measuring biodiversity - up · measuring biodiversity geometric series motomura, i. 1932. a...

Biodiversidade de Ecossistemas Aquáticos 13/04/11 1

Measuring Biodiversity



Species Site 1 Site 2 Site 3

Stenothoe monoculoides 122 28 4657

Arcturidae 0 0 16

Campecopea hirsuta 0 0 10

Cymodoce truncata 0 0 103

Dynamene bidentata 378 112 2224

Dynamene edwardsi 0 0 10

Dynamene magnitorata 14 0 1010

Gnathia sp. 20 0 12

Idotea baltica 0 0 22

Idotea granulosa 4 1011 3411

Idotea pelagica 0 107 627

Ischyromene lacazei 0 51 754

Are these communities equally diverse?



Commonness and rarity of species



Plotting species abundance data

For Site 3For Site 1

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Whittaker plots

1 2 3 4 5 6 7 8 9 10 11 12 130.000

0.100

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Site 1Site 3

Species Rank

Re

lati

ve A

bun

da

n ce



Relative Species Abundance Patterns

● Descriptive approaches– attempt to fit a mathematical model to real data sets

● Mechanistic approaches– create a mathematical model based on biological

principles and then test how well these models fit real data sets



Descriptive approaches



Geometric SeriesMotomura, I. 1932. A statistical treatment of associations, Jpn. J. Zool. 44: 379–383 (in Japanese)

Within the geometric series each species’ level of abundance is a sequential, constant proportion (k) of the total number of individuals in the community. Thus if k is 0.5, the most common species would represent half of individuals in the community (50%), the second most common species would represent half of the remaining half (25%), the third, half of the remaining quarter (12.5%) and so forth.

ni=NC k k 1−k i−1

ni = total number of individuals in the ith species

N = total number of individualsS = total number of speciesk = proportion of the remaining nicheC

k = [1-(1-k)S]-1



Geometric Series

Best fit to species-poor assemblages (harsh environments) in the very early stages of succession



Logseries (Fisher et al 1943)Fisher, R.A, Corbet A.S., Williams C.B. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12: 42-58.

S= ln 1N

N = total number of individualsS = total number of speciesα = proportion of the remaining nichex = a positive constant (0<x<1) which is derived from the sample data set and generally approaches 1 in value

n = total number of individuals for a given species

S n= xn

n

The number of species with 1,2,3,...,n individuals is x , x2

2, x3

3 , , xn

n



Log Normal (Preston 1948)Preston, F.W. 1948. The Commonness, and Rarity of Species. Ecology 29 (3): 254-283

S R=S 0 exp −a2 R2

S(R) = number of species in the Rth octave (class)S

0 = number of species in the modal class

a=22

Species abundances follow a Normal (Gaussian) distribution. According to this argument, the left-skew observed in species abundance frequency histograms (including those described by the logseries) is a sampling artifact. Given that species toward the left side of the x-axis are increasingly rare, they may be missed in a random species sample. As the sample size increases however, the likelihood of collecting rare species in a way that accurately represents their abundance also increases, and more of the normal distribution becomes visible



Log Normal (Preston 1948)



(Niche apportionment models)Mechanistic approaches (Niche apportionment models)

● MacArthur's Broken-stick model

● Tokeshi's models



MacArthur's Broken-stick modelMac Arthur, R. H. (1957). On the relative abundance of bird species. Proc. Natl. Acad. Sci. 43, 293-295.

ni=N T

S∑n=i

S1n

Niche space is divided randomly between S species (a stick is broken randomly and simultaneously in S parts)

ni = number of individuals in the ith most important species

NT = total number of individuals

S = total number of species



Tokeshi's models



Tokeshi's models (e.g., Power Fraction)

The probability with which a portion of the niche colonized is dependent on the relative sizes of the established niches, and is scaled by an exponent k. k can take a value between 0 and 1 and if k>0 there is always a slightly higher probably that the larger niche will be colonized. This model is thought as being more biologically realistic because one can imagine many cases where the species with the larger proportion of resources is more likely to be invaded because that niche has more resource space, and thus more opportunity for acquisition.



Fitting models to data

The AIC (Akaike Information Criterion)

Y = a + bx + cx²Y = a + bx



Species Richness (S)

How many species are there?



DMg=1−S ln N

Simple indices

Margalef's diversity index

DMn=S

NMenhinick's index

Both measures remain strongly influenced by sampling effort



Species accumulation curves

Plot the cumulative number of species recorded (S) as a function of sampling effort (n)

Sampling effort (n) can be an increase in area, volume, time, pitfalls, nets, etc.




Estimating the number of (unobserved) species

Parametric methods

S n=Smax n

BnS(n) = total number of species observed in n samplesS

max = total number of species in the assemblage

B = sampling effort required to detect 50% of Smax

Methods try to estimate the fitted constants Smax

and B





Non-parametric estimators

SChao 1

SChao 2

SACE

SJack 1

SJack 1

SBoot

Mostly base estimates on “rare” species





SChao1=S obsF 1

2

F 22

Sobs

= number of species in sample

F1 = number of singletons

F2 = number of doubletons

SChao2=S obsQ1

2

2Q22

Sobs


Q1 = number of species that occur in

one sample only

Q2 = number of species that occur in

two samples





S ACE=S abundS rare

C ACE

F 1

C ACE

×2ACE

Srare

= number of rare species (< 10 individuals)

Sabund

= number of common species (> 10 individuals)

Fi = number of species with i individuals

CACE

= 1-F1/N

rare

and 2ACE=max { S rare

C ACE

×

∑i=1

10

i i−1F i

N rare N rare−1 −1.0}





S Jack 1=S obsQ1m−1m

Sobs


Q1 = number of species that occur in one sample only

Q2 = number of species that occur in two samples

m = number of samples

S Jack 2=S obs[Q1 2m−3

m−Q2m−22

mm−1 ]





Sboot=Sobs∑k=1

S obs

1− pk m

Sobs


pi = frequency of species i

m = number of samples



Performance

Known universe Unknown universe



Rarefaction

Technique to standardize and compare species richness computed from samples of different sizes

The larger the number of individuals sampled, the more species will be found



Rarefaction

Rarefaction curves are necessary for estimating species richness. Raw species richness counts, which are used to create accumulation curves, can only be compared when the species richness has reached a clear asymptote.

