Most biodiversity conservation programs are forced to prioritise species in order to allocate their funding. This paper contains a mathematical proof that provides biological support for one common approach based on phylogenetic indices. Phylogenetic trees describe the evolutionary relationships between a group of taxa. Two indices for computing the distinctiveness of each taxon in a phylogenetic tree are considered here—the Shapley value and the Fair Proportion index. These indices provide a measure of the importance of each taxon for overall biodiversity and have been used to prioritise taxa for conservation. The Shapley value is the biodiversity contribution a taxon is expected to make if all taxa are equally likely to become extinct. This interpretation makes it appealing to use the Shapley value in biodiversity conservation applications. The Fair Proportion index lacks a convenient interpretation, however it is significantly easier to calculate and understand. It has been empirically observed that there is a high correlation between the two indices. This paper shows the mathematical basis for this correlation and proves that as the number of taxa increases, the indices become equivalent. Consequently in biodiversity prioritisation the simpler Fair Proportion index can be used whilst retaining the appealing interpretation of the Shapley value.
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Crozier RH (1992) Genetic diversity and the agony of choice. Biol Conserv 61:11–15
Faith DP (1992) Conservation evaluation and phylogenetic diversity. Biol Conserv 61:1–10
Haake C, Kashiwada A, Su FE (2008) The shapley value of phylogenetic trees. J Math Biol 56:479–497
Hartmann K, Steel M (2006) Maximizing phylogenetic diversity in biodiversity conservation: Greedy solutions to the Noah’s Ark problem. Syst Biol 55(4):644–651
Hartmann K, Wong D, Stadler T (2010) Sampling trees from evolutionary models. Syst Biol 59(4):465–476
Isaac NJB, Turvey ST, Collen B, Waterman C, Baillie JEM (2007) Mammals on the EDGE: conservation priorities based on threat and phylogeny. PLoS One 2(3):e296
McPeek MA, Brown JM (2007) Clade age and not diversification rate explains species richness among animal taxa. Am Nat 169:E97–E106
Redding D, Hartmann K, Mimoto A, Bokal D, DeVos M, Mooers AO (2008) The most “original species” often capture more phylogenetic diversity than expected. J Theor Biol 251:606–615
Redding DW, Mooers AO (2006) Incorporating evolutionary measures into conservation prioritization. Conserv Biol 20:1670–1678
Steel M, Mimoto A, Mooers AO (2007) Hedging our bets: the expected contribution of species to future phylogenetic diversity. Evol Bioinform 3:237–234
Weitzman ML (1998) The Noah’s Ark problem. Econometrica 66(6):1279–1298
Zoological Society of London (2011) EDGE of existence. http://www.edgeofexistence.org
I would like to thank Arne Mooers for motivating me to explore this problem. I would also like to thank Mats Gyllenberg and two anonymous referees for their comments which have substantially clarified this manuscript.
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Hartmann, K. The equivalence of two phylogenetic biodiversity measures: the Shapley value and Fair Proportion index. J. Math. Biol. 67, 1163–1170 (2013). https://doi.org/10.1007/s00285-012-0585-y
- Phylogenetic diversity
- Fair proportion
Mathematics Subject Classification (2000)