Biodiversity measures and the role of species similarity

Transcription

1 Biodiversity measures and the role of species similarity Christina Cobbold University of Glasgow, Boyd Orr Centre for Population and Ecosystem Health

2 Quantifying diversity Richness: How many species? pecies ite 1 ite 2 ite All sites have 4 species how do we distinguish them?

3 Quantifying diversity Richness: How many species? pecies ite 1 ite 2 ite All sites have 4 species how do we distinguish them? Answer: Include relative contribution of a species to the community Diversity captures richness, eveness and dominance

4 Hill numbers and the spectrum of view points Hill (1973) Ecology Let p i be the relative abundance of the ith species. p=(p 1,p 2 p s ). The diversity of order q of the community is exp (Rényi entropy): q D(p) =, 1/(1 q) " %. q $ p i ' if q 1,.# i=1 &. p 1/ (p 1 p 1 p 2 p - 2! p ) if q =1.. 1/max p i if q= 1 i. /

5 Why take the exponential of entropy? Jost (2006) Oikos Consider a continent of 1 million equally abundant species. A meteorite hits the continent wiping out 50% of the species. What happens to diversity? Expected answer: Diversity drops by 50%

6 Jost (2006) Oikos Why take the exponential of entropy? Consider a continent of 1 million equally abundant species. A meteorite hits the continent wiping out 50% of the species. What happens to diversity? Expected answer: Diversity drops by 50% BUT hannon entropy drops by just 5%! Entropies are not effective numbers, Hill numbers are.

7 What is the role of q? Diversity profile. view point parameter

8 What is the role of q? Diversity profile. view point parameter q=0, species richness rare species are important

9 What is the role of q? Diversity profile. view point parameter q=0, species richness rare species are important q=infinity, 1/Berger-Parker rare species are unimportant Importance of evenness

10 What are some of the limitations of Hill numbers associated with the idea of diversity is the concept of distance i.e. some measure of the dissimilarity of the resources in question OECD Handbook on biodiversity valuation: A guide for policy makers A community of ten species of barnacle should clearly be less diverse than a community of ten very different species The essence of a point made by E.C. Pielou, Ecological Diversity There is not always a good notion of what constitutes a species e.g. microbial communities

11 Including species similarity Leinster and Cobboold (2012) Ecology model of community imilarity data, Z Abundance data, p matrix ( = number of species) Z ij = similarity between ith and jth species totally dissimilar 0, 1 Z i j identical

12 Including species similarity Leinster and Cobboold (2012) Ecology model of community imilarity data, Z Abundance data, p matrix ( = number of species) Z ij = similarity between ith and jth species Example totally dissimilar This model assumes that distinct species are totally dissimilar 0, 1 Z i j 1 0 Z = # 0 identical 0! 1 " " "! 0 0 # 0 1

13 beak length (Zp) I low imilarity-sensitive diversity measures (Zp) I high The ordinariness of the i-th species is i ( Zp) = Z j= 1 ij Leinster and Cobboold (2012) Ecology p j toe length

14 beak length (Zp) I low imilarity-sensitive diversity measures (Zp) I high toe length The ordinariness of the i-th species is i ( Zp) = Z j= 1 Average ordinariness of an individual within the community i= 1 which measures lack of diversity. ij i Leinster and Cobboold (2012) Ecology p j p ( Zp) i

15 beak length (Zp) I low imilarity-sensitive diversity measures (Zp) I high toe length The ordinariness of the i-th species is i ( Zp) = Z j= 1 Average ordinariness of an individual within the community i= 1 which measures lack of diversity. o one measure of diversity is 1 i= 1 ij i Leinster and Cobboold (2012) Ecology p j p ( Zp) p ( Zp) i i i

16 pectrum of view points: general q Generalised weighted power mean of x=(x 1 x ) has the form i= 1 p i x q 1 i o the generalised power mean of ordinariness is i= 1 p i ( Zp) q 1 i Leinster and Cobboold (2012) Ecology 1/( q 1) 1/( q 1)

17 pectrum of view points: general q Generalised weighted power mean of x=(x 1 x ) has the form i= 1 p i x q 1 i o the generalised power mean of ordinariness is i= 1 p i ( Zp) q 1 i Diversity of order q is 1/ generalised power mean # q D Z (p) = % $ i:p i >0 1/( q 1) 1/( q 1) & q 1 p i (Zp) i ( ' Leinster and Cobboold (2012) Ecology 1 1 q

18 Example: Butterfly data in the Ecuadorian rainforest Abundance data of Charaxinae Butterflies at a rainforest site in Ecuador pecies Canopy Understorey Prepona laertes 15 0 Archaeoprepona demophon Zaretis itys Memphis arachne Memphis offa 21 3 Memphis xenocles 32 8 Taxonomic similarity matrix Z ij = if if if of different genera different but congeneric i = j, Data from DeVries et al (1997)

19 Butterfly data in the Ecuadorian rainforest effective No. of species No similarity data q, view point parameter imilarity data (taxonomic) Data from DeVries et al (1997)

20 Comparing sites: ite 1 ite k. pecies 1 p 1 pecies 2 p 2 pecies 3 p 3 pecies 4 p 4 Ecosystem g 1 g 2 g 3 g 4 pecies p g How does diversity of site k contribute to the ecosystem diversity? Is site k typical or distinct in some way?

