Geometric constraints and spatial pattern of species richness: critique of range-based null models

Does the shape of a biogeographical region influence its spatial patterns of species richness? A complete answer must include careful distinction between the distribution of a species, which is a complex geometric object, and the range of a species, which is relatively simple, especially when reduced to one dimension. We consider range-based models of species richness, in particular range overlap counts in one dimension, for which we give a unified mathematical treatment via the joint probability P(m,l) of midpoints and lengths of ranges. We discuss a number of difficulties, in practice and in principle, using range-based models, and show that the so-called mid-domain effect, a proposed null model for the effect of geometric constraint, is qualitatively a property of all biologically realistic models based on range overlap counts. As such, range-based models provide little insight into understanding or explaining biogeographical patterns in species richness. We characterize the quantitative null model for range overlap counts in one dimension, for which we give a simple and direct field test based on P(m,l). We apply this test to a large clade in a complete bioregion (the Proteaceae of the Cape Floristic Region): geometric constraint does not explain the spatial pattern in this case. We show that any geometric constraint on species richness, including range overlap counts, must act via edge effects. Thus, to understand biogeographical patterns, an understanding of the effects and consequences of edges is fundamental.