and Entropy, National Institute for
Mathematical and Biological Synthesis,
Knoxville, Tennessee, USA, 10 April 2015.
Abstract Different ecologists, shown two communities, will make different judgements on which is the more diverse. One axis of difference is the relative importance attached to rare and common species: one person might prioritize conservation of rare species, while another prioritizes overall community balance. This spectrum of viewpoints is captured by the family of measures known to ecologists as the Hill numbers and to information theorists as the exponentials of the Rényi entropies. It is a one-parameter family, the parameter q indicating one's position on the spectrum of viewpoints.
However, all these measures use a crude model in which the varying similarities between species are ignored. They behave as if distinct species have nothing whatsoever in common. Christina Cobbold's talk will show how to repair this defect, factoring in inter-species similarity.
A natural question then arises. Given a list of species with known similarities, and a choice q of viewpoint, which frequency distribution maximizes the diversity?
The big surprise is this: there is a single frequency distribution that maximizes diversity from all viewpoints simultaneously. No matter whether one's priority is rare species (low q) or common species (high q), this distribution is optimal. Moreover, the value of the maximum diversity is the same for all q. Thus, any list of species of known similarities has an unambiguous maximum diversity.
Slides In this pdf file.
Paper An early account of the main result (with a proof that has since been simplified) is at arXiv:0910.0906.