Advanced analytical approaches in ecological data analysis

The world comes in fragments

Early plant succession in the post brown cole mining area Chicken Creak

2005 2010

Succession starts with colonising species from a regional species pool and from the initial seed bank

Species abundance matrix MPhylogenetic distance matrix P

Species trait matrix T

Site GPS location matrix D

Environmental variable matrix V

Interdepen-dence

matrix X

Species

Spec

ies

Spec

ies

Spec

ies

Sites

Sites

Sites

Varia

bles

Varia

bles

Traits

Traits

Multivariate approaches to biodiversity

Why are species abundant or rare?What determines community composition?How does a community function in space and time? L

Species abundance matrix MPhylogenetic distance matrix P

Species trait matrix T

Species

Spec

ies

Spec

ies

Spec

ies

SitesTraits

The interplay between traits, phylogeny, and species occurrences

As species of the same genus have usually, though by no means invariably, some similarity in habits and constitution, and always in structure, the struggle will generally be more severe between species of the same genus, when they come into competition

with each other, than between species of distinct genera. (Darwin 1859)

Traits and phylogeny are closely related.

Trait distance correlates positively with phylogenetic distance

Niche conservatism

Competition: niche conservatism causes phylogenetically close species to

co-occur less common.Habitat filtering causes species with

similar traits (close in phylogeny) to co-occur more frequent.

Local colonizers

Regional pool of species

Environmental filters

Random colonizationRegional pool of potential colonizers

Regional pool of species

No phylogenetic structure

No phylogenetic structure

Phylogenetic clumping

Early succession

Facilitation

Phylogenetic segregation

Local community structure

Competition

No phylogenetic signal

Phylogenetic clumping

Later succession

Positive interactions

Phylogenetic segregation

Neutral interactions

Phylogenetic distance matrix P

Species trait matrix T

Species

Spec

ies

Spec

ies

Traits

Mantel test

A Mantel test is a correlation between two distance matrices.

We have to transform the trait matrix into a distance matrix and correlate with the phylogeny matrix.

S

Achillea_panno

nica

Agrostis_capill

aris

Agrostis_stolonifera_

agg.

Agrostis_vinea

lisAchillea_pannonica 0.0 179.0 179.0 179.0Agrostis_capillaris 179.0 0.0 2.5 2.5Agrostis_stolonifera_agg. 179.0 2.5 0.0 2.5Agrostis_vinealis 179.0 2.5 2.5 0.0

S

Achillea_panno

nica

Agrostis_capill

aris

Agrostis_stolonifera_

agg.

Agrostis_vineali

sAchillea_pannonica 0.0 179.0 179.0 179.0Agrostis_capillaris 179.0 0.0 2.5 2.5Agrostis_stolonifera_agg. 179.0 2.5 0.0 2.5Agrostis_vinealis 179.0 2.5 2.5 0.0

R = -0.04; P(r=0) = 0.37

Plant traits and phylogeny in the year 2011 plare not correlated for the species in 2011. Why?? We averaged over all traits

Mantel test

Blue eyes

Blue eyes

Blue eyes

Trait

Brown eyes

Brown eyes

Blue eyes

Brown eyes

Blue eyes

Blue eyes

Brown eyes

The trait ‚brown eyed’ is phylogenetically conserved.

Closely related species have the same eye colour

Eye colour is phylogenetically not conserved.

10

5

10

8

12

Blue eyes: (10+5+10+15)/3=10

15

Blue eyes: (10+5+15++10+ 8+12)/3=20

10

12

Trait

10

5

10

8

12

15

10

12

Compare these phylogenetic distances per trait with those expected from a random (Brownian motion ) phylogeny.

Brownian motion

Brownian motion is the standard method to generate random phylogenetic trees

1. Start with a pixel2. Take a second pixel and move at

random until it sticks to the first3. Repeat two until the desired number of

end-tips is reached.

