1. No category

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Neutral Model of Vegetation
– An Introduction to Hubbell –
Jari Oksanen
Department of Biology
University of Oulu
6622 for 2007
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The Book
Hubbell, S. P. (2001) The Unified Neutral Theory of Biodiversity and
Biogeography. Princeton University
Press.
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The World without Competition
• Radically neutral model
• No competitiion
• No differences among species
• No environmental differences or “gradients”
• Neutral model is certainly wrong but this does not matter
because the model works!
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The Lectures
• Introduction into Hubbell’s Unified Neutral Theory and related
aspects of plant community ecology
• Species abundance relations, (bio)diversity, species–area
relationship, spatial pattern, and species pool: All in the light of
Neutral Theory
• How to estimate Hubbell’s Fundamental Parameter (θ) with the
Data
• Simulation of Hubbell models and parameter estimation in
statistical environment
• Warning: Some new material will be added to these lectures!
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CONTENTS
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Contents
1 A Gentle Introduction to Hubbell
6
2 Species Abundance Models
18
3 Hardcore Hubbell
45
4 Hubbell’s Game
63
5 Species Richness and Sample Size
74
6 Hubbell’s Second Game: Metacommunity Landscape
88
7 Species Pool: What is a Metacommunity?
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A GENTLE INTRODUCTION TO HUBBELL
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A Gentle Introduction to Hubbell
In “The Unified Neutral Theory of Biodiversity and Biogeography’
S. P. Hubbell explains his neutral model that can explain most things
that are worth explaining in Biogeography. A crucial parameter is
Hubbell’s “fundamental biodiversity number” θ. In this lecture I give
a gentle introduction to the model — no matrix algebra at all!
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A GENTLE INTRODUCTION TO HUBBELL
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Neutral model
• All organisms are equal, irrespective of species or position in the
community
• Individuals can interact, as long as the rules are identical for all
individuals and species
• Recognized mechanisms
1. Random ecological drift
2. Stochastic, albeit limited distribution
3. Random speciation
• Neutral models are “wrong”, but if they work — who cares?
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A GENTLE INTRODUCTION TO HUBBELL
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Island Biogeography: Another Neutral Model
• Only the sizes and distances of islands important: No differences
among species, but the the toal number of species is interesting
• Logical consequence: Random collection of species
• Applications negated neutrality, and looked for differences among
species.
• Hubbell more radical: Neutral at the level of individual.
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A GENTLE INTRODUCTION TO HUBBELL
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The Earth is Neutral and Crowded
• Species multiply and fill the Earth, but communities won’t grow
without limit
• The number of individuals J increases (linearly) with area
• Community assembled in a zero-sum game: Total number of
individuals unchanged
• Species succeeds only if some other species perish
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A GENTLE INTRODUCTION TO HUBBELL
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Local Community Dynamics
1. Assume we have a local community of J individuals
2. D randomly picked individuals die
3. Surrounding Metacommunity sends in M immigrants
4. Remaining free lots are replaced from within the local community
with probabilities proportional to abundances after disturbance
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A GENTLE INTRODUCTION TO HUBBELL
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60
80
Ecological Drift in Local Community
0
20
40
N
1
0
50
100
150
200
t
Absorbing states: Monodominance, extinction to others
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A GENTLE INTRODUCTION TO HUBBELL
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Isolated Communities
Island: extinction
500
J=1600, D=400, m=0
100
50
t=0
t=10
20
t=500
10
• The ultimate fate of all
communities is monodominance.
t=1000
200
• Isolation leads to extinction and lower richness.
Abundance
t=25
t=100
2
• Local extinction is the
fate of most species.
5
t=250
1
1
t=50
0
10
20
30
40
Rank
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de 1
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A GENTLE INTRODUCTION TO HUBBELL
• Single Local Community in the Metacommunity Sea: θ defines
'
Metacommunity composition Pi and it is filtrated into Local
Communities asRescue:
m allows:Metacommunity
13
$
m
Ji
Pi
θ = 2JM ν
• Metacommunity as a Network of Local Communities: Migration
•mLocal
communityrates
J is surrounded
by aCommunities,
Metacommunity
and speciation
ν create Local
andP which
can send in new isspecies
Metacommunity
their sum:
• Immigration from a Metacommunity restores diversity
and cures
θ
Jki
Jki
ν= ! !
from monodominance
m
2
Jki
k
i
! entity: We only
• At the moment, Metacommunity is a mythical
Jki
Jki
Pi =
Jki
know that it is there because we see its effectsk in local
"community, but we do not know what is a Metacommunity
• At least Metacommunity is big so that extinctions are slow, but
still they should happen
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A GENTLE INTRODUCTION TO HUBBELL
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Speciation in Metacommunity
• Monodominance is the ultimate fate of a community
• In Metacommunity, random speciation balances extinction and
leads into stable species abundance relations
• The size JM of a Metacommunity is large, and (nearly)
impossible to measure, the rate of speciation ν is low, and
impossible to measure, but together they produce Hubbell’s
fundamental diversity number θ:
θ = 2JM ν
which is moderate in range and can be estimated
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A GENTLE INTRODUCTION TO HUBBELL
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Derivation of Ultimate Diversity. . .
• What is the probability F2t+1 that two random individuals belong
to the sampe species in generation t + 1 when this probability
was F2t in the previous generation?
• Probability that neither speciated at birth is (1 − ν)2 and
probability that they have a common parent is 1/JM .
• In the beginning of Times, all had one parent and recursively
1
1
t+1
2
F2 = (1 − ν)
+ 1−
F2t
JM
JM
and in stable community F2t+1 = F2t . . .
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A GENTLE INTRODUCTION TO HUBBELL
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. . . And The Solution is
• In stable community F2t+1 = F2t = F2 , with an approximate
solution:
F2
=
∼
=
∼
=
&
(1 − ν)2
JM − (1 − ν)2 (JM − 1)
1
1 + 2JM ν
1
1+θ
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A GENTLE INTRODUCTION TO HUBBELL
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Things to explain
• Species abundance distribution
• Slope of species–area curve
• Similarity decay with distance (“beta diversity”)
• Things to describe = things to estimate:
– Fundamental diversity number θ, and perhaps immigration
between communities (m) or its flip side, isolation ω
– Fitted models depend on community size J, but that we take
as known
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SPECIES ABUNDANCE MODELS
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Species Abundance Models
The Neutral Theory predicts some kind of abundance model and we
can —in principle— estimate Neutral Theory parameters from
abundances models. But what kind of model do we expect?
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SPECIES ABUNDANCE MODELS
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Abundance Distribution models
• Logarithmic series: Rare species never end.
• Log-Normal model: There is modal commonness, rare species get
rare.
• Pre-emption model: Species have a natural, hierarchic ordering.
• Brokenstick: Species divide community randomly.
• Zipf–Mandelbrot: Rare species have larger costs to invade the
community because they must cope with conditions and
abundant species.
