# Shannon diversity. Site Description. Hill numbers. Simpson diversity H =

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1 Vegetation Analysis ite Description lide 1 Vegetation Analysis ite Description lide 2 hannon diversity ite Description 1. Diversity indices 2. pecies abundance models 3. pecies area relationship H = p j log b p j j=1 Originally information theory with base b = 2: Average length in bits of code with shortest possible unique coding The limit reached when code length is log 2 p i : longer codes for rare species. Biologists use natural logarithms (base b = e), and call it H Information theory makes no sense in ecology: Better to see only as a variance measure for class data. Vegetation Analysis ite Description lide 3 impson diversity The probability that two randomly picked individuals belong to the same species in an infinite community is P = i=1 p2 i. Can be changed to a diversity measure (= increases with complexity): 1. Probability that two individuals belong to different species: 1 P. 2. umber of species in a community with the same probability P, but all species with equal abundances: 1/P. Claimed to be ecologically more meaningful than hannon diversity, but usually very similar. Vegetation Analysis ite Description lide 4 Hill numbers Common measures of diversity are special cases of Rényi entropy; H a = 1 1 a log Mark Hill proposed using a = exp(h a ) or the Hill number : H 0 = log() 0 = umber of species H 1 = i=1 pi log p1 1 = exp(h1) exp hannon H 2 = log i=1 p2 i 2 = 1/ i=1 p2 i Inverse impson ensitivity to rare species decreases with increasing a: 1 and 2 are little influenced and nearly linearly related. i=1 All Hill numbers in same units: virtual species. p a i

2 Vegetation Analysis ite Description lide 5 J H Vegetation Analysis ite Description lide 6 Choice of index Evenness Diversity indices are only variances of species abundances. It is not so important which index is used, since all sensible indices are very similar = exp(h) If everything else remains constant, diversity increases when 1. umber of species increases, or 2. pecies abundances p i become more equal. Evenness: Hidden agenda to separate these two components For a given number of species, diversity is maximal when all probabilities p i = 1/: in hannon index H max = log() Pielou s evenness is the proportion of observed and maximal diversity J = H H max Vegetation Analysis ite Description lide 7 ample size and diversity Vegetation Analysis ite Description lide 8 Logarithmic series = 10, = 154 With increasing sample size umber of species increases Diversity ( 1 or 2 ) stabilizes Evenness decreases Diversity little influenced by rare species: a variance measure. Evenness based on twisted idea Diversity pecies richness Evenness R.A. Fisher in 1940 s Most species are rare, and species found only once are the largest group In larger samples, you may find more individuals of rare species, but you find new rare species = 30, = = 110, =

3 Vegetation Analysis ite Description lide 9 Vegetation Analysis ite Description lide 10 Log-ormal model ed abundance diagrams Preston did not accept Fisher s log-series, but assumed that rare species end with sampling Plotted number of species against octaves : doubling classes of abundance Modal class in higher octaves, and not so many rare species Canonical standard model of our times Lajien lukumäärä R 0 = = 31.2 a = 3.98 a R Oktaavi Horizontal axis: ranked species Logarithmic abun- Vertical axis: dance The shape of abundance distribution clearly visible: Linear: Pre-emption model Log-normal or broken- igmoid: stick Runsaus Vegetation Analysis ite Description lide 11 Fitting RAD models Vegetation Analysis ite Description lide 12 Broken tick Pre-emption model pecies abundances decay by constant proportion. A line in the ranked abundance diagram. Log-normal model pecies abundances distributed ormally igmoid: excess of both abundant and rare species to pre-emption model. Carabid, site 6 pecies break a community ( stick ) simultaneously in pieces. o real hierarchy, but chips arranged in rank order: Result looks sigmoid, and can be fitted with log- ormal model. Carabid site 6

4 Vegetation Analysis ite Description lide 13 Vegetation Analysis ite Description lide 14 Ultimate diversity parameter θ Hubbell s abundance model θ = 2J M ν, where J M is metacommunity size and ν evolution speed θ and J define the abundance distribution imulations can be used for estimating θ. pecies generator θ/(θj 1) gives the probability that jth individual is a new species for the community. Carabid site 6! = 8 pecies richness: The trouble begins pecies richness increases with sample size: can be compared only with the same size. Rare species have a huge impact in species richness. Rarefaction: Removing the effects of varying sample size. ample size must be known in individuals: Equal area does not imply equal number of individuals. Plants often difficult to count. Vegetation Analysis ite Description lide 15 Rarefaction Vegetation Analysis ite Description lide 16 pecies richness and sample size Rarefy to a lower, equal number of individuals Only a variant of impson s index E( =4) pecies richness Rarefied to = E( =2)! Fisher log-series predicts: ( = α ln 1 ) α pecies never end, but the rate of increase slows down.! = umber of individuals impson index

5 Vegetation Analysis ite Description lide 17 pecies Area models Island biogeography: = ca z. Parameter c is uninteresting, but z should describe island isolation. Regarded as universally good: Often the only model studied, so no alternatives inspected. Assuming that doubling area A brings along a constant number of new species fits often better. Arrhenius 0.19 Doubling umber of individuals

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