Brownian motion generates random

trees and can simulate the evolution of traits

along this trees.

Blue eyes

Blue eyes

Blue eyes

Blue eyes

Brown eyes

Blue – brown boundary

Brownian motion (random walk) along a phylogeny

Compare the observed pattern of blue and brown with those generated by many Brownian motion trees.

𝑑𝑋 (𝑡)=𝜎𝑑𝐵 (𝑡)Start 0

1 0.0021682+B2+0.5*(LOS()-0.5)

Moran’s I as a test of phylogenetic signal

𝐼= 𝑛

∑𝑗∑𝑗

𝑤 𝑖𝑗

∑𝑗∑𝑗

𝑤𝑖𝑗 (𝑥 𝑖− 𝑥 ) (𝑥 𝑗−𝑥 )

∑𝑖

(𝑥 𝑖− 𝑥 )2 where n is the number of data points.

w defines the strength of distance effects.

I is similar to a weighed coefficient of correlation.M X-M

n X X-M -4.76 -2.21 3.59 4.26 -0.881 0.83 -4.76 22.68 10.52 -17.10 -20.27 4.182 3.38 -2.21 -50.09 -23.24 37.77 44.78 -9.223 9.18 3.59 -179.86 -83.44 135.62 160.80 -33.124 9.85 4.26 -765.71 -355.23 577.36 684.56 -140.995 4.71 -0.88 671.37 311.47 -506.23 -600.22 123.62Mean 5.59

W wij wij wij wij wijDiagonal sum 943.24 wij 0 0.40 0.74 0.18 0.40n 5 wij 0.69 0.00 0.15 0.53 0.47

wij 0.46 0.87 0.00 0.75 0.32wij 0.12 0.03 0.56 0.00 0.27wij 0.45 0.02 0.64 0.94 0.00

Sum 9.01WM 1 2 3 4 5

1 0.00 4.26 -12.70 -3.71 1.682 -34.38 0.00 5.75 23.53 -4.373 -83.09 -72.95 0.00 120.97 -10.514 -92.88 -9.94 325.45 0.00 -38.29

I -0.27 5 304.35 6.52 -324.49 -561.83 0.00Exp I -0.25

𝐸𝑥𝑝 𝐼=−1𝑛−1

Species K P(K)Lamb

daP(Lamb

da)Species K P(K)

Lambda

P(Lambda)

Morphological traits Reproductive traits

Canopy height (m) 1.20 0.001 1.00 <0.001Average month of seedling

0.11 0.004 0.00 1.000

Emergent attached to substrate

0.07 0.324 0.88 0.001 Duration of flowering 0.04 0.743 0.13 0.135

Leaf mass [mg] 0.11 0.155 1.00 0.001 Duration of seedling 0.04 0.505 0.00 1.000Leaf size [mm2] 0.10 0.138 0.98 0.003 Early month flowering 0.08 0.028 0.80 <0.001

Life span 0.07 0.014 0.50 0.020Early month seed shedding

0.05 0.226 0.00 1.000

Max releasing height [m] 2.11 0.001 1.00 <0.001 Latest month flowering 0.14 0.001 0.82 <0.001

Min releasing height [m] 0.78 0.003 1.00 <0.001Latest month seed shedding

0.19 0.001 0.52 0.085

Specific leaf area mm2/mg 0.06 0.048 <0.01 1.000 ln (Seeds per shoot) 0.06 0.054 0.63 0.002Stem ascending to prostrate %

0.10 0.012 0.89 <0.001 Mean seed weight 0.13 0.084 1.00 <0.001

Stem erect % 0.11 0.006 0.96 <0.001 Seed bank longevity 0.06 0.074 0.00 1.000Terminal velocity m/s 0.06 0.120 0.58 <0.001 Type reproduction 0.05 0.179 0.22 0.011Woodiness Stem 0.65 0.001 1.00 <0.001

Pagel’s lambda

Species K P(K)Lamb

daP(Lamb

da)Species K P(K)