• Hubbell’s model: Commonness from a zero-sum game
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SPECIES ABUNDANCE MODELS
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Logarithmic series
R.A. Fisher in 1940’s:
Barro Colorado Island, Site 1, alpha = 35.7
25
20
15
0
• In larger samples, you may
find more individuals of rare
species, but you find new rare
species, too
5
10
• Most species are rare, and
species found only once are the
largest group
30
αxn
fn =
n
Species
2
0
5
10
15
20
25
Frequency
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SPECIES ABUNDANCE MODELS
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Sample size and Log-series
10
5
0
• When you collect a larger sample, you will find new individuals of rare species,
Species
15
Barro Colorado Island, N = 12, alpha = 34.3
0
500
1000
1500
Frequency
20
15
Species
10
5
0
• But you will find new, previously unknown species — and
these are rare.
25
Barro Colorado Island, N = 25, alpha = 37.1
0
500
1000
1500
Frequency
10
Species
15
Barro Colorado Island, N = 50, alpha = 35.0
5
• Log-series: Shape and parameter α should remain constant
with increasing sample size.
0
2
0
500
1000
1500
Frequency
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%
SPECIES ABUNDANCE MODELS
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Fisher’s alpha
200
Barro Colorado Island, Accumulation of Plots
50
20
• Another parameter (x) is an
uninteristing nuisance parameter: x = N/(N + α)
10
• Should be independent of sample size.
Fisher alpha
S
exp(H)
1/Simpson
Berger−Parker
100
• Fisher’s α is used as a diversity
index.
Diversity
2
0
5000
10000
15000
20000
Number of Trees
Rug: Plots
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Log-Normal model
Preston did not accept Fisher’s
log-series, but assumed that rare
species end with sampling
• Canonical standard model of
our times
&
35
30
25
20
15
10
5
• Modal class in higher octaves,
and not so many rare species
0
• Plotted number of species
against ‘octaves’:
doubling
classes of abundance
Barro Colorado Island, All Sites
Species
2
1
2
4
8
16
32
64
128
512
2048
Frequency
%
SPECIES ABUNDANCE MODELS
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Sample Size and Log-Normal Model
20
10
0
• Mode moves to the right.
Species
30
Barro Colorado Island, N = 12
1
• The veil moves and reveals
new, rare species.
8
16
32
64
128
256
512
20
10
0
Species
30
Barro Colorado Island, N = 25
1
2
4
8
16
32
64
128
256
512
1024
Frequency
25
35
Barro Colorado Island, N = 50
15
• The shape of the fitted model
should remain unchanged.
4
Frequency
Species
• Fewer rare species found as
sampling proceeds.
2
0 5
2
1
2
4
8
16
32
64
128
256
512
1024
2048
Frequency
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%
SPECIES ABUNDANCE MODELS
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Preston vs. Fisher
10
15
20
25
30
Fisher
Preston
Octaves
• Higher octaves lump
number of frequencies.
large
0
5
• Actually Fisher’s logseries becomes humped if you plot it by
octaves.
35
• “Small data sets follow
logseries, but when you collect more data, they are
log-Normal”.
Species
2
1
2
4
8
16
32
64
128 256 512
2048
Frequency
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%
SPECIES ABUNDANCE MODELS
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Extrapolated Species Richness
250
200
150
100
• If Log-Normal height is S0 and
width σ, the total richness, including species behind the veil,
√
is ST = S0 σ 2π.
Observed
Extrapolated
50
• Normal density has a finite
area: It is possible to estimate
the number of species behind
the veil integrating the LogNormal model.
300
Barro Colorado Island, Accumulation of Plots
Number of Species
2
0
5000
10000
15000
20000
Number of Trees
Rug: Plots
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Ranked abundance diagrams
• Linear: Pre-emption model
• Sigmoid: Log-normal or brokenstick
&
100
50
10
5
The shape of abundance distribution
clearly visible:
1
• Vertical axis: Logarithmic abundance
500
• Horizontal axis: ranked species
Runsaus
2
0
50
100
150
200
250
Rank
Brokenstick
Preemption
Lognormal
Zipf
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SPECIES ABUNDANCE MODELS
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Log-Normal RAD model
20
Preemption
Lognormal
Veiled.LN
Zipf
Mandelbrot
• Veiled Log-Normal: Cut out rare
species.
5
1
2
• Sigmoid: excess of both abundant and rare species to preemption model.
10
• Ranked abundances Normal
Abundance
2
5
10
15
Rank
Brokenstick
Preemption
Lognormal
Zipf
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Preemption and Mandelbrot RAD
• Preemption pˆr = k(1 − k)r−1
– Zipf (β = 0): Strong dominance and a long tail of rare
species.
20
5
2
• Mandelbrot pˆr = pˆ1 (r + β)γ
10
decay
– Line in RAD.
Preemption
Lognormal
Veiled.LN
Zipf
Mandelbrot
1
– Ranked abundances
proportionally.
Abundance
2
5
– Mandelbrot: Weaker dominance but a long tail.
10
15
Rank
Brokenstick
Preemption
Lognormal
Zipf
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Selecting RAD
• Adding parameter means better fit: Pre-emption 1, Log-Normal
and Zipf 2, Veiled Log-Normal and Zipf–Mandelbrot 3
parameters.
• Fit to minimize −` or negative log-Likelihood, but penalize for
each estimated parameter p, and select model with smallest
penalized criterion.
• Akaike’s Information Criterion AIC = −2` + 2p, and Bayesian
Information Criterion BIC = −2` + log(S) × p.
• The penalty in AIC is mild, and BIC often more appropriate (but
Information Criteria have different goals than statistical testing).
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Best fitting RADs
0
5
Preemption
Lognormal
10
15
20
Veiled.LN
Zipf
0
Mandelbrot
5
10
15
20
0
5
10
R4P9V2
R1P12V3
R4P11V3
R3P8V2
R1P6V2
R4P6V2
R3P10V3
R2P16V4
R1P10V3
R1P1V1
R2P3V1
R1P3V1
15
20
100
30
10
3
1
100
Abundance
2
30
10
3
1
R4P4V1
R2P8V2
R1P4V1
R2P13V4
R2P5V2
R1P2V1
100
30
10
3
1
0
&
5
10
15
20
0
5
10
15
20
0
5
Rank
Carabids, Sites Ordered by Species Richness, Criterion BIC
10
15
20
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SPECIES ABUNDANCE MODELS
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Broken Stick
• Result looks sigmoid, and
can be fitted with log-Normal
model.
10
1
2
• No real hierarchy, but chips arranged in rank order:
5
• Species ‘break’ a community
(‘stick’) simultaneously in S
pieces.
20
Carabids, Trap 5
Abundance
2
5
10
15
Rank
Brokenstick
Preemption
Lognormal
Zipf
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SPECIES ABUNDANCE MODELS
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Hubbell’s abundance model
Ultimate diversity parameter θ
Carabids, Trap 5
Zipf–Mandelbrot usually the best
model (but Hubbell says Lognormal)
&
2
• Simulations can be used for estimating θ.
