Ecological Archives E088-103-A4

Karen E. Samis and Christopher G. Eckert. 2007. Testing the abundant center model using range-wide demographic surveys of two coastal dune plants. Ecology 88:1747–1758.

Appendix D. Implementation of constraint space analysis to test the abundant center model (ACM).

The ACM predicts that local abundance and individual performance should, on average, be higher at the range center and decline towards range margins. This prediction would seem amenable to testing by quadratic regression analysis. However, low quality habitat might occur throughout the range, whereas high quality habitat should occur primarily at the center (Lawton 1993, Brown et al. 1995). The variance in abundance and performance among sites should, therefore, be highest at the center and lowest at the margins. This makes regression analysis inappropriate for testing the ACM. Accordingly, we used a non-parametric constraint space analysis following Enquist et al. (1995) and Sagarin and Gaines (2002). We tested whether the pattern of geographic variation in each parameter (y = density or performance) fit a triangular constraint space predicted by the ACM. Each parameter (y) was expressed on a 0 to 1 scale:
(yiy<min)/(ymaxymin), where yi is the value for the ith site, and ymin and ymax are the lowest and highest values among all sites, respectively. Geographic position (km of coast from range center) of each site (xi) was expressed on a –1 to +1 scale, with 0 at the range center. The ACM constraint space was described by x:y co-ordinates: –1:0, 0:1, +1:0. Under the ACM, the x:y coordinates of all populations should fall within this space. We quantified the fit of the observed data to the constraint space by calculating the squared deviation from the boundaries of the constraint space of any point falling outside the constraint space boundary. The data fit the ACM if the sum of these squared deviations was smaller for the observed data (SSobs) than for 95% of 3000 random permutations of the data (SSrand).

In addition to the simple triangular constraint space, we tested the fit of the data to two alternative abundant center models, one based on a normal distribution (from Eq. 3 in Sagarin and Gaines 2002) and the other based on a platykurtotic distribution (from Eq. 13 in Garcia-Ramos and Kirkpatrick 1997); and two non-ACM models with maximum abundance at one or other range limit (Fig. 2 in Sagarin and Gaines 2002). We included these latter two models because ramped distributions have been detected empirically for other species (e.g., Graves 1997, Sagarin and Gaines 2002, Gilman 2005). If the data exhibited a better-than-random fit to more than one constraint space, we used parsimony as a criterion (following Hilborn and Mangel 1997) to select the best model, which yielded the lowest value of SSobs/(npops – 2m), where npops = the number of populations sampled, and m = the number of parameters in the model (m = 2 for linear and normal models, and 1 for platykurtotic model).

LITERATURE CITED

Brown, J. H., D. W. Mehlman, and G. C. Stevens. 1995. Spatial variation in abundance. Ecology 76:2028­–2043.

Enquist, B. J., M. A. Jordan, and J. H. Brown. 1995. Connections between ecology, biogeography, and paleobiology: relationship between local abundance and geographic distribution in fossil and recent molluscs. Evolutionary Ecology 9:586–604.

Garcia-Ramos, G., and M. Kirkpatrick. 1997. Genetic models of adaptation and gene flow in peripheral populations. Evolution 51:21–28.

Gilman, S. 2005. A test of Brown's principle in the intertidal limpet Collisella scabra (Gould, 1846). Journal of Biogeography 32:1583–1589.

Graves, G. R. 1997. Geographic clines of age ratios of black-throated blue warblers (Dendroica caerulescens). Ecology 78:2524–2531.

Hilborn, R., and M. Mangel. 1997. The ecological detective. Confronting models with data. Princeton University Press, Princeton, NJ, USA.

Lawton, J. H. 1993. Range, population abundance and conservation. Trends in Ecology and Evolution 8:409–413.

Sagarin, R. D., and S. D. Gaines. 2002. Geographical abundance distributions of coastal invertebrates: using one-dimensional ranges to test biogeographic hypotheses. Journal of Biogeography 29:985–997.

TABLE D1. Test results (P values) for the parameters of interest for each species, Camissonia cheiranthifolia and Abronia umbellata for three constraint models relevant to the abundant center model and two alternative models. P values less than 0.05 (bolded) indicate a better-than-random fit of the parameter data to the tested constraint space. In cases where the data showed a significant fit to more than one constraint space, the best model, chosen using the parsimony criterion described above, is underlined.

Species/Variable
n
Sites
Triangular
Normal
Platykurtotic
Southern
maximum
Northern
maximum
Camissonia cheiranthifolia
Radial density (plants/m2)
64
0.24
0.49
0.17
0.33
0.42
Average density (plants/m2)
63
0.13
0.54
0.24
0.24
0.87
Plant size (cm2)
64
0.00058
0.043
0.013
0.071
0.61
Seeds/plant
36
0.38
0.16
0.32
0.059
0.52
Seeds/m2
36
0.059
0.038
0.0009
0.51
0.32
Abronia umbellata
Radial density (plants/m2)
35
0.69
0.54
0.67
0.79
0.43
Average density (plants/m2)
38
0.57
0.69
0.55
0.31
0.77
Plant size (cm2)
34
0.018

0.029

0.018
0.12
0.51
Seeds/plant
30
0.74
0.58
0.69
0.97
0.51
Seeds/m2
29
0.64
0.77
0.43
0.20
0.80


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