SupplementAL Materials
The supplemental materials have two parts: 1) examples of break-down in fractal structure of species distributions; and (2)aggregation vs. randomness in 2-D and fractal occurrence maps.
SECTION 1.Examples of fractal structure of species distributions breaking down
If self-similarity for a species holds over all spatial scales, there is a power-law relationship between species occupancy and grain size (scale–area relationship in Kunin 1998) that is linear in a log–log plot. However, the assumption of self-similarity is often criticized in ecology, and self-similarity may only hold for a givenspecies over some spatial scales. Hui and McGeoch (2006) showed an example in which self-similarity broke down. The scale–area relationship was not strictly linear in a log–log plot. In this paper, two tree species on Barro Colorado Island (BCI), Panama, AnaxagoreapanamensisandOenocarpusmapora, are selected to illustrate the break-down of the self-similarity assumption in species spatial distributions.
Fig.A1Scale–area relationships in log–log plots of two BCI tree species; is the area of one grid and Area is the total area of all occupied grids.
In Fig.A1, the nonlinear scale–area relationship (log–log) implies that self-similarity did not hold over all spatial scales.For other BCI tree species, such as Beilschmiediapendula, Alchorneacostaricensis, and Brosimumalicastrum, the self-similarity of fractal structure also did not hold for all spatial scales (data not shown).
SECTION 2.aggregation vs. randomness in 2-D fractal occurrence maps
The spatial patterns of most tree species on BCI are aggregated (Condit et al. 2000; Zillio and He 2010) within a regular two-dimensional (2-D) study area. The regular two-dimensional study area is gridded into cells (also called grids or lattices), and the state of each grid is recorded (presence or absence of the species), resulting in occurrence maps (Kunin 1998; He and Gaston 2000; Harte et al. 2001; He and Hubbell 2003). If we consider occurrence maps (only occupied cells) rather than the regular 2-D study area, the detected spatial distribution patterns of tree species on the occurrence maps also change. In contrast, occurrence mapsat fine scales that more or less represent the spatial structure of tree species also show self-similar structures. In this section, we show how the detected spatial patterns change in regular 2-D study areas and occurrence maps. The probability distribution models are Poisson and negative binomial distribution (NBD) models, and the tree species Beilschmiediapendulais examined.
First, when the Poisson and NBD models are each used to fit the quadrat count data (quadrat size: 25×25m), the NBD model performs better than the Poisson model (Fig.A2). This result is trivial, because the spatial pattern is obviously aggregated in the regular 2-D study area. When the regular 2-D study area is gridded into cells (25 × 25m), we get the occurrence map (Fig.A2). After randomly sampling25 × 25m square quadrats fromthe occurrence map, we get another set of quadrat count data.If we fit the quadrat count data to the Poisson and NBD models, we find that Poisson model performs better than NBD model (Fig.A3).This result implies that the spatial point pattern in the occurrence map is random rather than aggregated. For many other tree species, such as Calophyllumlongifolium and Drypetesstandleyi, we can obtain similar results (data not shown).
Fig.A2 Spatial distribution of the tree species Beilschmiediapendula.(a) Spatial point pattern. The small squares are sampling quadrats (25×25m); (b) frequency distributions of quadrat count data in a regular 2-D study area. The histogram represents the empirical frequency distribution, circles represent the fitted Poisson model (, ), and triangles represent the fitted NBD model (, ).
Fig.A3 (a) Spatial point pattern of the tree species Beilschmiediapendula and its occurrence map (25 × 25m); (b) Pink squares representoccupied grids and other squares represent unoccupied grids.
Fig.A4 Spatial distribution of the tree species Beilschmiediapendula.(a) Spatial point pattern and sampling quadrats in the occurrence map in Fig.A3(b); (b) frequency distributions of quadrat count data in the occurrence map. The histogram represents the empirical frequency distribution, circles represent the fitted Poisson model (, ) and triangles represent the fitted NBD model (, ).