Supporting Information – Methods
Bird Data. We used a global database on the breeding ranges of 9,626 extant, recognized
bird species following a standard avian taxonomy (Sibley & Monroe 1990). We converted the
species’ vector range maps into a gridded format using the equal-area Behrman projection,
with a cell size of 96.486 km, equal to 1° longitude at 30° latitude, and conducted our analysis
on the gridded data. A smaller grain size would permit analysis of range shapes at a finer
resolution and reduce the overestimation in range size, particularly for species with linear
distributions (Rahbek 2005). However, these concerns need to be weighed both against
computational tractability of conducting simulations at this scale, and the accuracy of global
species’ ranges (Hurlbert & Jetz 2007). A scale of around 1° is generally considered to be the
finest resolution appropriate for global scale distribution data (Hurlbert & Jetz 2007). Using an
equal area projection was necessary because of the need to conserve species’ range sizes in
our cellular based simulations (Storch et al. 2006). Although the shapes and distances
between cells in our grid are not constant, this distortion is only marked at very high latitudes
where few species live (see Fig. S1 in Gaston et al. 2007). We discuss the implications of this
for our measurement of shape below.
Quantifying Range Shape. Previous studies investigating patterns in range shape have
employed simple shape measures, such as N-S and E-W extent (Brown & Maurer 1989;
Sandel 2009), but these fail to capture much of the detail in range configuration. The
perimeter to area ratio has also been used to measure the shape of isodensity maps (Rapoport
1982; Ruggiero et al. 1998; Ruggiero 2001), a rough proxy for spatial patterns in range
shape. Shape indices incorporating the perimeter are, however,
highly sensitive to
irregularities in the range boundary (Maceachren 1985; McGarigal & Marks 1995; Brown et al.
1996). As a result, indices based on maximum linear (l) and areal dimensions (a), are known
to be more appropriate for quantifying shape circularity (Maceachren 1985). Of the various
potential measures we calculated range shape (Rs) according to the following formula; Rs =
l2/a (Fig S2) This is equivalent to the ratio of the long axis of a range to its average width
(Graves 1988; Niemi et al. 1990). We selected this index for a number of reasons. First, the
1
index is scale invariant which was crucial given the enormous variation in range size amongst
birds (Orme et al. 2006). Second, it produces an intuitive and easily conceptualised shape
parameter i.e. a value of 3 indicates a shape 3x as long as it is wide. Finally, in contrast to
other shape indices based on maximum linear dimensions (Clark & Gaile 1973; Maceachren
1985), it is linearly related to shape elongation. We note that although more complex shape
indices, based on distances between all grid cells, exist (Maceachren 1985) these are not scale
invariant
and
are
difficult
to
standardise
where
the
distances
between
cells
vary
geographically.
To quantify the shape of the total gridded species range, we first calculated the shape of each
of the constituent range fragments (McGarigal & Marks 1995) using graph theory,
implemented in the “graph” and “RBGL” packages in R (http://cran.r-project.org/). Fragments
were identified as clusters of occupied cells connected by cell edges; cells connected only by
their vertices were assumed to be separate fragments (Fig. S2). To calculate the longest axis
for each fragment, we used the shortest path distance through the range (i.e. passing through
adjacent occupied cell centres) (Mcrae & Beier 2007) between all cells lying on the convex hull
of the shape, and selected the longest of these (Fig. S2). This measure is preferable to a
simple Euclidian distance (Graves 1988), because it accounts for concavity in range fragments
(Mcrae & Beier 2007). In calculating the path length, the distances between adjacent cell
centres were measured using great circle distances in the R package “fields” (http://cran.rproject.org/) (cells sharing edges or vertices were treated as adjacent). The use of great circle
distances allowed the true distance over the Earths surface between cells to be calculated and
so our measure of the range long axis was robust to the distortion of shape that occurs when
the globe is viewed on the equal-area Behrman projection. In addition, a constant was added
to the path length representing the distance from final cell centres to the actual external cell
vertices. The overall range shape for a species was calculated as the average shape across
fragments, weighted by fragment area (McGarigal & Marks 1995) (Fig. S2). Fragments smaller
than 3 cells have a constant shape and were therefore not included in the calculation of range
shape.
2
We calculated an alternative measure of range shape based on the scaled difference in the
length of the eigen values of the variance-covariance matrix of distances between occupied
cells (Maurer 1994). For full details of the general methodology we refer the reader to Maurer
(1994). Here we briefly outline the details of how we applied this method to our global study.
We first calculated the mean latitude and longitude of each range fragment and then
calculated the great circle distance along the meridians and parallels from each occupied cell
centre to these latitudinal and longitudinal averages. We then extracted the eigen values (e1
and e2) from the variance-covariance matrix of these internal range distances. Finally, the
index of shape (Rs2) was calculated as the scaled difference between the length of the eigen
values according to the following formula; Rs2 = (e1 - e2)/ (e1 + e2).