Lambda

P(Lambda)

Habitat requirements Molecular traitsLight 0.04 0.506 0.22 0.063 Polyploidy 0.05 0.251 0.40 0.001Soil fertility 0.06 0.154 <0.01 1.000 Chromosome number 8.62 0.001 1.00 <0.001pH 0.05 0.251 <0.01 1.000 DNA content 0.39 0.001 0.75 <0.001Nitrogen 0.05 0.126 <0.01 1.000Life strategy type 0.06 0.133 0.39 1.000Grazing tolerance 0.03 0.855 <0.01 1.000Hemerobic level 0.05 0.142 <0.01 1.000

Morphological and genetic plant traits are phylogenetically more conserved than life history, reproductive, and ecological traits.

Large size

Large size

Large size

Body size

Small size

Small size

10

5

10

8

12

15

10

12

Small home range

Small home rangeLarge home range

Large home range

Small home range

Species daily home rangeUngulates

Mammal predators

Without phylogenetic knowledge we would link body size to home range in a functional manner.

Home range and body size are linked by common phylogeny. They are phylogenetically preserved.

Phylogenetic pseudoreplication

Phylogenetic regression accounts for the phylogenetic non-independence of variables

Phylogenetic distance matrix P

Species trait matrix T

Species

Spec

ies

Spec

ies

Traits

Eigenvector methods

M U U= l

x

Every square matrix M has a vector U so that

𝑴𝑼=𝜆𝑼 (𝑴 − 𝜆 𝑰)𝑼=0U: Eigenvectorl: EigenvalueI: Identity matrix

Because both sides of the equation are equal the right side contains the same information as the left side.The eigenvector U contains the information in M in a condensed form.

SpeciesLeaf mass [mg]

Leaf size

[mm2]Life

span Light Soil fertility pH Nitro

gen

ln (Seeds

per shoot)

Specific leaf area

mm2/mg

DNA content

Mean seed

weight

Grazing toleran

ce

Achillea_pannonica 82.3 567.8 5 7 3 6 2 6.3 8.1 19.1 0.1 5.1Agrostis_capillaris 61.0 1147.9 5 7 0 4 4 5.3 26.8 7.1 0.1 5.0Agrostis_stolonifera_agg. 61.0 1147.9 5 8 7 0 5 12.1 22.8 7.0 0.1 9.0

Agrostis_vinealis 1.4 22.8 5 9 9 3 2 4.3 15.6 6.9 0.1 2.0Ajuga_genevensis 18.0 365.5 5 8 3 7 2 12.1 25.8 5.9 1.8 5.1Apera_spica_venti 61.0 1147.9 1 6 6 5 0 8.5 0.0 10.8 0.1 5.1Arenaria_serpyllifolia_agg. 0.2 3.6 0.5 8 4 7 0 6.0 16.1 1.7 0.1 5.1

Artemisia_vulgaris_agg. 61.0 1147.9 5 7 6 0 8 12.8 0.0 6.0 0.1 5.1Species

Leaf mass [mg]

Leaf size

[mm2]Life

span Light Soil fertility pH Nitro

gen

ln (Seeds

per shoot)

Specific leaf area

mm2/mg

DNA content

Mean seed

weight

Grazing toleran

ce

Achillea_pannonica 1.32 -0.26 0.58 -0.58 -0.66 0.76 -0.35 -0.66 -0.63 2.31 -0.38 -0.05Agrostis_capillaris 0.60 0.94 0.58 -0.58 -1.80 0.00 0.45 -0.98 1.23 -0.20 -0.38 -0.10Agrostis_stolonifera_agg. 0.60 0.94 0.58 0.58 0.85 -1.51 0.84 1.13 0.83 -0.22 -0.38 2.17

