On random placement and species-area relations

Abstract

A collection C of individuals from several species residing in a region R which is the union of a number of nonoverlapping subregions is considered. 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 .alpha. of the area of r to the area of R. When this hypothesis of random placement holds, the mean .hivin.s and variance .sigma.2 of the number of species from C represented in r are given by explicit functions of .alpha., provided the numbers ni are known. If all the species in C have been censused, species-area data permit a test of the hypothesis of random placement. The nature of the dependence of .hivin.s and .sigma.2 on .alpha. is discussed in detail for special cases in which the numbers ni are given by such theoretical 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 to deduce consequences of the hypothesis of random choice are rigorous and exact. No use is made of the assumption, commonly made heretofore, 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 that the 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.