On random placement and species-area relations

Consider a collection C of individuals from several species residing ina region R which is the union of a number of nonoverlapping subregions. Let ni be the number of individuals from species i which belong to C and hence reside somewhere in R. The simplest hypothesis about the spatial distribution throughout R of the members of C is that their placement in dwelling sites is random and noninteractive, with the probability of a given individual of C residing in a particular subregion r of R equal to the ratio α ofthe area of r to the area of R. It is shown here that when this hypothesis of random placement holds, the mean s̄ and variance σ2 of the number of species from C represented in r are given by explicit functions of α, provided the number ni are known. Thus, if all the species in C have been censured, species-area data permit a test of the hypothesis of random placement. The nature of the dependence of s̄ and σ2 on α is discussed in detail for special cases in which the numbers ni are given by such theoritical abundance relations as the logarithmic series distribution, the “broken stick” distribution, the lognormal distribution, the Poisson lognormal distribution, and the gamma distribution. The arguments employed here to deduce consequences of the hypothesis of random choice are rigorous and exact. No use is made of the assumption, commonly made heretofore (but not in general correct, even under the hypothesis of random placement), that a species-area curve (giving the number of species expected to be found in a sample of known area) must have the same form as the corresponding collector's curve (giving the number of species expected in a sample of a known number of individuals). Nor is it assumed in advance, as is often done in the theory of island biogeography, that the distribution of individuals throughout the subregions of R is such thatthe species abundance relations for subregions of different areas must be of a preassigned type, i.e., must share a common form, such as that associated with a truncated lognormal distribution.