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distance to the first nearest neighbor, the second nearest neighbor, and so on, to the nth nearest neighbor? For a species with a total population size of n individuals, then the average radius of its range will be given by the mean distance to the nth nearest neighbor. In taking this approach, one makes no assumptions about the dispersion or degree of species aggregation of tropical tree species, but we know that most tropical tree species are clumped in distribution (Hubbell, 1979; Condit et al., 2000).

In a population with random (Poisson) dispersion, Thompson (1956) proved that the mean distance to the nth nearest neighbor rn is given by

where δ is the mean density of trees per unit area. The distance E[rn] as a function of n is asymptotically a power law for large n. The above approximation is derived from Sterling’s formula, which holds very well even for small n. Therefore, the slope of the log log relationship between distance and rank of nearest neighbor approaches 0.5 as n → ∞ in a Poisson-distributed population. Power laws are convenient because of their scale independence, which means that we can compute E[rn] for any arbitrarily large population size. But this result was obtained for a randomly distributed population. What about nonrandomly distributed tropical tree populations?

To a very good approximation, the relationship between log E[rn] and log n is also a power law for nonrandomly distributed tropical tree populations. We computed the relationship between log distance to the nth nearest neighbor and log rank of nearest neighbor for all tree species with total abundances ≥102 individuals (155 species) in the 50-ha plot on Barro Colorado Island (BCI), Panama. Virtually all of these are very good power laws, illustrated for two arbitrarily chosen species in Fig. 6.4, for all stems >1 cm DBH (Fig. 6.4a and c) and for canopy adult trees >20 cm DBH (Fig. 6.4b and d). Based on available data, these power law relationships also appear to hold on spatial scales >>50 ha. For example, Tabebuia guayacan (Bignoniaceae), a canopy-emergent species whose individual adults can be accurately censused by using hyperspectral data from the Quickbird satellite, exhibits a very precise log–log relationship over the entire 15.2 km2 area of BCI (Fig. 6.5) (J.K. and S.P.H., unpublished data). Therefore, we assume that this relationship also holds on larger scales. John Harte has indicated that this result can now be proven (J. Harte, unpublished work). To calculate range sizes of the 11,200 tree species in the Brazilian Amazon, we adjusted the intercept of the log–log regression to reflect the effect of rarity on the first nearest-neighbor distance (Fig. 6.4f),



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