Maximizing Diversity in Biology and Beyond
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 (201...
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MDPI AG
2016-03-01
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Online Access: | http://www.mdpi.com/1099-4300/18/3/88 |
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author | Tom Leinster Mark W. Meckes |
author_facet | Tom Leinster Mark W. Meckes |
author_sort | Tom Leinster |
collection | DOAJ |
description | 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. |
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format | Article |
id | doaj.art-bfb77e6ef5644071b0a44941c9e9006e |
institution | Directory Open Access Journal |
issn | 1099-4300 |
language | English |
last_indexed | 2024-04-11T13:10:46Z |
publishDate | 2016-03-01 |
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series | Entropy |
spelling | doaj.art-bfb77e6ef5644071b0a44941c9e9006e2022-12-22T04:22:35ZengMDPI AGEntropy1099-43002016-03-011838810.3390/e18030088e18030088Maximizing Diversity in Biology and BeyondTom Leinster0Mark W. Meckes1School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, UKDepartment of Mathematics, Applied Mathematics, and Statistics, Case Western Reserve University, Cleveland, OH 44106, USAEntropy, 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.http://www.mdpi.com/1099-4300/18/3/88diversitybiodiversityspecies similarityentropyRényi entropymaximum entropymetric entropyHill numbermaximum clique |
spellingShingle | Tom Leinster Mark W. Meckes Maximizing Diversity in Biology and Beyond Entropy diversity biodiversity species similarity entropy Rényi entropy maximum entropy metric entropy Hill number maximum clique |
title | Maximizing Diversity in Biology and Beyond |
title_full | Maximizing Diversity in Biology and Beyond |
title_fullStr | Maximizing Diversity in Biology and Beyond |
title_full_unstemmed | Maximizing Diversity in Biology and Beyond |
title_short | Maximizing Diversity in Biology and Beyond |
title_sort | maximizing diversity in biology and beyond |
topic | diversity biodiversity species similarity entropy Rényi entropy maximum entropy metric entropy Hill number maximum clique |
url | http://www.mdpi.com/1099-4300/18/3/88 |
work_keys_str_mv | AT tomleinster maximizingdiversityinbiologyandbeyond AT markwmeckes maximizingdiversityinbiologyandbeyond |