species-abundance distribution: neutral regularity or idiosyncratic stochasticity? fangliang he...
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Species-Abundance Distribution:
Neutral regularity or idiosyncratic stochasticity?
Fangliang He
Department of Renewable Resources
University of Alberta
Law Order
Species-Abundance Relationships
Nu
mb
er o
f sp
ecie
s
Abundance
Species abundance
Sp1 1
Sp2 1
Sp3 1
Sp4 2
Sp5 2
Sp6 5
Sp7 6
Sp8 6
Sp9 10
Sp10 50
Sp11 500
……
Why species cannot be equally abundant?
Logseries distribution
bSp /0
1/ dbx
where
n
xnf
n
)( ...,2,1n
(the biodiversity parameter, Fisher’s )
Lognormal distribution
2)ln(
2
1
2
1)(
x
ex
xf 0x
n
x
Neutral Niche
w
d d
x
Idiosyncrasy
xx
Species
Ecological equivalence.
Individuals are identical
in vital rates.
Coexistence is
determined by drift
Each species is unique in
its ability to utilize and
compete for limiting
resources and follows a
defined pattern.
Niche differentiation is
prerequisite for coexist.
Any factor can contribute
to population dynamics.
Each species is unique
and follows no defined
patterns.
Coexist. is determined by
multiple factors
Logseries Distribution Derived From Neutral Theory
(the biodiversity parameter)bSp /0
1/ dbx
where
n
xnf
n
)(
Volkov, Banavar, Hubbell & Maritan. 2003. Neutral theory and relative species abundance in ecology. Nature 424:1035-1037.
Maximum Entropy
• Predict species abundance from life-history traits
• Derive logseries distribution
Entropy:
Linking microscopic world to macroscopic worlds
n1
n3
n2
N
Nu
mb
er o
f sp
ecie
s
Abundance
WkH log
H: macroscopic quantity
W: microscopic degrees of freedom (multiplicity)
Entropy: the Probability Perspective
ii ppH log
Entropy measures the
degree of uncertainty.p1
p3
p2
N
Tossing a Coin
WkH log
!!
!
21 nn
NW
ii ppH log
5.021 pp
A fair coin has the maximum degrees of freedom (largest W), thus max entropy.
The 2nd Law of Thermodynamics: Systems tend toward disorder
n1n3
n2
N
n1
n3
n2
N
n1
n3
n2
N
f(x)
x
f(x)
x
ii ppH log
Maximum
n1
n3
n2
The 2nd Law Constraints
Without any prior knowledge,
the flattest distribution is most
plausible. This is the 2nd law of
thermodynamics.
f(x)
x
Two Opposite Forces
i =1, 2, …, 6
?ip
5.3i
Predicting Dice Outcome Using MaxEnt
The Boltzmann Distribution Law
621 ,...,, ppp621 ,...,, xxx
Probabilities:
Scores:
ii
i
pxx
p 1
!!...!
!
621 nnn
NW
The total # of ways that N can be partitioned into a particular set of
{n1, n2, …, n6}, e.g., {2, 3, 1, 4, 0, 2}:
6
1)log(
log
iii pp
N
WH
nnnn )log()!log(Stirling’s approximation:
The Boltzmann Distribution Law
ii
i
pxx
p 1
6
1)log(
iii ppH
6
1
6
10
6
11)log(
iii
ii
iii pxxpppH
Entropy Constraints
Math constraints
Objective function using Lagrange multipliers:
The Boltzmann Distribution Law
6
1
6
10
6
11)log(
iii
ii
iii pxxpppH
Entropy Constraints
Math constraints
6
1i
x
x
ii
i
e
ep
The Boltzmann Distribution Law
6
1i
x
x
ii
i
e
ep
371.0,5.2 x 371.0,5.4 x0,5.3 x
Shipley et al’s work
Shipley, Vile & Garnier. 2006. From plant traits to plant community: A statistical mechanistic approach to biodiversity. Science 314:812-814.
Use 8 life-history traits to predict abundance for 30 herbaceous species in 12 sites
along a 42-yr chronosequence in a vineyard in France.
S
iikijj xptxt
1)()(Community-aggregated traits:
Probability constraint: 1 ip
trait jsp i
site ktime x
Entropy (degrees of freedom): ii ppH
S
iikijj xptxt
1)()(Community-aggregated traits:
Probability constraint: 1 ip
Entropy (degrees of freedom): )log( ii ppH
T
j
S
iiijjjiii pttpppH
1 10 )1()log(
Objective function using Lagrange multipliers:
S
i
T
jijj
T
jijj
i
t
t
p
1 10
10
exp
exp
ˆ
The predicted abundance:
Criticisms
• Circular argument
• Entropy is not important
• Random allocation of traits to
species would also predict
abundance
• Species abundance does not
follow exponential distribution
Roxhurgh & Mokany. 2007. Science 316:1425b.Marks & Muller-Landau. 2007. Science 316:1425c.
The Boltzmann Law = Logistic Regression
S
i
T
jijj
T
jijj
i
t
t
p
1 10
10
exp
exp
ˆ
The Idiosyncratic Theory
N individuals belong to S species
112
11 ...,,, Snnn
222
21 ...,,, Snnn
iS
ii nnn ...,,, 21
.
.
.
.
.
.
x
SnS
n
S
ppnnn
NW ...
!!...!
!1
121
The total # of ways that N can be partitioned into a particular
set of S species:
)(log)()(log)( 0 nPnPnPnPH
Relative Entropy
Prior
The two most basic constraints
Nns
nP
n
1)(
)(log)()(log)( 0 nPnPnPnPH
nnnPNns
nP
n )(
1)(Maximize H subject to constraints:
nenPnP 2110 )()(
nnP
)(0
nennP 1)(
Pueyo, He & Zillio. The maximum entropy formalism and the idiosyncratic theory of biodiversity. Ecol. Lett. (in press).
nenPnP 2110 )()(
Prior
Geometric distribution as prior:
1. Species-abundance is
invariant at different scales.
2. log(n) is uniform distribution.
Logseries Distribution
n
xn
n
Lognormal distribution
22 log)(log
log)(log
1)(
nnPn
nnnP
nP
Maximize H subject to constraints:
nn
ennP
2
2
2
)(log1)(
)(log)()(log)( 0 nPnPnPnPH
Conclusions
1. Ecological systems are structured by two opposite forces. One is the Second
Law of thermodynamics which drives the systems toward disorder (maximum
degrees of freedom). The other is constraints that maintain order by reducing
the degrees of freedom.
2. The Boltzmann Law provides a tool to model abundance in terms of traits.
The Law is equivalent to logistic regression.
3. Logseries and lognormal distributions are the emerging patterns generated
by the balance.
4. Logseries arises from complete noise in idiosyncratic theory, but from strict
regularity (identical demographics) in neutral theory. It therefore does not
contain information about community assembly. The MaxEnt shows that the
neutral theory is just one of a large number of plausible models that lead to
the same patterns of diversity.
5. Many biodiversity patterns (Pareto, lognormal) can be readily explained by
the idiosyncratic theory.
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