1
• θ and J define the abundance
distribution
5
10
20
• θ = 2JM ν, where JM is metacommunity size and ν evolution speed
Abundance
2
5
10
15
Rank
%
SPECIES ABUNDANCE MODELS
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High Diversity
10
5
2
1
1
2
5
Abundance
10
20
Barro Colorado Island, Site 1 & Hubbell
20
Barro Colorado Island, Site 1 & Brokensticks
Abundance
2
0
20
40
60
Rank
&
80
0
20
40
60
80
Rank
%
2
SPECIES ABUNDANCE MODELS
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Hubbell’s Species Generator
Species generator
θ
θ+j−1
gives the probability that jth collected individual belongs to a new
species
1. Collect J individuals, or j = 1 . . . J
2. Each individual is a new species at probability θ/(θ + j − 1): If
this happens, add the species to the community
3. In other case, add abundance 1 to some of the old species, with
probabilities proportional to abundances
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R code is crystal clear
h u b b e l l . b u i l d <− function ( t h e t a , J )
{
comm <− NULL
for ( j in 1 : J ) {
i f ( runif ( 1 ) < t h e t a / ( t h e t a+j −1))
comm <− c (comm , 1 )
else {
sp <− sample ( length (comm ) , 1 , prob=comm/ ( j −1))
comm [ sp ] <− comm [ sp ] + 1
}
}
return (comm)
}
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SPECIES ABUNDANCE MODELS
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Expected RAD from simulations
Simulated, J=64
Mean of 100 simulations
θ=∞
100
Abundance
1000
5.00
10000
50.00
J=100000
0.50
Abundance
0.05
10
θ = 100
θ=3
θ=1
θ = 10
1
0.01
2
θ = 0.1
0
5
10
15
Rank
&
20
25
30
θ = 2 θ = 5 θ = 10
0
100
θ = 50
θ = 20
200
300
400
500
Rank
%
SPECIES ABUNDANCE MODELS
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Supports Preston. . . or Fisher?
θ = 100
Simulation J=100000
θ = 20
80
60
5
10
Species
60
40
20
Species
15
80
20
θ = 100
6
8
10
12
2
4
6
Octave
8
10
12
14
Octave
θ=5
θ = 10
40
4
Species
2
20
5
4
Species
3
6
2
4
Species
8
6
10
7
All from simulation J=100000
2
4
6
8
Octave
10
12
14
0
1
2
2
2
4
6
8
Octave
10
12
14
0
500
1000
1500
2000
2500
3000
Abundance
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SPECIES ABUNDANCE MODELS
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Difference is in the eye of the beholder
The same data look like logarithmic series or log-Normal, depending
how you plot them
15
0
0
5
5
10
10
Species
20
25
15
30
35
Barro Colorado Island, All Sites
Species
2
0
500
1000
Frequency
&
1500
1
2
4
8
16
32
64
128
512
2048
Frequency
%
SPECIES ABUNDANCE MODELS
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The Hubbell RAD
Carabids
Average rad (line) vs. All traps (points)
100
50
1e+00
Abundance
500
1e+01
1e+02
100 Simulations, theta=4, J=100
Abundance
5
10
1e−01
Individual Communities
Mean
1
1e−02
2
5
10
15
Rank
20
0
10
20
30
40
Rank
Hubbell suggests averaging RADs, but biased because averages ranks
instead of species.
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SPECIES ABUNDANCE MODELS
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Typical RAD
• From preemption to Zipf–Mandelbrot, perhaps through Zipf with
increasing J, and faster with high θ.
• Lognormal rare: Often an averaging artefact.
Preemption
Lognormal
Veiled.LN
Zipf
32
128
theta=4
Mandelbrot
512
2048
theta=15
theta=40
40
Counts
2
30
20
10
0
32
128
512
2048
32
128
512
2048
J
&
%
SPECIES ABUNDANCE MODELS
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Diversity Indices
1nverse
10
8
6
4
2
m
C
Carabids
M
Fisher
F
IInverse
M
0
isher
arabids
alpha
Simpson
• It is best to see diversity indices
as variance estimator of abundances
• Fisher’s α is also a diversity index, and it is pretty close to
Simpson’s index
&
4
2
• All sensible indices are pretty
similar: it does not matter
which you use, unless you use
species richness
6
8
10
Carabids
Inverse Simpson
2
2
4
6
8
10
Fisher alpha
%
2
SPECIES ABUNDANCE MODELS
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Hill numbers
Common measures of diversity are special cases of R´enyi entropy;
S
X
1
Ha =
log
pai
1−a
i=1
Mark Hill proposed using Na = exp(Ha ) or the “Hill number”:
H0
=
H1
=
H2
=
log(S)
PS
− i=1 pi log p1
PS
− log i=1 p2i
H∞
=
− log(max pi )
N0
=
S
Number of species
N1
=
Shannon
N2
=
exp(H1 )
PS
1/ i=1 p2i
Simpson
N∞
=
1/ max pi
Berger–Parker
All Hill Numbers in the same units: Number of “virtual species”,
assuming all species are equal.
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Choice between indices
1.5
2.0
2.5
0.5
1.0
1.5
3.0
1.0
2.0
2.5
1.5
2.0
0
1.5
1
1.5
2.0
1.0
• It is not so important which index is used, since all sensible
indices are very similar.
2.5
• Diversity indices are only variances of species abundances.
0.5
1.5
1.0
• Sensitivy to rare species decreases with increasing scale of
R´enyi diversity.
1.0
2
Inf
0.5
2
1.5
2.0
2.5
3.0
0.5
1.0
1.5
2.0
Carabids
&
%
3
HARDCORE HUBBELL
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Hardcore Hubbell
In this section we have a more formal introduction into Hubbell’s
radical theory. This chapter is heavy. . . you were warned. . .
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HARDCORE HUBBELL
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Isolated local community: No immigrants
• Species have two absorbing states with no return:
1. Monodominance: all replacements from one species.
2. Extinction: no replacement – fate of all but one species.
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HARDCORE HUBBELL
47
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Transition matrix for µ = 1 and D = 1
M=
&

1
0
0
















1
J
J−2
J
1
J
..
.
..
.
..
.
···
0
···
..
.
(Jk )−2(J−2
k−1 )
(Jk )
..
.
(J−2
k−1 )
(Jk )
..
.
0
..
.
0

0
..
.
0
..
.
···
0
..
.
..
.
















0
0
···
..
.
..
.
(J−2
k−1 )
(Jk )
..
.
0
0
0
···
1
J
J−2
J
1
J
0
0
0
···
0
0
1
%
3
HARDCORE HUBBELL
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48
$
The analysis of the transition matrix
• Matrix M can be arranged into a canonical form:


I 0


M=
R Q
• I is the identity matrix for absorbing states, R gives the
probability of transitions to absorbing states, Q gives transitions
between non-absorbing states, and 0 is zero matrix.
