Quadratic diversity: Its maximization can reduce the richness of species

Abstract

In recent decades numerous diversity indices have been introduced. Among them the quadratic entropy index Q expresses the mean difference between two individuals chosen from the community at random. Differing from diversity indices habitually employed, Q does not satisfy a property postulated earlier for those measures. Namely, the uniform distribution of species does not necessarily yield the maximal index value. Q is based on the difference matrix of species. For a given matrix one can seek for the vector yielding the maximum quadratic entropy. This task leads to a quadratic programming problem. Using the appropriate program of a program package, we determined the maximum vector for a genetic difference matrix of crane species, as published in the literature. We discovered that some components (frequencies) in the maximum vector are equal to zero. That is, by maximizing the quadratic diversity some species can be eliminated. We discuss briefly the possible implications of this observation. Moreover, even if all elements in the maximum vector are positive, the elements can differ.