Rarefaction curves produce smoother lines that facilitate point-to-point or full dataset comparisons.



Rarefaction

Rarefaction only works well when no taxon is extremely rare or common, or when beta diversity is very high.

Rarefaction assumes that the number of occurrences of a species reflects the sampling intensity, but if one taxon is especially common or rare, the number of occurrences will be related to the extremity of the number of individuals of that species, not to the intensity of sampling.

It also assumes individuals disperse randomly



Rarefaction



Indices of α diversity



Logseries α

Lognormal λ

“Parametric” measures of diversity

S= ln 1N

The number of species with 1,2,3,...,n individuals is x , x2

2, x3

3 , , xn

n



“Non-parametric” measures of diversity

Shannon-Wiener H'

Brillouin's Index (HB)

Simpson's Index (D) and variants

Taxonomic diversity



Shannon-Wiener* H'

*Note: Incorrectly spelled “Shannon-Weaver” because it was published in a book edited by Shannon and Weaver

H '=−∑

i=1

n

pi ln pi

n = number of species in samplepi = proportion of species i in sample



Shannon-Wiener* H'

Site 1 Site 2 Site 1 -p*lnp

Site 2 -p*lnp

Sp A 12 49 0.34 0.06

Sp B 11 1 0.33 0.08

Sp C 14 1 0.36 0.08

Sp D 13 1 0.35 0.08

sum 50 52 1.38 0.28

12/38*Ln(12/38)



Shannon Evenness J'

H '=−∑

i=1

n

pi ln pi

n = number of species in samplepi = proportion of species i in sampleS = total number of species observed

J '=

H '

H max

H max=ln S



Simpson's Index D

n = number of species in samplepi = proportion of species i in sample

D=∑i=1

n

p2i

Simpsons's D measures the probability of any two individuals drawn at random from an infinitely large community belonging to the same species

Sometimes also reported as 1-D or 1/D



Simpson's Index D

ni = total number of individuals of the ith species

N = total number of individualsS = total number of species

D=∑i=1

S niN

×[ni−1]

[N−1]

This version is for finite samples...



Simpson's Evenness


E1 /D=1/DS



Taxonomic Diversity



Taxonomic Diversity

=[∑i=1

S−1

∑j=i1

S

ij xi x j ][ nn−1

2 ]S = total number of speciesx

i = proportion of ith species in sample

n = Σxi



Taxonomic Diversity



Taxonomic Distinctness

*=

[∑i=1

S−1

∑j=i1

S

ij x i x j ][∑i=1

S−1

∑j=i1

S

x i x j ]S = total number of speciesx

i = proportion of ith species in sample

Effects of proportionsof individuals are “removed”

Measures pure taxonomicrelatedness



Average Taxonomic Path Length

+=

[∑i=1

S−1

∑j=i1

S

ij ][ S S−1

2 ]S = total number of species

This measure is a particular case of both Δ and Δ* when data is reduced to presence absence



a) Δ+ = 3.0 b) Δ+ = 1.0 c) Δ+ = 1.56 d) Δ+ = 1.2

Average Taxonomic Path Length



Taxonomic Diversity: invariant to sample size!

H' Pielou's JMergalef's D

ΔΔ Δ* Δ+



Taxonomic Diversity: testing hypothesis



Measuring β diversity



Whittaker's βw

w=S /


= average sample diversity

Samples should have a standard size and diversity is measured as species richness



Other measures

C=g H lH

2Cody's β

C

R=S2

2rS−1 Routledge's β

R

I=logT−[1/T ∑ ei log e i ]−[1/T ∑ Sj log Sj ] Routledge's βI

E=e I Routledge's β

E



Indices of similarity and dissimilarity

CM=1−a

abc

a = number of species in both samplesb = number of species exclusive of sample 1c = number of species exclusive from sample 2

C j=a

abcCM is the complement of Jaccard's (1908) index




CM=1−a

abc

a = 7b = 2c = 2

CM=1−7

722=0.364

Species Site 1 Site 2 Site 3

Stenothoe monoculoides 122 28 4657

Arcturidae 0 0 16

Campecopea hirsuta 0 0 10

Cymodoce truncata 0 0 103

Dynamene bidentata 378 112 2224

Dynamene edwardsi 0 0 10

Dynamene magnitorata 14 0 1010

Gnathia sp. 20 0 12

Idotea baltica 0 0 22

Idotea granulosa 4 1011 3411

Idotea pelagica 0 107 627

Ischyromene lacazei 0 51 754

Between Site 1 and 2




CS=2a

2abc

sim=1− aaminb ,c

CN=2jNN aN b

Plenty of them...



Multivariate Analysis

●Cluster Analysis (UPGMA, K-means, etc.)

●PCA (Principal Component Analysis)

●PCo (Principal Coordinate Analysis)

●CA (Correspondence Analysis)

●MDA (Multidimensional Scaling)

●etc.



Non-metric Multidimensional Scaling

measuring biodiversity - up · measuring biodiversity geometric series motomura, i. 1932. a...

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