Relative abundance of species: A collection of animals may contain a number of species. Some rare species may be represented by only 1 individual in the collection. Others may be represented by more....

Relative abundance of species: A collection of animals may contain a number of species. Some rare species may be represented by only 1 individual in the collection. Others may be represented by more. In 1943 Fisher, Corbert, and Williams37 proposed a model for finding the number of species in certain types of collections represented by a fixed number of individuals. They associate with an animal collection two constants, α and x, with the following property: The number of species in the sample represented by a single individual is αx; the number in the sample represented by 2 individuals is α(x2 /2); and, in general, the number of species represented by n individuals is α(xn/n). If we add all these numbers up, we get the total number S of species in the collection. It is an important fact from advanced mathematics that this sum also yields a natural logarithm:

If N is the total number of individuals in the sample, then it turns out that we can find x by solving the equation