial Distribution of Species

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Science  27 Oct 2000:
Vol. 290, Issue 5492, pp. 671
DOI: 10.1126/science.290.5492.671a

Condit et al. (1) examined the spatial aggregation of individuals of tree species in six different tropical forest sites. They found that most species were aggregated, that rare species tended to be more aggregated than abundant ones, and that smaller individuals of a species tended to be more aggregated than larger individuals. The observed clustering pattern, they pointed out, was inconsistent with a random distribution of individuals. We show here that the clustering described in (1) is similar to that expected for a species with a self-similar spatial distribution. Such a distribution has been demonstrated to correctly predict range characteristics across a wide variety of taxa and spatial scales (2) and is the analog to the community-level self-similarity shown in (3) to be equivalent to the power-law species-area relationship (SAR).

To measure the clustering of a species, Condit et al.(1) used an index called the relative neighborhood density, Ωx1 , x2. For a given species, Ωx1 , x2is 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:Embedded Image(1)where〈nx 1 , x 2〉 is the number of conspecifics located between a distancex 1 and a distance x 2 from each individual, averaged over all individuals of the species;Ax 1 , x2is the area of the annuli defined by the radiix 1 and x 2; andn is the number of individuals of the species in a plot of area A 0 (4). If Ω > 1 at distances that are short relative to the plot size, the species is considered clustered, whereas Ω < 1 at short distances indicates spacing or dispersion of individuals.

Self-similarity in the distribution of a single species is defined as follows. Let A 0 be a rectangular plot whose dimensions have the ratio Embedded Image, and letAi =A 0/2i be the size of areas obtained from A 0 by ishape-preserving bisections (5). Given that a species is in a particular area of size Ai , let αi be the average probability that it is in at least a particular one of the twoAi  + 1 contained inAi . The distribution of the species is self-similar, or scale invariant, if αi = α 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 relatesi , here defined as the average abundance of a species in the Ai that it occupies, and n, the total abundance of the species in the plot A 0:Embedded Image(2) Embedded Imagewhere α depends on the species but not oni. Eq. 2 can be used to obtain an expression for Ωx1 , x2in terms of the α for each species. Let Ωirefer to Ωx1 , x2in the case where x 1 is the radius of a circle of size Ai  + 1 andx 2 is the radius of a circle of sizeAi . For this case, the numerator in Eq. 1 can be closely approximated (6) by [1/(Ai  + 1)](i i  + 1), and henceEmbedded Image(3)Ωi can be expressed directly in terms of the distance ri , defined as the average of the radii of Ai andA i + 1. Note that Ωi = (1/α)iΩ0, where Ω0 = 2 − (1/α), and thatri = (1/Embedded Image)i r 0, where r 0 = (1/2)(1 + 1/Embedded Image)(Embedded Image 0/π). By defining (Embedded Image)w ≡ α, we can writeEmbedded Image(4)where C = Ω0/r 0 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 α increases with abundance (8), Ω 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 yieldedr 2 > 0.9 for half of the species, andr 2 > 0.8 for all but two (10). Fig. 2 shows three of these analyses.

Figure 1

Relative neighborhood density, Ωi, as a function of distance,ri , as expected for self-similarly distributed species with α = 0.65, 0.75, 0.85, and 0.95, in a plot of sizeA 0 = 50 ha. These values of α correspond to species with abundances n ≈ 102, 103, 104, and 105, respectively, if there are 300,000 individuals in the plot and a simplifying assumption is made (15). The smallest distance plotted is 5.3 m (this point is off the graph for α = 0.65).

Figure 2

Relative neighborhood density, Ω, versus distance, r, for three species in (1). The lines are the result of linear regression on the log-transformed data (to eliminate heteroscedasticity) (9): ln Ω = 480 − 1.17 ln r, r 2 = 0.914, α = 0.667 (±0.077) for Spondias mombin; ln Ω = 10.3 − 0.466 ln r, r 2= 0.946, α = 0.851 (±0.022) for Chrysochlamys eclipes; and ln Ω = 1.95 − 0.128 ln r,r 2 = 0.959, α = 0.978 (±0.008) for Garcinia intermedia, where number in parentheses are standard errors.

The exponent w in Eq. 4 is directly related to the exponenty′ that appears in the relationship between the box-counting measurement of range size R of a species and the areaA of the grid cell used to measure the range,R = A 0(A/A 0)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 wis related to the range-area exponent y′ byEmbedded Image(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.

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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 Conditet 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 Ω = 1. Because Ω > 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.

Figure 1

Aggregate neighborhood density functions from the Pasoh 50-ha plot. The steepest line, with wiggles, is the aggregate neighborhood function for all 89 species with 10 to 24 individuals in 50 ha. The aggregate function was calculated by taking the arithmetic average of all 89 individual neighborhood functions. The gray line, with intermediate slope, is the aggregate neighborhood function for all 73 species with 200 to 299 individuals, and the flattest line the aggregate for the seven species with ≥5000 individuals. At each of the plots, species were aggregated into abundance categories and the neighborhood functions were aggregated; in nearly all cases, the aggregate functions were very close to linear on the log-log scale, and always steeper for less common species.

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?

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