Macroecology …characterizing and explaining patterns of abundance, distribution, and diversity

The Feasible Set: A New Understanding of Constraints

on Ecological Patterns of Abundance

CHAPTER 1: How species richness and total abundance constrain the

distribution of abundance

CHAPTER 2: Efficient algorithms for sampling feasible sets

Rank-abundance curve (RAC)

Rank in abundance

Abun

danc

e

Frequency distribution

Species abundance distribution (SAD)

Abundance class

freq

uenc

y

Frequency distribution

The ubiquitous hollow-curve

Abundance class

freq

uenc

y

Rank-abundance curve (RAC)

Rank in abundance

Abun

danc

e104

103

102

101

100

Rank in abundance

Abun

danc

e104

103

102

101

100

Predicting the SAD

Observed

Predicted

Rank in abundance

Abun

danc

e104

103

102

101

100

N = 1,700S = 17

Rank in abundance

Abun

danc

e104

103

102

101

100

How many forms of the SAD for a given N and S?

Integer Partitioning

Integer partition: A positive integer expressed as an unordered sum of positive integers

e.g. 6 = 3+2+1 = 1+2+3 = 2+1+3

Written in non-increasing ordere.g. 3+2+1

Rank-abundance curves are integer partitions

Rank-abundance curve

N = total abundanceS = species richness

S unlabeled abundancesthat sum to N

Integer partition

N = positive integerS = number of parts

S unordered +integersthat sum to N=

Combinatorial Explosion

N S Shapes of the SAD

1000 10 > 886 trillion

1000 100 > 302 trillion trillion

Random integer partitions

Goal: Random partitions for N = 5, S = 3:

54+13+23+1+12+2+12+1+1+11+1+1+1+1

Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York.

SAD feasible sets are dominated by hollow curves

Fre

quen

cy

log2(abundance)

The SAD feasible setln

(abu

ndan

ce)

Rank in abundance

N=1000, S=40

Question: Can we explain the SAD based solely on how N and S constrain

observable variation?

DATAEthan P. White, Katherine M. Thibault, and Xiao Xiao 2012. Characterizing species

abundance distributions across taxa and ecosystems using a simple maximum entropy model. Ecology 93:1772–1778

Dataset Number of sites

Christmas Bird Count 1992

North American Breeding Bird Survey 2769

Gentry’s Forest Transect 222

Forest Inventory & Analysis 10356

Mammal Community Database 103

TOTAL 15442

Dataset Number of sites

Indoor Fungal Communities 128

Terrestrial metagenomesChu Arctic Soils, Lauber 88 Soils 128

Aquatic metagenomesCatlin Arctic Waters, Hydrothermal Vents 252

TOTAL METAGENOMES 512

GRAND TOTAL 15954

Microbial metagenomic datasetsobtained from MG-RAST metagenomics.anl.gov

TOOL LOGO COOLNESS

Sage mathematical software 8

Amazon Web Services 2

Weecology Servers (in-house) 10

TOTAL COMPUTING CORES 180

Generating random samples of the feasible set

Dataset total sites analyzable sites

Christmas Bird Count 1992 129 (6.5%)

North American Breeding Bird Survey 2769 1586 (57%)

Gentry’s Forest Transect 222 182 (82%)

Forest Inventory & Analysis 10356 7359 (71%)

Mammal Community Database 103 42 (41%)

Indoor Fungal Communities 128 124 (97%)

Terrestrial metagenomes 128 92 (72%)

Aquatic metagenomes 252 48 (19%)

TOTAL 15950 9562 (60%)

The center of the feasible setln

(abu

ndan

ce)

Rank in abundance

N=1000, S=40

R2 = 0.93

100 101 102

102

101

100

Obs

erve

d ab

unda

nce

Abundance at center of the feasible set

North American Breeding Bird Survey(1583 sites)

Abundance at center of the feasible set

Obs

erve

d ab

unda

nce

Obs

erve

d ab

unda

nce

Abundance at center of the feasible set

DOI: 10.1111/ele.12154

Public code and data repository

https://github.com/weecology/feasiblesets

General Conclusions

Feasible set: A primary way to account for how variables constrain ecological patterns…before attributing a pattern to a process

General Conclusions

Extending the feasible set approach:○ Spatial abundance distribution○ Species area relationship○ Distributions of wealth and abundance

The ubiquitous hollow curve

0.91

Obs

erve

d

Urban population sizesamong nations

(1960-2009, rescaled)

Oil related CO2 emission among nations

(1980-2009, rescaled)