Agrostis_vinealis -1.41 -1.39 0.58 1.73 1.61 -0.38 -0.35 -1.27 0.12 -0.24 -0.38 -1.80Ajuga_genevensis -0.85 -0.68 0.58 0.58 -0.66 1.13 -0.35 1.13 1.13 -0.45 2.65 -0.05

Apera_spica_venti 0.60 0.94 -1.59 -1.73 0.47 0.38 -1.14 0.04 -1.43 0.57 -0.38 -0.05

Arenaria_serpyllifolia_agg. -1.45 -1.43 -

1.86 0.58 -0.28 1.13 -1.14 -0.75 0.17 -1.33 -0.38 -0.05

Artemisia_vulgaris_agg. 0.60 0.94 0.58 -0.58 0.47 -1.51 2.03 1.36 -1.43 -0.43 -0.38 -0.05

=(S12-ŚREDNIA(S$4:S$11))/ODCH.STAND.POPUL(S$4:S$11)

We use Z-transforms to normalize the trait values

Species Achillea_pannonica

Agrostis_capillaris

Agrostis_stolonifera_agg

.

Agrostis_vine

alisAjuga_gen

evensisApera_spica_venti

Arenaria_serpyllifolia_agg.

Artemisia_vulgaris_

agg.

Achillea_pannonica 0.00 3.79 5.37 5.58 5.43 3.77 5.58 5.13Agrostis_capillaris 3.79 0.00 4.54 5.61 4.79 4.75 5.06 4.76Agrostis_stolonifera_agg. 5.37 4.54 0.00 6.01 5.45 5.44 6.18 3.61Agrostis_vinealis 5.58 5.61 6.01 0.00 5.35 6.04 4.28 5.96Ajuga_genevensis 5.43 4.79 5.45 5.35 0.00 5.93 4.69 5.98Apera_spica_venti 3.77 4.75 5.44 6.04 5.93 0.00 4.81 4.74Arenaria_serpyllifolia_agg. 5.58 5.06 6.18 4.28 4.69 4.81 0.00 6.52

Artemisia_vulgaris_agg. 5.13 4.76 3.61 5.96 5.98 4.74 6.52 0.00

The Euclidean distance matrix

Eigenvalues Squared eigenvalues

Explained variance

-8.80 77.43 0.05-6.78 45.96 0.03-5.67 32.16 0.02-4.74 22.44 0.01-4.09 16.70 0.01-3.44 11.81 0.01-2.84 8.08 0.0136.35 1321.43 0.86

Dominant eigenvector

2nd eigenvector

0.340 0.1390.328 0.1300.356 0.4060.374 -0.4490.364 -0.2830.347 0.1390.360 -0.5130.357 0.484

The dominant eigenvector

(DEV) contains 86% of variance in the distance

matrix.

R² = 9E-05

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

DEV

phy

loge

ny

DEV traits

Traits 2011 PhylogenyEigenvalue 104786 Eigenvalue 19855S DEV S DEVAchillea_pannonica 0.249299 Achillea_pannonica 0.296259Agrostis_capillaris 0.202592 Agrostis_capillaris 0.318521Agrostis_stolonifer 0.202402 Agrostis_stolonifer 0.318521Agrostis_vinealis 0.331781 Agrostis_vinealis 0.318521Ajuga_genevensis 0.271579 Ajuga_genevensis 0.323584Apera_spica_venti 0.202506 Apera_spica_venti 0.318354Arenaria_serpyllifo 0.336897 Arenaria_serpyllifo 0.318607

Trait Spearman's rLeaf_mass_[mg] 0.12525Leaf_size_[mm2] 0.055584Life_span 0.14348Light -0.13865Soil_fertility 0.007277pH -0.01992Nitrogen -0.06805ln_(Seeds_per_shoot) 0.10293Specific_leaf_area_mm2/mg 0.049542DNA_content 0.022556Mean_seed_weight 0.097499Grazing_tolerance -0.03174

None of the traits is correlated to phylogenetic distance.

