Range size in mid-domain models of species diversity

Geographical patterns of species diversity have been examined using mid-domain null models, in which the ranges of individual species are simulated by randomly arranging them on a bounded one- or two-dimensional continent. These models have shown that structured patterns in the geographical distribution of biodiversity can arise even under a fully stochastic procedure. In particular, mid-domain models have demonstrated that the random generation of ranges of different sizes and locations can produce a gradient of species diversity similar to the one found in real assemblages, with a peak at the middle of a continent. A less explored feature of mid-domain models is the pattern of range-size frequency distribution. Numerical simulations have provided some insights about the geographic pattern of average range size, but no exploration of the shape of range-size frequency distributions has been carried out. Here I present analytical and numerical models that generate explicit predictions for patterns of range size under the assumptions of mid-domain models of species diversity. Some generalizations include: (1) Mid-domain models predict no geographic gradient of average range size; the mean range size of species occurring at any point on a continent is constant (0.5 of the extent of the continent in the one-dimensional model, 0.25 of the area of the continent in the two-dimensional case); (2) Variance in range size is lowest at the middle of a continent and highest near the corners of a square-shaped continent; (3) The range-size frequency distribution is highly right-skewed at any point of a continent, but the skewness is highest near the corners. Despite their alleged weaknesses, mid-domain models are adequate null models against which real-world patterns can be contrasted.