0.92

Center of the feasible set

Center of the feasible set

Obs

erve

d ho

me

runs

0.93 0.88

0.91 0.91

0.94 0.93

http://mlb.mlb.com

General Conclusions

● The integer partitioning approach needs improvement

CHAPTER 2: Efficient algorithms for sampling feasible sets

Generate a random SADfor N=5 and S=3

54+13+23+1+12+2+12+1+1+11+1+1+1+1

Combinatorial Explosion

N S SAD shapes

1000 10 > 886 trillion

1000 1,...,1000 > 2.4x1031

Probability of generating a random partition of 1000 having 10 parts: < 10-17

Generate a random SADfor N=5

1) 52) 4+13) 3+24) 3+1+15) 2+2+16) 2+1+1+17) 1+1+1+1+1

Task: Generate random partitions of N=9 having S=4 parts

4+3+2

Task: Generate random partitions of N=9 having S=4 parts

4+3+2

4+3+2

4+3+2

3+3+2+14+3+2

1. Generate a random partition of N with S as the largest part

2. Conjugate the partition

A recipe for random SADsN = total abundanceS = species richness

Generate a random partition of N with S as the largest part

Divide & Conquer

54+13+23+1+12+2+12+1+1+11+1+1+1+1

Multiplicity

Top down

Bottom up

Un(bias)

Skewness of partitions in a random sample

De

nsi

ty

Speed

Number of parts (S)

Sag

e/al

gorit

hm

N = 50 N = 100

N = 150 N = 200

Old Apples: probability of generating a partition for N = 1000 & S = 10: < 10-17

New Oranges: Seconds to generate a partition for N = 1000 & S = 10: 0.07

Integer partitions

S positive integers that sum to N in without respect to order

What if a distribution has zeros?• subplots with 0 individuals• people with 0 income • publications with 0 citations

Abundance class

freq

uenc

y

0 1 2 3 4 5

Intraspecific spatial abundance distribution (SSAD)N = abundance of a species

S = number of subplots

SSAD

N = total abundanceS = no. subplots

S non-negative abundances that sum to N without respect to order

(weak) Integer partition

N = positive integerS = number of parts

S non-negative integersthat sum to N without

respect to order=

Intraspecific spatial abundance distribution (SSAD)

Intraspecific spatial abundance distribution (SSAD)

Abundance class

Freq

uenc

y

Abundance class Abundance class

Freq

uenc

y SAD

“…frequency distributions of intraspecific abundance among sample sites resemble distributions … that have been used to characterize the distribution of abundances among species” (Brown et al. 1995)

Species abundance = 1KSubplots = 100

Community abundance =1KSpecies = 50

SSAD

Abundance class Abundance class

Conclusions

•How do empirical SSADs compare to the feasible set of possible SSAD shapes?

•Other ecological patterns/distributions:

–Occupancy frequency distribution–Collector’s curve–Species-area curve–Species-time relationship

Public code repository

https://github.com/klocey/partitions

PeerJ Preprint

https://peerj.com/preprints/78/

Locey KJ, McGlinn DJ. (2013) Efficient algorithms for sampling feasible sets of macroecological patterns. PeerJ PrePrints 1:e78v1

Acknowledgements

For collecting, managing and providing datasets:North American Breeding Bird SurveyChristmas Bird CountGentry’s Forest Transect DataForest Inventory and Analysis datasetMicrobial metagenomic datasets accessed from MG-RASTMammal Community Database

My committee: Morgan Ernest, David Koons, Jeannette Norton, Jacob Parnell Past: Mike Pfrender, Paul CliftenColleagues: Justin Kitzes, James O’Dwyer, Bill Burnside, Jay Lennon, Paul Stone and the Stone CrewFaculty and Staff of the Biology Dept: esp. Brian Joy, Kami McNeil

Funding: W. L. Eccles Graduate Research Fellow 2008-2011James A. and Patty MacMahon ScholarshipJoseph E. Greaves Scholarship in BiologyDissertation FellowshipCAREER grant from NSF to Ethan White (DEB-0953694)Research grant from Amazon Web ServicesAmerican Museum of Natural History Theodore Roosevelt Memorial Grant

Weecology

I you guys

Sampling the SAD feasible Set

Den

sit

y

Evenness Evenness Evenness

Den

sit

y

Den

sit

y

Sample size = 300 Sample size = 500 Sample size = 700

Future Directionsin Feasible Sets

Evenness and diversity metrics

Evenness and diversity metrics

The ubiquitous hollow-curve

New feasible sets:

integer composition: all ordered ways that S positive integers can sum to N

New feasible sets:

integer composition: all ordered ways that S positive integers can sum to N