&
%
3
HARDCORE HUBBELL
49
'
$
Ecological Random Drift
• The matrix of ecological random drift is A = (I − Q)−1
• Fixation time is T (N ) = A1T , and in our very special case
T (Ni ) =
"
#
N
J−1
i
X
X
−1
(J − 1) (J − Ni )
(J − k) + Ni
k −1
k=1
&
k=Ni +1
%
3
HARDCORE HUBBELL
50
'
$
Numerical example
A very small community (J = 8)
 1
0
0
0







&
0
0
0
0
0
0.125
0.75
0.125
0
0
0
0
0
0
0
0.214
0.571
0.214
0
0
0
0
0
0
0
0.286
0.429
0.286
0
0
0
0
0
0
0
0.268
0.464
0.268
0
0
0
0
0
0
0
0.268
0.429
0.268
0
0
0
0
0
0
0
0.214
0.571
0.214
0
0
0
0
0
0
0
0.125
0.750
0.125
0
0
0
0
0
0
0
0
1








%
3
HARDCORE HUBBELL
51
'
$
Small Community
• Row is the population size at time t, and column at time t + 1
• First row, first column: size 0.
• Time step chosen so that D = 1: one change per time step.
• Principal diagonal gives the probability that nothing changes, the
other elements give the probability of increase or decrease by one
individual.
J
• The binomial coefficient is k : how many ways do we have to
8
8
8
8
pick k individuals from J: 1 = 8, 2 = 28, 3 = 56, 4 = 70,
8
5 = 56 . . .
&
%
3
HARDCORE HUBBELL
52
'
$
Further numerical analysis
• Let’s take the submatrix Q (7 × 7) of previous (9 × 9) matrix.
• Matrix A = (I − Q)−1 ={ajk } : If species has abundance Nj how
many times it passes abundance Nk before absorptions.






A=



7
3.5
2.33
1.75
1.4
1.67
1
6
7
4.67
3.5
2.8
2.33
2
5
5.83
7
5.25
4.2
3.5
3
4
4.67
5.6
7
5.6
4.67
4
3
3.5
4.2
5.25
7
5.83
5
2
2.33
2.8
3.5
4.67
7
6
1
1.17
1.4
1.75
2.33
3.5
7








• The expected time to fixation T (N ) = A1T is
T = [ 18.2
&
28.3
33.8
35.5
33.8
28.3
18.2 ]
%
3
HARDCORE HUBBELL
53
'
$
Back to business (forget math)
• Time to extinction or to monodominance is dependent on
community size J and population size Ni of species i.
• For rare species (Ni J) time to fixation (extinction or
monodominance) is:
T (Ni ) ∼
= Ni (J − 1)[1 + ln(J)]
• Three multiplications for J: Potentially a large number.
• Based on special case D = 1: If mortality is higher, things can
develop faster.
&
%
3
HARDCORE HUBBELL
54
'
$
Open community
• New species can immigrate from the metacommunity.
• Monodominance no longer an absorbing state: “ergodic”
community.
• The transition matrix for ergodic community B gives the
probability that species remains at abundance Ni (principal
diagonal) or that its abundance changes with one individual
(subdiagonal elements).
&
%
3
HARDCORE HUBBELL
55
'
$
Descriptive Statistics
The first eigenvector ψ of the matrix B describes species dynamics:
E(Ni )
=
J
X
n · ψ(n)
n=1
Var(Ni )
=
j
X
(n − E{Ni })2 · ψ(n)
n=1
&
%
3
HARDCORE HUBBELL
56
'
$
Ergodic community
• The simplest possible ergodic community is of size J = 1:


1 − mPi
mPi


B=
m(1 − Pi ) 1 − m(1 − Pi )
where ψ = [1 − Pi , Pi ]
• “After six weeks of algebra, we find that the general solution
eigenvector is. . . ” (Hubbell, p. 88)
&
%
3
HARDCORE HUBBELL
57
'
$
. . . And the solution is
• Let Pi bet the expected probability of occurrence in the
metacommunity, so its expected abundance in the local
community will be:
E(Ni ) =
J
X
ψ(k) · k = JPi
k=0
or equal to the metacommunity, and independent of immigration
rate.
• Variance, though, is dependent on immigration rate and
community size.
&
%
3
HARDCORE HUBBELL
58
'
$
Many deaths
• Previously D = 1: one death per time step.
• If more deaths per time step, everything is much more
complicated, so we take a shortcut and go directly to conclusions
• Everything is faster, but similar.
• In closed communities, time to fixation inversely related to
mortality in large community, and in small communities even
stronger relation.
• In ergodic communities D does not influence exptected
population size E(Ni ) = JPi , but variance increases.
&
%
3
HARDCORE HUBBELL
'
59
$
Speciation and Metacommunity
• Metacommunity feeds new species to local community to replace
local extinctions.
• With time, species go extinct even in metacommunity.
• Speciation maintains diversity.
• Hubbell uses a simple, mechanistic model – in lack of theory – of
random point mutation
• Rate of speciation ν: Probablilty that offspring belongs to
different species (!) than her parent – we may guess this is very
small (e.g. ν ≈ 10−10 )
• The universal occurrence of log-Normal abundance model follows
from speciation in metacommunity.
&
%
3
HARDCORE HUBBELL
60
'
$
Stable abundances in metacommunity
• Let us inspect stable abundaces (when t → ∞) in a
metacommunity of size JM .
• What is the probability F2t+1 that two random individuals belong
to the same species in generation t + 1, when this probability was
F2t in the previous generation?
• Probability that neither speciated in birth is (1 − ν)2
• Probability that they have a common parent is
&
1
JM
%
3
HARDCORE HUBBELL
61
'
$
Balance of Speciation and Extinctions
• In the beginning of Times, all had one parent, and so we get a
recursive equation:
1
1
F2t+1 = (1 − ν)2
+ 1−
F2t
JM
JM
• In stable community nothing changes F2t+1 = F2t = F2 or
speciation balances extinctions
• An approximate solution is:
F2
=
∼
=
&
(1 − ν)2
JM − (1 − ν)2 (JM − 1)
1
1 + 2JM ν
%
3
HARDCORE HUBBELL
62
'
$
Hubbell’s Fundamental Diversity Number
• Hubbell’s fundamental number θ is
θ = 2JM ν
• This number is central in the rest of the Theory.
• JM is very large and ν is very small, but θ is moderate.
• JM is difficult to measure and ν may be impossible to measure,
but θ can be estimated.
&
%
4
HUBBELL’S GAME
'
4
63
$
Hubbell’s Game
Most useful results of Hubbell’s theory are best found with
simulations, or playing Hubbell Games.
&
%
4
HUBBELL’S GAME
'
64
$
Metacommunity and local community
• If species is very common in Metacommunity, it is dominant in
the local community as well. . . most of the time.
• Isolated local community: local abundance profile U-shaped:
(Temporary) dominance or (temporary) extinction.
