Tom S. Romdal, Robert K. Colwell, and Carsten Rakbek. 2005. The influence of band sum area, domain extent, and range sizes on the latitudinal mid-domain effect. Ecology 86:235–244.

Appendix A. Resolving the band sum area bias.

This appendix describes and discusses how band sums were adjusted for latitudinal variation in area.

That total species richness is a nonlinear function of area is one of the fundamentals of ecology. The species-area curve will typically be approximately linear in log-log space, following the Arrhenius Equation (Preston 1960):

where S is number of species, A is area, and z (the slope) and c are fitted constants. In a typical dataset, estimates of z and c can therefore be found from the regression of logS on logA. If species-area curves are to be used to adjust species richness for the effect of area, the exact value of z is extremely important. However, the value of z has been shown to depend on the size of the surrounding region, on the degree of isolation, and on evolutionary history (Anderson and Marcus 1993, Rosenzweig 1995, Rosenzweig and Ziv 1999). Possibly z also varies systematically with latitude, but Lyons and Willig (2002) found contradictory evidence for latitudinal variation in z when analyzing equal-area quadrat data on New World mammals. Another cause for variation (and in some contexts, bias) is that number of species also varies substantially and systematically with the shape of the sampled area. This has been shown both theoretically (Harte et al. 1999) and empirically (Condit et al. 1996, Kunin 1997, Harte et al. 1999), but not addressed in latitudinal studies before. A state or country of compact shape will have fewer species expected than a narrow latitudinal band across the continent with the same area, because the band covers much more distance. (For instance, Texas, with 691,000 km2 has 335 breeding bird species, while a continent-wide band through Texas of only 358,000 km2 has 356 species). Likewise, Fig. A1 shows how the direction of variation (latitudinal or longitudinal) of the size of the bands themselves influences z, and an empirical example of the importance of shape of bands appears in Fig. A2. When complete latitudinal bands in Canada are added together to create species-area curves, the value of the slope, z, is low, because even the narrowest (1º) bands include a substantial turnover of species. The value of z for the curve based on band segments that vary in longitudinal extent, only, is three times higher, because the smallest segments all have few species. The curve based on provinces, which do not vary systematically in shape, is intermediate. It follows that extreme caution is called for when constructing species-area curves. When comparing a band in, e.g., Central America with a neighboring but longer band in northern South America, the difference in shape and longitudinal variation of extent of the bands is identical to what we find within segments of a single band, and the appropriate area adjustment should be a species-area curve with a high z-value. Therefore species/area curves for comparing latitudinal bands should be created only from data derived from single bands and segments of bands.

For the present study we produced six species-area curves between 40º S and 56º N from the New World bird dataset. Species counts for each species-area curve were based on segments of a single latitudinal band. Since the shortest of our empirical bands, i.e., 53–54º S - has three 1º cells and the spatial dimensions 1 × 1.79 (111 × 199 km), we included band segments that ranged from small segments with dimensions of around 1 × 2 , up to entire bands. The smallest segments thus consisted of 2 or 3 cells, depending on latitude. All non-overlapping 3-cell, 4-cell etc. segments entered the species-area correlation. The z values of these curves are shown in Fig. A3 and lie between 0.338 and 0.427.

DISCUSSION

Some studies (e.g., Pagel et al. 1991) have standardized area in latitudinal bands by simply dividing number of species by area, but this approach neglects the fact that species-area relations are not linear (Gotelli and Colwell 2001). Kaufman (1995) eliminated area by analyzing residuals from a species vs. area regression, but such use of residuals is likely to produce biased estimates of test parameters (Freckleton 2002). Estimation of species-area relations is a more cumbersome, but transparent way of standardizing. If only data from complete bands were available, rather than cell data, S-A curves could be created in two ways, but both are problematic. First, one could make a curve from a collection of single bands. Because considerable variation in band area is essential to create a robust S-A curve, one could not limit this dataset to, for example, 35–45º N for the Americas, because all these bands are of roughly similar size. A species-area curve based on single band sums, specifically representing 35–45º N or even representing all of North America would thus not allow for standardization to a smaller area. The second possibility would be to create more bands with larger areas by combining adjacent bands. As demonstrated above, this results in species-area curves that are fundamentally inapplicable to adjustment of area differences among single bands. Curves from latitudinal bands that differ only in North-South extent and curves from bands that differ only in East-West extent have z values that represent the extremes of the imaginable spectrum. The most appropriate way of creating estimates of z for bands is to partition bands into band-shaped segments, as illustrated in this study.

The z-values we calculated for the latitudinal band area adjustments are much higher than those traditionally reported for continental data (e.g., Connor and McCoy 1979) and are more like z-values for island archipelagos (Rosenzweig 1995). In a meticulous analysis of latitudinal variation in z for equal-area square data on New World mammals, Lyons and Willig (2002) found conflicting results. They reported that, for South America, z was significantly smaller towards the tropics, with the opposite trend in North America. With only six different z values calculated there is no evident systematic latitudinal trend in our data. The z-values from Lyons and Willig (2002), derived from a square lattice grid, are generally below 0.15, while our values based on latitudinal band sums all exceed 0.3. This reflects the relation of polygon shape and z demonstrated in Appendix A. Harte et al. (1999) described theoretically how z must vary with polygon shape. It can be predicted, for instance, that a z of 0.23 for a quadrat series of increasing area would correspond to a z of 0.37 for a series of elongate bands of constant width. This was confirmed with empirical small-scale plot data (Harte et al. 1999, see also Condit et al. 1996 and Kunin 1997 for similar results). Empirical band sum datasets must thus have higher expected z-values than traditionally have been used. The theoretical z of 0.26 (Preston 1962) has also been challenged in recent work that has employed randomized sampling of simulated distributions, which generated much higher expected values (Leitner and Rosenzweig 1997, Bell 2001, Cam et al. 2002). Again, this illustrates the need for careful consideration of assumptions regarding parameters in species-area relations.