21 Reeve et al Average ordinariness of site j from the ecosystem point of view The ordinariness of the i-th species in the ecosystem (Zg) i = Z ij g j j=1

22 Reeve et al Average ordinariness of site j from the ecosystem point of view The ordinariness of the i-th species in the ecosystem (Zg) i = Z ij g j Generalised power mean of ordinariness of species in site k when viewed from the ecosystem perspective: p i i=1 j=1 # q 1 & % (Zg)i ( $ ' 1/(q 1)

23 Reeve et al Average ordinariness of site j from the ecosystem point of view The ordinariness of the i-th species in the ecosystem (Zg) i = Z ij g j Generalised power mean of ordinariness of species in site k when viewed from the ecosystem perspective: p i i=1 j=1 # q 1 & % (Zg)i ( $ ' Renyi s cross entropy # 1 ( 1 q) ln % p i $ i =1 1/(q 1) & q 1 (Zg) i ( '

24 Cross diversity exp (cross entropy) Which site/s has highest cross diversity? pecies ite 1 ite 2 ite Ecosystem (my back garden)

25 Cross diversity exp (cross entropy) Which site/s has highest cross diversity? pecies ite 1 ite 2 ite Ecosystem (my back garden)

26 Reeve et al Relationship between entropy and cross entropy For Z=I and q=1 Relative entropy=cross entropy Entropy # 1 ( 1 q) ln % p i $ i =1 & q 1 (Zp / Zg) i ( = 1 1 q ' # ( ) ln % p i $ i =1 & q 1 (Zg) i ( 1 1 q ' # ( ) ln % p i $ i =1 & q 1 (Zp) i ( '

27 Reeve et al Relationship between entropy and cross entropy For Z=I and q=1 Relative entropy=cross entropy Entropy # 1 ( 1 q) ln % p i $ i =1 & q 1 (Zp / Zg) i ( = 1 1 q ' # ( ) ln % p i $ i =1 & q 1 (Zg) i ( 1 1 q ' # ( ) ln % p i $ i =1 & q 1 (Zp) i ( ' This does not hold for general q and Z, but does relative entropy tell us something about diversity?

28 Relative entropy and ordinariness The ordinariness of the i-th species at site j compared to the ordinariness of the i-th species in the ecosystem (Zp / Zg) i = Z ij p j / Z ij j=1 j=1 g j

29 Relative entropy and ordinariness The ordinariness of the i-th species at site j compared to the ordinariness of the i-th species in the ecosystem (Zp / Zg) i = Z ij p j / Z ij g j j=1 Averaging relative ordinariness weighted by relative abundance of species i at site j p i i=1 Relative diversity = 1/ Av relative ordinariness j=1 # q 1 & % (Zp/Zg)i ( $ ' 1/(q 1)

30 Relative diversity exp (relative entropy) Which site/s has highest relative diversity? pecies ite 1 ite 2 ite Ecosystem (my back garden)

31 Relative diversity exp (relative entropy) Which site/s has highest relative diversity? pecies ite 1 ite 2 ite Ecosystem (my back garden)

32 ummary Generalised entropy allows us to consider evenness and dominance pecies ite 1 ite 2 ite imilarity sensitive diversity functional diversity, genetic diversity etc Cross entropy allows us to consider sites containing rarity Relative entropy allows to consider distinctness of sites

33 Acknowledgements to. Tom Leinster chool of Mathematics University of Edinburgh Louise Matthews Boyd Orr Centre University of Glasgow Richard Reeve Boyd Orr Centre University of Glasgow

34 Microbe communities: notion of a species is problematic Expected similarity between a randomly chosen pair of individuals: 1 2 D Z (p) = µ 2 Now consider q>2 (whole number). Given q individuals of species i 1,i 2, i q then a measure of group similarity is: Z i i Zi i! Z i 1 i q Let µ q be the expected similarity of a randomly chosen group of q individuals (sampled with replacement). Then q D Z 1/(1 q) (p) = µ q

35 Properties of diversity of order q Elementary properties: 1. If two species are nearly identical, then merging them into one leaves the diversity nearly unchanged. 2. Diversity is unchanged by adding a new species of abundance 0 3. The order we list species in does not change diversity.

36 Properties of diversity of order q Partitioning properties: 4. It is an effective number: diversity of a community of equally abundant, totally dissimilar species is. 5.Replication: If m islands of equal size each have diversity d, and species on different islands are totally dissimilar, the the diversity of the whole is md 6.Modularity: If the m islands have different sizes and diversities (species on different islands totally dissimilar), the diversity of the whole is determined by the sizes and diversities of the islands

37 Properties of diversity of order q imilarity properties: 7. Monotonicity: increasing species similarity decreases diversity 8. Naïve model: ignoring species similarity increases diversity. 9. Range: The diversity of a community of species is between 1 and