• With increased migration, local abundance profile approaches
metacommunity probability
&
%
4
HUBBELL’S GAME
65
'
$
Make your own local communities
To estimate both θ and migration m, you must play many rounds of
the Hubbell game to estimate resulting abundance distribution and
its variance
• Explicit simulation of a local community with known size J
• Run several simulation timesteps with immigration from a
Metacommunity
• Look for abundance patterns similar to the observed community
1. Vary migration rate m
2. Vary Metacommunity θ
&
%
Hubbell’s Two Models
4
HUBBELL’S GAME
'
• Single Local Community in the Metacommunity Sea: θ defines
Metacommunity composition Pi and it is filtrated into Local
Parameters in Simulation
Communities as m allows:
66
$
m
Ji
de 1
Pi
θ = 2JM ν
• Metacommunity as a Network of Local Communities: Migration
•mJ:and
Sizespeciation
of local community
(fixed)
rates ν create
Local Communities, and
is theirper
sum:
•Metacommunity
D: Number of deaths
time step (fixed)
θ
• m: Probability
of
immigration
from
the
Metacommunity
Jki
Jki
ν= ! !
m
2 k i Jki
(random)
!
Jkiproportions
Jki in the Metacommunity
Pi = k Jki
• Pi : Species
(fixed) — these
may be simulated for a given Metacommunity size JM and
"
Ultimate Diversity θ
&
$
%
HUBBELL’S GAME
67
'
$
1
5
10
50 100
500
Start with any community
Abundance
4
0
10
20
30
40
Rank
&
%
HUBBELL’S GAME
68
'
$
Isolated Community
1
5
10
50 100
500
θ = 20, m = 0.005, J = 1600
Abundance
4
0
10 20 30 40 50 60
Rank
&
%
HUBBELL’S GAME
69
'
$
Open Community
1
5
10
50 100
500
θ = 20, m = 0.5, J = 1600
Abundance
4
0
10 20 30 40 50 60
Rank
&
%
HUBBELL’S GAME
70
'
$
Expected Abundance in Local Community
• Expected proportion in a local community = proportion in
Metacommunity
• Incidences vary, and variance higher in isolated communities (low
m).
0.0
0.1
0.2
0.3
Proportion in Metacommunity
&
0.4
0.8
0.6
0.4
0.0
0.2
Proportions in Local Communities
0.8
0.6
0.4
0.0
0.2
Proportions in Local Communities
0.0
0.2
0.4
0.6
0.8
Observed
Mean
1:1
1.0
m = 0.005
1.0
m = 0.05
1.0
m = 0.5
Proportions in Local Communities
4
0.0
0.1
0.2
0.3
Proportion in Metacommunity
0.4
0.0
0.1
0.2
0.3
0.4
Proportion in Metacommunity
%
HUBBELL’S GAME
71
'
$
• The level of variance, or the
height of the parabola, increases with decreasing immigration rate m.
&
10
m=0.005
m=0.05
m=0.5
5
• Metacommunity variance is a
parabolic function of P .
0
• Variance of local abundances related to metacommunity variances JPi (1 − Pi ).
15
Variance of Local Abundance
SD in Local Communities
4
0.0
0.2
0.4
0.6
0.8
1.0
Proportion in Metacommunity
%
HUBBELL’S GAME
72
'
$
Incidence Function
&
0.8
0.0
0.2
0.4
Incidence
0.6
0.8
0.6
0.4
0.2
0.0
10
15
20
25
30
0
5
10
15
20
Frequency
Frequency
m=0.005
m=0.005
25
30
25
30
0.6
0.4
0.0
0.2
0.4
Incidence
0.6
0.8
5
0.8
0
0.2
• In isolated communities, ecological random drift dominates, and
causes U–shaped incidence functions: Species either in all plots
or in none.
m=0.5
0.0
• Ecological Random Drift towards monodominance and extinctions.
m=0.5
Incidence
• The average incidence = proportion in Metacommunity (Pi ).
Incidence
4
0
5
10
15
20
Frequency
25
30
0
5
10
15
20
Frequency
Simulated, J=32
%
HUBBELL’S GAME
73
'
$
Isolation and RAD
m=1
m=0.005
0
20
Observed
theta=4
40
m=0.05
60
80
m=0.5
100
Observed
theta=15
Observed
theta=40
2^ 6
2^ 4
2^ 2
Abundance
4
2^ 0
Fitted
theta=4
Fitted
theta=15
Fitted
theta=40
2^ 6
2^ 4
2^ 2
2^ 0
0
20
40
60
80
100
0
20
40
60
80
100
Rank
J=400, D=100, timelag=500
&
%
5
SPECIES RICHNESS AND SAMPLE SIZE
'
5
74
$
Species Richness and Sample Size
Species richness increases with sample size, and Neutral Theory
predicts a specific relationship with number of individuals and
number of species. Moreover, we can try to estimate the joint effect
of isolation and Ultimate Diversity from Species—Area curves.
&
%
5
SPECIES RICHNESS AND SAMPLE SIZE
'
75
$
Species richness: The trouble begins
• Species 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.
• Sample size must be known in individuals: Equal area does not
imply equal number of individuals.
• Plants often difficult to count.
&
%
SPECIES RICHNESS AND SAMPLE SIZE
76
'
$
Rarefied Species Richness
• Results may not be clear:
Rarefaction curves can
cross.
10
8
2
4
• Diversity indices do about
the same.
6
• Rarefaction
compares
species richness in samples
rarefied to equal number
of individuals.
12
Carabids, Traps 12 and 63
Species Richness
5
0
50
100
150
200
250
300
Sample Size
&
%
SPECIES RICHNESS AND SAMPLE SIZE
77
'
$
Assume that plants are cubes. . .
• Sparse stand: Biomass increases when number of plants increases
• Crowded stand: Biomass increases when size of plants increases
and number decreases
5
10
15
20
25
Kokemäenjoki meadows
Species Richness
5
+
++
+
+
++
+
+ +
+
++ + +
+
++ + + + +
+
+ ++++ + ++
++ +++++ +++ ++++ +
++
++ +++++ +
+ + +++++++ + +
+
++++++ +++ + +++ ++
+++++++++++++++++ + +++ +
+
+++++++ ++ +++ + ++++ +
++++++ ++++++ +++ + + +++
+ +++++++ +++ ++ + ++++ + + ++ +
++ ++++++ + + +++ ++++++++
++++++ ++ +++ +++ + + +
+
++++++ +++ +++ + + ++ ++
++
+++++++
+
+
+ +++++
+
+
+
++
+ ++ +
++
+
+ +
0
500
+
+
+
+
+
+
++
1000
++
+
1500
Biomass g/m2
&
%
SPECIES RICHNESS AND SAMPLE SIZE
78
'
$
Species Richness and Area: Classical models
200
150
65.3Lakes0.3
100
• Doubling area brings along
a constant number of new
species: E(S) = c + log(A)
36.2 + 31.8log2(Lakes)
50
– Usually the only model
used, therefore regarded as
universally best
250
• Arrhenius model most popular:
E(S) = cAz , probably because
easy to fit
S
5
0
20
40
60
80
100
Lakes
&
%
SPECIES RICHNESS AND SAMPLE SIZE
79
'
$
Species Richness and Fisher
Carabids
Species never end, but the rate
of increase slows down.