LITERATURE CITED

Anderson, S., and L. F. Marcus. 1993. Effect on quadrat size on measurements of species density. Journal of Biogeography 20:421–428.

Bell, G. 2001. Neutral Macroecology. Science 293:2413–2418.

Cam, E., J. D. Nichols, J. E. Hines, J. R. Sauer, R. Alpizakar-Jara, and C. H. Flather. 2002. Disentangling sampling and ecological explanations underlying species-area relationships. Ecology 83:1118–1130.

Condit, R., S. P. Hubbell, J. V. Lafrankie, R. Sukumar, N. Manokaran, R. B. Foster, and P. S. Ashton. 1996. Species-area and species-individual relationships for tropical trees: a comparison of three 50-ha plots. Journal of Ecology 84:549–562.

Connor, E. F., and E. D. McCoy. 1979. The statistics and biology of the species-area relationship. The American Naturalist 113:791–833.

DeSante, David F., and P. Pyle. 1986. Distributional checklist of North American birds. Artemisia Press, Lee Vining, Californai, USA.

Freckleton, R. P. 2002. On the misuse of residuals in ecology: regression of residuals vs. multiple regression. Journal of Animal Ecology 71:542–545.

Gotelli, N. J., and R. K. Colwell. 2001. Quantifying biodiversity: procedures and pitfalls in the measurement and comparison of species richness. Ecology Letters 4:379–391.

Harte, J., S. McCarthy, K. Taylor, A. Kinzig, and M. L. Fischer. 1999. Estimating species-area relationships from plot to landscape scale using species spatial-turnover data. Oikos 86:45–54.

Kaufman, D. M. Diversity of New World mammals: universality of the latitudinal gradients of species and bauplans. Journal of Mammalogy 76:322–334.

Kunin, W. E. 1997. Sample shape, spatial scale and species counts: implications for reserve design. Biological Conservation 82:369–377.

Leitner, W. A., and M. L. Rosenzweig. 1997. Nested species-area curves and stochastic sampling: a new theory.Oikos 79:503–512.

Lyons, S. K., and M. R. Willig. 2002. Species richness, latitude, and scale-sensitivity. Ecology 83:47–58.

Pagel, M. D., R. M. May, and A. R. Collie. 1991. Ecological aspects of the geographical distribution and diversity of mammalian species. American Naturalist 137:791–815.

Preston, F. W. 1960. Time and space and the variation of species. Ecology 41:785–790.

Preston, F. W. 1962. The canonical distribution of commonness and rarity. Ecology 43:185–215, 410–432.

Rosenzweig, M. L. 1995. Species Diversity in Space and Time. Cambridge: Cambridge University Press, Cambridge, UK.

Rosenzweig, M. L., and Y. Ziv. 1999. The echo pattern of species diversity: pattern and process. Ecography 22:614–628.

FIG. A1. The relationship of polygon shape and turnover of species for latitudinal bands that vary in latitudinal or longitudinal extent, respectively. The term polygon is used for a geographical area of any size or shape for which number of species can be tallied. Distance is the maximum linear distance across a given polygon. Distance is a surrogate for inherent species turnover, and thus related to polygon richness. Because distance varies for polygons of similar area, depending upon their shapes, shape also influences species richness. For polygons of invariant shape, distance increases with the square root of x when area increases with x, but polygons that are latitudinal bands vary in shape. (a) The relationship of area and distance for theoretical bands that vary in latitudinal but not longitudinal extent, as when adding bands together to create a species-area curve. The first band has area A and distance D (distance is measured as the diagonal). The second band has twice the latitudinal extent. It has the area A' = 2A and the distance D'. Because species turnover in both bands is mainly along the longitudinal dimension, D' is almost identical to D. (b) The relationship for bands that vary in longitudinal extent only. This is similar to constructing species-area curves from single latitudinal bands of different length or from segments of bands. The first band again has area A and distance D. The second band has twice the longitudinal extent. It has the area A' = 2A and the Distance D'. Because of the direction of variation in size, D' is almost 2*D.

FIG. A2. Variation in species-area curves with shape and direction of area increment, as seen in a plot of logS vs. logA. All data in the graph are from a latitudinal zone from 49 to 60º N, covering the southern part of Canada. The "Provinces" curve is constructed from six Canadian provinces (data from DeSante and Pyle 1986) and the total for the zone. The "complete bands" curve is compiled by joining adjacent complete bands, covering from 1º of latitude to the entire zone (49 to 60ºN). The "within bands" curve is constructed from segments of a 2 º latitudinal band between 54 and 56º N, with area of segments ranging from 2 by 8º to the entire band (2 by 66º). This curve has large variation in S among small samples, because band segments that cover the Rocky Mountains have much higher species richness than other segments. All regressions are significant at P < 0.05, the regression equations are: Provinces: y = 1.01 + 0.236x. Complete bands: y = 1.685 + 0.129x. Within bands: y = 0.154 + 0.380x.

FIG. A3. Interpolation of z values of species-area functions. Species-area curves were based on latitudinal bands at six latitudes: 39-40º S, 19-20º S, 1º N-0º, 16-17º N, 32-34º N (2º cells), 54-56º N (2º cells). Data are from segments of entire latitudinal bands. The six z values are plotted vs. latitude. They are assumed to be representative and used to estimate z values for all other latitudes by simple interpolations, as shown by the lines. At extreme latitudes values are extrapolated by continuation of the trend of variation.