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.