Transcript Species-Abundance Distribution

Species-Abundance Distribution:
Neutral regularity or idiosyncratic stochasticity?
Fangliang He
Department of Renewable Resources
University of Alberta
Law
Order
Species-Abundance Relationships
Sp1
1
Sp2
1
Sp3
1
Sp4
2
Sp5
2
Sp6
5
Sp7
6
Sp8
6
Sp9
10
Sp10
50
Sp11
500
……
Number of species
Species abundance
Abundance
Why species cannot be equally abundant?
Logseries distribution
xn
f ( n)  
n
where
  Sp0 / b
n  1, 2, ...
(the biodiversity parameter, Fisher’s )
x  b / d 1
n
Lognormal distribution
1
f ( x) 
e
2 x
1  ln(x )   
 

2 

2
x0
x
Neutral
Niche
d
Species
d
w
x
x
Ecological equivalence.
Individuals are identical
in vital rates.
Coexistence is
determined by drift
Idiosyncrasy
Each species is unique in
its ability to utilize and
compete for limiting
x
Any factor can contribute
to population dynamics.
resources and follows a
defined pattern.
Each species is unique
and follows no defined
patterns.
Niche differentiation is
prerequisite for coexist.
Coexist. is determined by
multiple factors
Logseries Distribution
Derived From Neutral Theory
xn
f ( n)  
n
where
  Sp0 / b
(the biodiversity parameter)
x  b / d 1
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:
N
n3
H  k log W
n1
n2
Number of species
Linking microscopic world to macroscopic worlds
Abundance
H: macroscopic quantity
W: microscopic degrees of freedom (multiplicity)
Entropy: the Probability Perspective
H   pi log pi
N
p3
p1
p2
Entropy measures the
degree of uncertainty.
Tossing a Coin
H  k log W
H   pi log pi
N!
W
n1!n2 !
p1  p2  0.5
A fair coin has the maximum degrees of freedom (largest
W), thus max entropy.
The 2nd Law of Thermodynamics: Systems tend toward disorder
N
n1
n2
H   pi log pi
f(x)
n3
N
x
n3
n1
n2
f(x)
Maximum
N
n3
n1
n2
x
Two Opposite Forces
The 2nd Law
Constraints
Without any prior knowledge,
the flattest distribution is most
plausible. This is the 2nd law of
thermodynamics.
n3
f(x)
n1
n2
x
Predicting Dice Outcome Using MaxEnt
i =1, 2, …, 6
pi  ?
i  3.5
The Boltzmann Distribution Law
Scores:
x1 , x2 , ... , x6
Probabilities:
p1 , p2 , ... , p6
 pi  1

 x   xi pi
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}:
N!
W
n1!n2 !... n6 !
Stirling’s approximation:
6
log W
H
  pi log( pi )
N
i 1
log( n!)  n log( n)  n
The Boltzmann Distribution Law
6
H   pi log( pi )
i 1
 pi  1

 x   xi pi
Objective function using Lagrange multipliers:
Math constraints
6
6




H   pi log( pi )   0 1   pi     x   xi pi 
i 1
i 1
 i 1 


6
Entropy
Constraints
The Boltzmann Distribution Law
Math constraints
6
6




H   pi log( pi )   0 1   pi     x   xi pi 
i 1
i 1
 i 1 


6
Entropy
Constraints
pi 
e
6
xi
e
i 1
xi
The Boltzmann Distribution Law
pi 
e xi
6
e
xi
i 1
x  3.5,   0
x  2.5,   0.371
x  4.5,   0.371
Shipley et al’s work
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
Community-aggregated traits:
t j ( x)   tij pik ( x)
i 1
trait j
time x
site k
sp i
Probability constraint:
 pi  1
Entropy (degrees of freedom):
H   pi  pi
Shipley, Vile & Garnier. 2006. From plant traits to plant community: A
statistical mechanistic approach to biodiversity. Science 314:812-814.
S
Community-aggregated traits:
t j ( x)   tij pik ( x)
i 1
Probability constraint:
Entropy (degrees of freedom):
 pi  1
H   pi log( pi )
Objective function using Lagrange multipliers:
S


H   pi log( pi )  0 (1   pi )    j  t j   tij pi 
j 1 
i 1

T
The predicted abundance:
T


exp  0    j tij 
j 1


pˆ i 
S
T


 exp  0    j tij 
i 1
j 1


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
T


exp  0    j tij 
j 1


ˆpi 
S
T


 exp  0    j tij 
i 1
j 1


The Idiosyncratic Theory
N individuals belong to S species
n11, n12 , ..., n1S
n12 , n22 , ..., nS2
x
.
.
.
n1i , n2i , ..., nSi
.
.
.
Relative Entropy
The total # of ways that N can be partitioned into a particular
set of S species:
N!
nS
n1
W 
p1 ... pS
n1! n2 !... nS !
H   P(n) log P(n)    P(n) log P0 (n)
Prior
The two most basic constraints
 P ( n )  1


 ns  N
 n
H   P(n) log P(n)    P(n) log P0 (n)
Maximize H subject to constraints:
 P ( n )  1

 nsn  N   nP (n)  n
P(n)  P0 (n)e
11 2n
P(n)  P0 (n)e112n
Logseries Distribution
Prior
Geometric distribution as prior:
P0 (n) 
1 n
P(n)  n e

n
1.
Species-abundance is
invariant at different scales.
2.
log(n) is uniform distribution.
xn
 n  
n
Pueyo, He & Zillio. The maximum entropy formalism and the
idiosyncratic theory of biodiversity. Ecol. Lett. (in press).
Lognormal distribution
H   P(n) log P(n)    P(n) log P0 (n)
Maximize H subject to constraints:
 P ( n )  1

 log nP (n)  log n

 log n 2 P (n)  log n 2
P(n)  n 1e

(log n   ) 2
2
2
n
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.