5
10
Fisher log-series predicts:
N
S = α ln 1 +
α
15
20
α = 3.82
S
5
0
100
200
300
400
500
600
700
Number of individuals
&
%
5
SPECIES RICHNESS AND SAMPLE SIZE
80
'
$
Unified, Neutral prediction
JM
X
θ
E(S|θ, J) =
θ+j−1
j=1
JM − 1
≈ 1 + θ · log 1 +
θ
• Exact estimator and its variance estimator available
• Often applied on area basis: The shape similar, but θ can be
estimated only with counts
• Hubbell’s estimator very close to Fisher’s
JM
E(S|α, J) = α · log 1 +
α
indicating α ∼
= θ = 2JM ν
&
%
SPECIES RICHNESS AND SAMPLE SIZE
81
'
$
80
60
40
20
• Both indicate “infinite” species
richness: Fisher simply said that
rare species never end, Hubbell
said that new species evolve at
rate ν, since θ = 2JM ν.
0
• Fisher’s α is almost identical to
Hubbell’s θ: Approximate θ by
estimating α.
100
Hubbell’s theta and Fisher’s alpha
Estimated Fisher alpha
5
0
20
40
60
80
100
theta in simulation
&
%
SPECIES RICHNESS AND SAMPLE SIZE
82
'
$
Species Accumulation Curves
Carabids
• Simulation and exact method
possible, like in rarefaction.
30
20
10
• Related to rarefaction but,
number of individuals unknown. . .
0
• Related to Species–Area models, but area unknown. . .
40
• Number of species against
number of sampling units
Number of Species
5
0
10
20
30
40
50
60
Number of Traps
&
%
SPECIES RICHNESS AND SAMPLE SIZE
83
'
$
Hubbell and Classical Models
200
• In large data sets similar to Fisher, in small similar to Arrhenius,
says Hubbell
150
100
S
10
S
S
20
100
50
150
100
θ = 30
50
50
5
θ = 10
2
θ=3
0
2000
4000
6000
J
&
8000
10000
0
1
θ=1
0
5
1
10
100
J
1000
10000
1
10
100
1000
10000
J
%
5
SPECIES RICHNESS AND SAMPLE SIZE
'
84
$
Local Communities & Dispersal Limitation
• Basic model applies to the Metacommunity: Due to local
extinctions and isolation (m < 1), Species–Area curve flatter
• Hubbell claims (wrongly) that S–A curve asymptotically
E(S) = J z , and immigration depends on the size of local
community, so that a model for isolation would be m(J) = J −ω ,
where ω ≥ 0 is isolation
• Modified Species Generator for Dispersal Limitation:
J
X
θ · j −ω
E(S|θ, ω, J) =
θ+j−1
j=1
which cannot be solved in closed form
&
%
SPECIES RICHNESS AND SAMPLE SIZE
85
'
$
Fitting Hubbell’s curves
Myllylampi
50
40
30
θ = 105.7, ω = 0.26
20
• In principle, we can estimate isolation ω, but in
practice too strongly correlated with θ
θ = 18.6
10
• With Area instead of J,
shape OK, but θ wrong
60
• Non-linear regression to
find θ
S
5
0
100
200
300
400
J
&
%
SPECIES RICHNESS AND SAMPLE SIZE
86
'
$
All Models for Diatoms
250
• Hubbell bad, with θ = 33.8
200
• Hubbell
with
Dispersal
Limitation
good
(θ = 7.8), but non-sensical
ω = −0.20
• Log–Area excellent: 31.5
new species for doubling
sample size
150
100
• Arrhenius poor, with z =
0.30
Hubbell
ω
Arrhenius
log−A
50
S
5
0
10000
20000
30000
40000
N
&
%
SPECIES RICHNESS AND SAMPLE SIZE
87
'
$
Diversity and Isolation
• Diversity θ and Isolation ω correlated: Almost similar model
with high θ and high ω, or with
low θ and low ω.
Point Estimate
95% Joint Confidence Region
0.5
1.0
1.5
+
0.0
+
• The same problem with migration rate m and diversity θ earlier.
&
−1.0
−0.5
• Almost impossible to estimate
θ and ω simultaneously: One
should be known independently.
Carabids
omega
5
−100
−50
0
50
100
theta
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
6
88
$
Hubbell’s Second Game:
Metacommunity Landscape
Hubbell has two different games. In the first game, we studied a
single local community surrounded by a mythical Metacommunity.
The definition of Metacommunity was left open, though. In the
second game, Hubbell defines the Metacommunity as a network of
local communities.
&
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
89
$
Building Communities from scratch
Start with a Metacommunity made of a network of local communities,
where species can migrate to their neighbour community
1. Kill D of J plants in each local community
2. Fill vacated slots with immigrant offspring from neighbour
communities at probability m
3. Fill the remaining slots from within the local community using
proportions of remaining species as probabilities
4. Mutate some of the new plants to a new species
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
90
'
$
1
10
100
1000
10000
In the beginning, Earth is gray
Abundance
6
1.0
1.5
2.0
2.5
3.0
Rank
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
91
'
$
Cells of Evolution in Isolated Landscape
1
5
10
50 100
500
5000
θ = 14, JM = 7056, m = 0.005, t = 100
Abundance
6
0
5
10
15
20
25
Rank
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
92
'
$
Isolated Landscape
1
5
10
50
100
500
5000
θ = 14, JM = 7056, m = 0.005, t = 1000
Abundance
6
0
20
40
60
80
Rank
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
93
'
$
Open Landscape
1
5
10
50
100
500
5000
θ = 14, JM = 7056, m = 0.5, t = 1000
Abundance
6
0
5
10 15 20 25 30 35
Rank
&
%
List
of Slides
6 HUBBELL’S
SECOND GAME: METACOMMUNITY LANDSCAPE
!
'
!
Slide 1
Hubbell’s Two Models
Hubbell’s Two Models
Hubbell’s Two Models
943
#
$
#
• Single Local Community in the Metacommunity Sea: θ defines
• Single
Local Community
in the PMetacommunity
Sea: θ defines
Metacommunity
composition
i and it is filtrated into Local
• Single Local Community
in the Metacommunity Sea: θ defines
Metacommunity
Communities ascomposition
m allows: Pi and it is filtrated into Local
Metacommunity
Pi and it is filtrated into Local
Communities
as mcomposition
allows:
Communities as m allows:
m
Ji
Pi
θ = 2JM ν
m
Slide 1
Pi of Local
θ=
2JM ν
• MetacommunityJas
Communities:
Migration
i a Network
• Metacommunity
as arates
Network
of Local
Communities:
Migration
and speciation
ν create
Local Communities:
Communities,Migration
and
• m
Metacommunity
as a Network
of Local
m Metacommunity
and speciation rates
ν create
Communities, and
their
sum: Local
m and speciation is
rates
ν create
Local Communities, and
Metacommunity is their sum:
Metacommunity is their sum:
θ
Jki
Jki
ν= ! !
m
2 θk i Jki
Jki
Jki
ν= ! !
m
2 !k i Jki
Jki
Jki
Pi = k Jki
!