Sources of environmental variables. Mean annual Temperature and Precipitation were
obtained from New (2002) for the period 1961–1990 at 10 min resolution. Net Primary
Productivity was obtained from Imhoff (2004). Mean elevation was calculated from 30-arcsecond
resolution
data
developed
by
the
USGS
EROS
Data
Center
available
at
http://eros.usgs.gov/products/ elevation/gtopo30.php). All environmental variables were
resampled using the equal-area Behrman projection, with a cell size of 96.486 km to match
the resolution of our species grid.
SI References
Brown J.H. & Maurer B.A. (1989). Macroecology - the division of food and space among
species on continents. Science, 243, 1145-1150.
Brown J.H., Stevens G.C. & Kaufman D.M. (1996). The geographic range: Size, shape,
boundaries, and internal structure. Annu Rev Ecol Syst, 27, 597-623.
Clark W.A.V. & Gaile G.L. (1973). The analysis and recognition of shapes. Geo Ann Ser B-Hum
Geo, 55, 153-163.
Gaston K.J., Davies R.G., Orme C.D.L., Olson V.A., Thomas G.H., Ding T.S., Rasmussen P.C.,
Lennon J.J., Bennett P.M., Owens I.P.F. & Blackburn T.M. (2007). Spatial turnover in
the global avifauna. Proc Roy Soc B, 274, 1567-1574.
Graves G.R. (1988). Linearity of geographic range and its possible effect on the population
structure of Andean birds. The Auk, 105, 47-52.
Hurlbert A.H. & Jetz W. (2007). Species richness, hotspots, and the scale dependence of range
maps in ecology and conservation. Proc Natl Acad Sci USA, 104, 13384-13389.
Imhoff M.L., Bounoua L., Ricketts T., Loucks C., Harriss R. & Lawrence W.T. (2004). Global
patterns in human consumption of net primary production. Nature, 429, 870-873.
Maceachren A.M. (1985). Compactness of geographic shape - comparison and evaluation of
measures. Geo Ann Ser B-Hum Geo, 67, 53-67.
3
Maurer B.A. (1994). Geographical population analysis: tools for the analysis of biodiversity
Blackwell Science, Cambridge, MA.
McGarigal K. & Marks B.J. (1995). FRAGSTATS: spatial pattern analysis program for
quantifying landscape structure. In: U.S. Department of Agriculture, Forest Service,
Pacific Northwest Research Station., p. 122.
Mcrae B.H. & Beier P. (2007). Circuit theory predicts gene flow in plant and animal
populations. Proc Natl Acad Sci USA, 104, 19885-19890.
New M., Lister D., Hulme M. & Makin I. (2002). A high-resolution data set of surface climate
over global land areas. Climate Res, 21, 1-25.
Niemi R.G., Grofman B., Carlucci C. & Hofeller T. (1990). Measuring compactness and the role
of a compactness standard in a test for partisan and racial gerrymandering. J Politics,
52, 1155-1181.
Orme C.D.L., Davies R.G., Olson V.A., Thomas G.H., Ding T.S., Rasmussen P.C., Ridgely R.S.,
Stattersfield A.J., Bennett P.M., Owens I.P.F., Blackburn T.M. & Gaston K.J. (2006).
Global patterns of geographic range size in birds. PLoS Biol, 4, 1276-1283.
Rahbek C. (2005). The role of spatial scale and the perception of large-scale species-richness
patterns. Ecol Lett, 8, 224-239.
Rapoport E.H. (1982). Areography: Geographical strategies of species. Pergamon Press, New
York.
Ruggiero A. (2001). Size and shape of the geographical ranges of Andean passerine birds:
spatial patterns in environmental resistance and anisotropy. J Biogeogr, 28, 12811294.
Ruggiero A., Lawton J.H. & Blackburn T.M. (1998). The geographic ranges of mammalian
species in South America: spatial patterns in environmental resistance and anisotropy.
J Biogeogr, 25, 1093-1103.
Sandel B. (2009). Geometric constraint model selection - an example with New World birds
and mammals. Ecography, 32, 1001-1010.
Sibley C.G. & Monroe B.L. (1990). Distribution and Taxonomy of the Birds of the World. Yale
University Press, New Haven.
Storch D., Davies R.G., Zajicek S., Orme C.D.L., Olson V., Thomas G.H., Ding T.S.,
Rasmussen P.C., Ridgely R.S., Bennett P.M., Blackburn T.M., Owens I.P.F. & Gaston
K.J. (2006). Energy, range dynamics and global species richness patterns: reconciling
mid-domain effects and environmental determinants of avian diversity. Ecol Lett, 9,
1308-1320.
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