DEV contains information on the average distance of species in niche space

Ecological niches

SpeciesLeaf mass [mg]

Leaf size [mm2]

Life span Light Soil

fertility pH Nitrogen

ln (Seeds

per shoot)

Specific leaf area

mm2/mg

DNA content

Mean seed

weight

Grazing toleranc

e

Achillea_pannonica 1.32 -0.26 0.58 -0.58 -0.66 0.76 -0.35 -0.66 -0.63 2.31 -0.38 -0.05Agrostis_capillaris 0.60 0.94 0.58 -0.58 -1.80 0.00 0.45 -0.98 1.23 -0.20 -0.38 -0.10Agrostis_stolonifera_agg. 0.60 0.94 0.58 0.58 0.85 -1.51 0.84 1.13 0.83 -0.22 -0.38 2.17

Agrostis_vinealis -1.41 -1.39 0.58 1.73 1.61 -0.38 -0.35 -1.27 0.12 -0.24 -0.38 -1.80Ajuga_genevensis -0.85 -0.68 0.58 0.58 -0.66 1.13 -0.35 1.13 1.13 -0.45 2.65 -0.05Apera_spica_venti 0.60 0.94 -1.59 -1.73 0.47 0.38 -1.14 0.04 -1.43 0.57 -0.38 -0.05Arenaria_serpyllifolia_agg. -1.45 -1.43 -1.86 0.58 -0.28 1.13 -1.14 -0.75 0.17 -1.33 -0.38 -0.05

Artemisia_vulgaris_agg. 0.60 0.94 0.58 -0.58 0.47 -1.51 2.03 1.36 -1.43 -0.43 -0.38 -0.05

Dominant eigenvector

2nd eigenvector

0.340 0.1390.328 0.1300.356 0.4060.374 -0.4490.364 -0.2830.347 0.1390.360 -0.5130.357 0.484

Convex hulls, eigenvector ellipses, and functional attribute diversity

0

0.5

1

1.5

0.4 0.6 0.8 1 1.2 1.4 1.6

Soil

ferti

lity

Light

EV1EV2l1

𝐴=𝜋 𝜆1 𝜆2

l1

The area of the eigenvector ellipse is a measure of niche space

Axes length are given by the respective eigenvalues

3.47=𝜋×1.58×0.70

The niche space spanned by light and soil fertility spans 3.47 units.

Eigenvalues1.58 0.70EV1 EV20.71 0.71-0.71 0.71

Compare these value with those obtained from a null model

The larger the niche space is the higher is the functional diversity of a community or a species.

Eigenvectors are always orthogonal.

0

0.5

1

1.5

0.4 0.6 0.8 1 1.2 1.4 1.6

Soil

ferti

lity

Light

Convex hulls CH

The area of a convex hull is a measure of total niche space.

High dimensional convex hull are difficult to obtain.

Two dimensional convex hulls are easy to calculate.

-…

Functional attribute diversity FAD

SpeciesAchillea_pannonic

aAgrostis_c

apillarisAgrostis_stolonifera_agg.

Agrostis_vine

alis

Ajuga_geneve

nsis

Apera_spica_v

enti

Arenaria_serpyllifol

ia_agg.Achillea_pannonica 0.00 3.79 5.37 5.58 5.43 3.77 5.58Agrostis_capillaris 3.79 0.00 4.54 5.61 4.79 4.75 5.06Agrostis_stolonifera_agg. 5.37 4.54 0.00 6.01 5.45 5.44 6.18Agrostis_vinealis 5.58 5.61 6.01 0.00 5.35 6.04 4.28Ajuga_genevensis 5.43 4.79 5.45 5.35 0.00 5.93 4.69Apera_spica_venti 3.77 4.75 5.44 6.04 5.93 0.00 4.81Arenaria_serpyllifolia_agg. 5.58 5.06 6.18 4.28 4.69 4.81 0.00

Sites

1001101

𝐹𝐴𝐷=5.58+5.43+5.58+5.35+4.28+4.69

6=5.15

𝐹𝐴𝐷=2∑ 𝑑𝑖𝑗

𝑛(𝑛−1); 𝑗>1

Raw FAD scores are meaningless.You have to compare these scores with an appropriate null model

of species occurrences.