Jki
Jki
Pi = k Jki
"
$
"
$
&
%
de 1
6
m
HUBBELL’S SECOND GAME:JMETACOMMUNITY
LANDSCAPE
Pi
θ = 2JM ν
i
'
• Metacommunity as a Network of Local Communities: Migration
m and speciation rates ν Parameters
create Local Communities, and
Metacommunity is their sum:
Jki
"
Jki
m
Jki
Jki
ν=
2
Pi =
θ
! !
!
k
k
i
95
$
Jki
Jki
• k: Number of local communities (fixed)
$
• Jk : Size of each local community (fixed)
• D: Number of deaths in each local community (fixed)
• m: Migration probability among neighbouring local communities
(random)
P
• ν: Speciation probability ν = θ/(2JM ), JM = k Jk
&
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
96
$
Is World big enough for Hubbell?
• Speciation rate ν can be found from θ and JM :
θ
ν=
2JM
• In Hubbell’s metacommunity simulation θ = 10, JM = 7056 or
ν = 10/2/7056 = 0.0007: one child of 1411 belongs to different
species than her parent.
• If JM is unknown, but we know the metacommunity area AM ,
we can get the density ρ needed for a realistic speciation rate, say
ν = 10−10
• In Panama: θ = 50, AM = 4.3 · 105 ha:
ρ = 2AθM ν = 2×4.3·10505 ×10−10 = 5.8 · 105 individuals per hectare
(obvserved ca. 400 trees/ha).
&
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
97
$
Diversity and Metacommunity Landscape
• Isolated communities (low m)
– Local extinctions: Monodominance usual
– Local communities unique: Rare species locally dominant
– Local diversity low, regional high
• Open communities (high m)
– Local communities like miniature of Metacommunity
– Local communities similar ot each other
– Local diversity high, regional low
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HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
98
'
$
• In multi-indivdiual community
persistence makes sense: highest
when m = 0 or m = 1.
80
60
40
20
• Counterintuitively
immigration
decreases persistence.
P=0.9
P=0.5
P=0.01
0
• When J = 1, persistence to
expected time to extinction is
1
TNi =1→Ni =0 = m(1−P
i)
100
Persistence and Immigration
Time to extinction T
6
0.0
0.2
0.4
0.6
0.8
1.0
Immigration rate m
&
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
99
$
Alpha and Beta Diversity
• α–diversity is the local species richess (or Arrhenius c)
• β–diversity is the change is species composition when exiting the
local community (or Arrhenius z)
• High migration m increases α but decreases β
• Fundamental θ increases both α and β
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
100
'
$
0
y
−3
−1
CCA2
1
1
0
−1
−3
−2
0
DCA1
&
x
2
2
3
3
Ordination finds order in a Neutral Universe
DCA2
6
2
4
−4
−2
0
2
CCA1
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
101
$
Putting things in even better order
. . . Whether they want or not
> anova(hubb.cca)
Permutation test for CCA under reduced model
Model: cca(X = hubb, Y = hubb.coord)
Df
Chisq
F N.Perm Pr(>F)
Model
2 0.6220 0.6845
100 < 0.01 ***
Residual 46 20.8995
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
&
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HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
102
$
Three-Phase Species–Area Curve
1. Local Scale. Local community abundance relations
2. Regional Scale: Speciation and dispersal rates
3. Continental Scale: Exceeds metacommunity limits
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HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
103
$
Metacommunity Diameter
• Rate of increase in species number slows down with increasing
area.
• After crossing the metacommunity border, number of new species
increases.
• The limit is the correlation length: L intersection of tangents of
species – area curve.
• Varies among plant groups and areas, in trees of Panama
d = 30 . . . 100 km
&
%
6
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
104
$
Correlation Length
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
105
'
$
Diameter of Lichen-rich Pine Forests
60
80
100
Lichen−rich pine forests
+
40
Species richness
6
100
200
300
400
600
Circle diameter (km)
&
%
HUBBELL’S SECOND GAME: METACOMMUNITY LANDSCAPE
'
106
$
Community Similarity and Distance
• The lower is m, the faster species composition changes with
distance.
• The smaller the local communities, the faster the change in
species composition.
0.2
0.3
0.4 0.5
0.7
Lichen−rich pine forests
Jacccard index
6
+
+++
++ +
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+
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++ ++ +++++ +++++ +
++
+
+++++++++
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+++++
+++++++++++++
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++ ++
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+ ++ +++++++++++ +++++ +
++ +++++++++++++ ++
++ +
+++++++++++
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+++++++
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+ + + ++++
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+
++++ ++++++
++ ++++++
+
++++++++++
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+ ++
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+++++++++ +
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++ ++ ++++++++++++
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+ ++ ++
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++++++ ++ + ++ +
+++++++ +
+++++
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+ ++ ++++ +
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+ ++
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++ +
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+ +++++++++ +++++ +++
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++
+ +++
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++++ +++ + +++ ++++++ + +++
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+ +
+ ++++ + + +++
++ ++++ ++
++
+
+ +
++
+++ ++++ +++++ ++
+ + +++
+ +++ ++
+++++ ++ ++ ++ ++ +++ ++++++++
+++++ + ++++ ++++
+
+ +
+
++ +
+ +
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+ ++
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++++++ + +
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+ ++
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+
+
++++ + + + + +++ +++ ++ +
+
++
+ + +++ ++
+ ++++
+++
+
+
++
++++++ + +
+ +
+ + + ++ + + ++++ + + +
+++
+
+
+
+++ ++ + + ++++++ + + +++ + + ++ +
++ + + +
+
+
+
+
+
+
++
+
+++
+ ++
++ +
+
+
++++ ++ +
++ +
+
+
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+
+
+
+
++ ++ + + + +
+ + ++ +
+
+
+ +
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++
+ ++
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+ ++
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+
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+
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+ + +++
++
+
+
0
200
400
600
800
1000
Distance (km)
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
'
7
107
$
Species Pool: What is a
Metacommunity?
Metacommunity can be seen as the species pool feeding species to
local communities. Neutral Model predicts a certain relationship
between pool size and local richness — and that relation is similar to
“niche limitation” model, although niches are not needed in Neutral
Theory. . .
&
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SPECIES POOL: WHAT IS A METACOMMUNITY?
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108
$
Species pool and local species richness
• The local richness is determined by the available species pool
1. Actual species pool: The species really occurring in the
community.
2. Local species pool: The species that can live in the community
and occur in the landscape.
3. Regional species pool: The species that can live in the
community and occur in the geographic region.
• Habitat is a filter that selects species from the species pool.
• There is no local control of richness, but the actual species
richness follows from the pool size.
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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$
Filtering species from the pool
Speciation
migration
Regional
species
Ecological filter
Isolation
Regional
pool
Local
species
Ecological filter
Isolation
Local
pool
Community
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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110
$
Null Model
• Pa¨rtel – Zobel – Zobel – van der Maarel (PZZM) hypothesis:
Community richness what ever between 0 and (local) species
pool: Testing concerns the variance of species richness.
