John Harte, UC BerkeleyINTECOLLondon

August 20, 2013

Maximum Entropy and Mechanism: Prospects

for a Happy Marriage

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8

ln(N/S(N))

z(A

)

MaxEnt Prediction

Data from over 50 sites

MaxEnt Approach to Macroecology To predict patterns in:

• abundance• distribution• energetics• network structure

• across taxonomic groups • across spatial scales

• across habitat categories • without adjustable parameters, • without arbitrary choice of governing mechanisms• and thereby to reach insight into mechanism.

Here “entropy” refers to information entropy,

not thermodynamic entropy.

Information entropy is a measure of the lack of structure or detail in the probability distribution describing your knowledge of a system.

P(x) P(x)

x x

Lower EntropyHigher Entropy

Maximum Entropy? Just what is being

maximized?

Ingredients of a Fundamental Theory of Macroecology

PREDICTIONS (Metrics of Ecology) Species-Area Relationships Endemics-Area Relationships Abundance & Body Size

Distributions Spatial Aggregation Patterns Web Structure & Dynamics Species Distribution across

Genera, Families, etc.

INPUT DATAState Variables:

S N E

THEORY MaxEnt: An inference procedure based on information theory

APPLICATIONS♣ Species Loss under Habitat Loss ♣ Reserve design

♣ Web Collapse under Deletions

♣ Scaling up Biodiversity

A Candidate Macroecological Theory: The Maximum Entropy Theory of Ecology

(METE)

Examples of Validated Predictions

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

MaxEnt Prediction

Data from over 50 sites

z(A

)z = 1/4

Harte et al., Ecology Letters, 2010; Harte, Oxford U. Press, 2011

MaxEnt predicts: all species-area

curves collapse onto a universal curve

log(N(A)/S(A))

2 3 4 5 6 72

3

4

5

6

7

f(x) = 0.984369380015343 x + 0.0357247800108356R² = 0.998980447497264

# species in each order with n < 10Averaged over plots

ln(observed)

ln(p

redi

cted

)

MaxEnt predicts: the fraction of

species that are rare

S, N, E

Resource constraints:

Evolutionary constraints: taxonomy/ phylogeny

Water, Phosphorus,..

Order, Family, Genus

At the Frontier of METE

Core theory

Linkages

Trophic interaction constraints:

Original Theory

Alters size-abundance distribution

Alters predicted rarity

Extending and Generalizing METE

If (S,N,E) (F,S,N,E), then the energy-abundance relationship is

modified:

Including higher taxonomic levels as constraints

m labels the species richness of the family (or order, …) that the species with abundance n is in.

Log(metabolic rate)

Log(abundance) Families of

differing species richness

(F = family or other higher order category)

The Damuth rule splits apart!

r - 1 = # additional resources

Including additional resource constraints (in addition to energy, E)

The log-seriesSAD becomes:

The inclusion of additional resource constraints predicts increased rarity

Kempton and Taylor (1974)

Abundance Distribution of Rothampsted MothsRelatively undisturbed fields:

Fisher log series distribution (predicted by METE)Fields recently left to fallow and in transition: Lognormal distribution

0

1

2

3

4

5

6

7

0 50 100 150 200 250

rank

log(

abun

danc

e)

predicted

observed

0

1

2

3

4

5

6

0 50 100 150 200

rank

log(

abun

danc

e) predicted

observed

test of abundance distribution

150 y

Test of abundance distributionArthropod abundance

distributions from Hawaiian sites of

different ages and stages of speciation

4 My

Data from Dan Gruner

The theory fails to predict patterns in ecosystems undergoing relatively rapid change

Similar pattern of success and failure for body size distributions!

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

ln(N/S)

slop

e

theory: anchor N/S = 10theory: anchor N/S = 2wet tropical foresttemperate understoryserpentine grasslandS. Af. birdsdry tropical forest

X X

Species-area slopes for plants in successional sites (aftermath of an erosion event) lie well above the scatter around the universal curve

1. 2.

3.

SUMMARY:

• METE is a relatively successful theory of macroecology.

• Success does not imply mechanism does not matter! Mechanisms are incorporated into the values of the state variables, and we still need to understand what they are.

• Failure of the core theory tells us that more mechanistic information than is captured by the state variables is needed to predict patterns in ecology.

• Testing various extensions of the theory allow us to identify the role of particular mechanisms.

Thanks: To my Collaborators: Erin Conlisk Adam Smith Xiao Xiao Mark Wilber Justin Kitzes Andrew Rominger Ethan White Chloe Lewis Erica Newman David Storch Tommaso Zillio Xiao Xiao

To Other Sources of Data: J. Green R. Krishnamani J. Godinez W. Kunin R. Condit P. Harnik K. Cherukumilla E. White D. Gruner J. Goddard STRI D. Bartholomew

To the Funders: NSF, Miller Foundation,Gordon and Betty Moore Foundation

To my Hosts during the development of METE: Santa Fe Institute, Rocky Mountain Biological Laboratory, NCEAS, The Chilean Ecological Society, Charles University, University of Padua

Deviations from the MaxEnt theory

x x x x

Measure of rapidity of change

Hypothesis:

But the pattern of deviation of abundance distributions from the predicted Fisher log series depends on whether the system is collapsing or diversifying.

This is just the first step in relating the mechanisms that disrupt an ecosystem to patterns predicted by macroecological theory.