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Maximizing Diversity in Biology and Beyond

by 1,2,* and 3,*
1
School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UK
2
Boyd Orr Centre for Population and Ecosystem Health, University of Glasgow, Glasgow G12 8QQ, UK
3
Department of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA
*
Authors to whom correspondence should be addressed.
Academic Editors: John Baez, John Harte and Marc Harper
Entropy 2016, 18(3), 88; https://doi.org/10.3390/e18030088
Received: 19 December 2015 / Revised: 22 February 2016 / Accepted: 24 February 2016 / Published: 9 March 2016
(This article belongs to the Special Issue Information and Entropy in Biological Systems)
Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or lesser importance to rare species. Leinster and Cobbold (2012) proposed a one-parameter family of diversity measures taking into account both this variation and the varying similarities between species. Because of this latter feature, diversity is not maximized by the uniform distribution on species. So it is natural to ask: which distributions maximize diversity, and what is its maximum value? In principle, both answers depend on q, but our main theorem is that neither does. Thus, there is a single distribution that maximizes diversity from all viewpoints simultaneously, and any list of species has an unambiguous maximum diversity value. Furthermore, the maximizing distribution(s) can be computed in finite time, and any distribution maximizing diversity from some particular viewpoint q > 0 actually maximizes diversity for all q. Although we phrase our results in ecological terms, they apply very widely, with applications in graph theory and metric geometry. View Full-Text
Keywords: diversity; biodiversity; species similarity; entropy; Rényi entropy; maximum entropy; metric entropy; Hill number; maximum clique diversity; biodiversity; species similarity; entropy; Rényi entropy; maximum entropy; metric entropy; Hill number; maximum clique
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Leinster, T.; Meckes, M.W. Maximizing Diversity in Biology and Beyond. Entropy 2016, 18, 88.

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