• Pool limitation model: Constant proportion of species in the pool
occur in the community: Tested by linear regression.
• Niche-limitation model: Pool limitation at low richness, but niche
limitation at high richness (“competition”), and this leads into
saturating curve: Tested (approximately) by quadratic regression.
&
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
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$
Alternative Null Models
0
50
100
150
200
Species Pool
&
250
300
250
200
150
Species Richness
0
50
100
250
200
150
100
50
0
0
50
100
150
Species Richness
200
250
300
Niche Limitation
300
Pool Limitation
300
PZZM
Species Richness
7
0
50
100
150
200
Species Pool
250
300
0
50
100
150
200
250
300
Species Pool
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
112
'
$
Species richness in Hubbell communities
• Theoretical form:
Expected
species richness in the Metacommunity of size JM against local
community of size J
&
40
30
20
• Indistinguishable from the “pool
limitation”.
JM = 64000, J = 64
Theoretical
Fitted GAM
Linear fit
10
• Species richness in a local community is dependent on the
species richness in the Metacommunity: Species pool = Metacommunity.
Local Richness
7
0
50
100
150
200
250
300
350
Pool Size
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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113
$
Does species richness exist?
• PZZM and pool limitation: Species richness is an incident and
actually does not exist: There are as many species as there can
live, but incidences vary.
• Niche-limitation: Species richness is controlled and so strictly
exists:
• Removing a species gives space for immigration.
• Adding a species causes local extinctions.
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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$
Which Species make up the Pool?
• All species that could live in a site and could get there belong to
the (regional) species pool.
• The Estonian strategy:
1. Assess the “ecological character” of a site using Ellenberg
indicator values for all species in the site
2. Find the species that are “close” to the assessed character of
the site
3. Their number is the size of the species pool
&
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
115
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$
Generating Communities from Gradients
• Assume that gradients alone
determine the species composition
&
30
40
50
60
70
+
+
+
+
+
+
+ + +++ + + +++ +
+
+
+ ++ + +++++ +++ + + +
+ + + ++
++
+
+
+
+
+
−−
−−
−− −− −− −−
−
−
−−
−−
20
• Interaction: Gradient information alone insufficient, but
we must know which other
species already are in the community.
10
• Adequate if gradient model is
appropriate and there are no
interactions between species.
pH filter in Diatom species pool
Species Richness
7
−−
4.5
−−
5.0
5.5
6.0
6.5
pH
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
'
116
$
Biological Interactions. . . and Statistical
• The variance of sum of independent random variates is the sum
of their variances:
Var(X + Y ) = Var(X) + Var(Y )
• If there are interactions, then
Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y )
• Covariance is a compund of correlation ρ and standard deviations
σ:
Cov(X, Y ) = ρX,Y σX σY
• Positive interactions: Sum has larger variance.
• Negative interactions (“competition”): Sum has lower variance.
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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$
Zobel’s quasineutral model
• Filtering from regional pool to the actual (local) pool and
filtering from actual pool to local community is a neutral and
stochastic process.
– Inter-specific competition typically asymmetric: Has a species
ever replaced another of similar size?
– Local community richness defined for number of shoots:
Constant proportion from the pool.
• Species are replaced from the community by light competition
during succession: Gives a regular change in species richness ven
per shoot.
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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118
$
Implications from the quasineutral model
• Local communities are open and immigration is important
• Succession is not a regular change in species compositon but
change in growth form
• Spatial scale is important.
• Similar species co-exist and do not out-compete each other.
• Species richness must be evaluated per shoot number, not per
area.
&
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
'
119
$
Species richness and biomass: Neutral model
• Low biomass because plants are small and sparse.
• When the number of shoots increases, species richness grows
lilnearly: N ∝ B
• In closed vegetation, biomass can increase only if the number of
shoots decreases because plant size incerases: Self thinning model
predicts N ∝ B −2
• Species richness follows from shoot number, either from Fisher or
Hubbell model.
&
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
120
'
$
Richness productivity hump
5
10
15
20
25
Kokemäenjoki meadows
Species Richness
7
+
++
+
+
++
+
+ +
+
++ + +
+
++ + + + +
+
+ ++++ + ++
++ +++++ +++ ++++ +
++
++ +++++ +
+ + +++++++ + +
+
++++++ +++ + +++ ++
+++++++++++++++++ + +++ +
+
+++++++ ++ +++ + ++++ +
++++++ ++++++ +++ + + +++
+ +++++++ +++ ++ + ++++ + + ++ +
++ ++++++ + + +++ ++++++++
++++++ ++ +++ +++ + + +
+
++++++ +++ +++ + + ++ ++
++
+++++++
+
+
+ +++++
+
+
+
++
+ ++ +
++
+
+ +
0
500
+
+
+
+
+
+
++
1000
++
+
1500
Biomass g/m2
&
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
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$
Plant Size, Pool Size and Plot Richness
• Grazing, e.g., can reduce
plant size and/or change
the spatial pattern.
2
3
4
5
15
20
25
30
1
10
• Plot level richness may increase although grazing introduces no new species to
the site
Given : grazing
Species Richness
7
1
• Local species pool may remain unchanged.
&
2
3
4
5
Year
%
7
SPECIES POOL: WHAT IS A METACOMMUNITY?
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$
Extrapolated Species Richness: Unseen Species
• Jackknife: There are almost as many unseen species as there
were species seen only once
N −1
SP = SO + a1
N
where SP is the pool size, SO the number of species seen, a1
number of species seen only once, and N the number of sites.
• Bootstrap: Number of unseen species equals number of species
not seen if sample resampled with replacement
SP = SO +
N
X
(1 − pi )N
i=1
• Other ideas: Chao SP = S0 +
&
a1
2a2
, 2◦ Jackknife, . . .
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
123
'
$
Unseen Species in Open Communities
• Assumption:: There is a fixed
number of all species — or a
species pool
&
260
240
220
200
180
• In open communities, all methods
tend to add a constant number of
“unseen” species to the observed
species.
Observed
Chao
Jackknife 1
Jackknife 2
Bootstrap
160
• Open communities: Accumulation
of species never ends, but new rare
species are found as sampling proceeds
Barro Colorado Island
Number of species
7
10
20
30
40
50
Number of plots
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
124
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$
Rarefaction: Biased Variance
Barro Colorado Island
Site 3
60
40
Rarefaction
Its 95% CI
Hubbell accumulator
Its 95% CI
• Number of unseen species: Unseen to variance, too.
0
20
• Current sample assumed errorfree: All error should come
from the resampling
80
• Rarefaction: Sampling without
replacement from the current
sample
Species
7
0
100
200
300
400
Trees
&
%
SPECIES POOL: WHAT IS A METACOMMUNITY?
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Species Accumulation Curves
• Similar to rarefaction, but accumulates sites instead of individuals.
&
200
+
++
++++
+
++++
100
150
+
0
• Variance biased.
++ +
50
• Again assumed that the complete collection of sites is errorfree.
+ +
0
• Randomized resampling without replacement or “exact” accumulator
Barro Colorado Island
exact
7
10
20
30
40
50
Sites
%