Total raw functional attribute diversity (grey bars) increased

while the respective SES scores of FAD (red bars) and convex hulls

(blue bars) decrease during succession when compared to a

neutral null model .

Regression coefficients

High plant cover decreased an species richness increases standardized effect sizes of FAD (neutral null model) in all study years.

Soil characteristics did not significantly influence SES FAD (functional diversity).

2006

2011

The evolutionary dimension of species occurrences

Eigenvector mapping, eigenvector regression, logistic eigenvector regressiuon

Species abundance matrix

M

Phylogenetic distance matrix P

SpeciesSp

ecie

s

Spec

ies

SitesEV1EV2

Significant eigenvectors

𝑨=𝑼 𝜷

The explained variance r2 of this regression is a measure of the influence of evolutionary history on species abundances.

Net relatedness index

S Achillea_pannonica

Agrostis_capillaris

Agrostis_stolonifera_

agg.Agrostis_vi

nealisAjuga_gen

evensisApera_spic

a_venti

Achillea_pannonica 0.0 179.0 179.0 179.0 117.0 179.0Agrostis_capillaris 179.0 0.0 2.5 2.5 179.0 6.7Agrostis_stolonifera_agg. 179.0 2.5 0.0 2.5 179.0 6.7

Agrostis_vinealis 179.0 2.5 2.5 0.0 179.0 6.7Ajuga_genevensis 117.0 179.0 179.0 179.0 0.0 179.0Apera_spica_venti 179.0 6.7 6.7 6.7 179.0 0.0

M7-2

0.50.500

0.50

Phylogenetic distance matrix Abundance vector

𝑅𝑎𝑤𝑁𝑅𝐼=179+117+179

3=158

The raw net relatedness index is the phylogenetic distance of all species present at

a focal site.

M7-2

00

0.50.50.50

Randomized bundance vector

𝑁𝑅𝐼=− 𝑟𝑎𝑤𝑁𝑅𝐼 −𝑟𝑎𝑛𝑑𝑜𝑚𝑖𝑧𝑒𝑑𝑟𝑎𝑤𝑁𝑅𝐼𝜎𝑟𝑎𝑛𝑑𝑜𝑚𝑖𝑧𝑒𝑑 𝑟𝑎𝑤 𝑁𝑅𝐼

Calculate the raw NRI from 1000

randomized abundance

matrices

NRI increases with increasing phylogenetic clustering

NTI2011 4

3.5

3

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

-3

-3.5

-4

-4.5

-5

-5.5

-6

-6.5

-7

-7.5

-8

-8.5

-9

-9.5

NRI2011 4

3.5

3

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

-3

-3.5

-4

-4.5

-5

-5.5

-6

-6.5

-7

-7.5

-8

-8.5

-9

-9.5

Relating phylogenetic patterns to environmental variables

NRI Phylogenetic eigenvector regression r2

Variables B Error B t p B Error B t pConstant 1305.600 121.690 10.729 <0.0001 -15.484 3.2963 -4.697 <0.0001Study year -0.651 0.061 -10.746 <0.0001 0.008 0.0016 4.748 <0.0001Species richness 0.193 0.016 11.947 <0.0001 0.001 0.0004 2.538 0.011

Abundance -0.023 0.002 -13.173 <0.0001 0.001 0.0000 22.931 <0.0001Soil carbon 0.345 0.096 3.588 <0.0001 -0.008 0.0026 -3.118 0.002Sand 0.008 0.011 0.699 0.484 -0.001 0.0003 -3.746 <0.0001pH -0.008 0.127 -0.066 0.947 -0.002 0.0034 -0.610 0.542