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Macroecology …characterizing and explaining patterns of abundance, distribution, and diversity.
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Transcript of Macroecology …characterizing and explaining patterns of abundance, distribution, and diversity.
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Macroecology …characterizing and explaining patterns of abundance, distribution, and diversity
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The Feasible Set: A New Understanding of Constraints
on Ecological Patterns of Abundance
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CHAPTER 1: How species richness and total abundance constrain the
distribution of abundance
CHAPTER 2: Efficient algorithms for sampling feasible sets
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Rank-abundance curve (RAC)
Rank in abundance
Abun
danc
e
Frequency distribution
Species abundance distribution (SAD)
Abundance class
freq
uenc
y
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Frequency distribution
The ubiquitous hollow-curve
Abundance class
freq
uenc
y
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Rank-abundance curve (RAC)
Rank in abundance
Abun
danc
e104
103
102
101
100
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Rank in abundance
Abun
danc
e104
103
102
101
100
Predicting the SAD
Observed
Predicted
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Rank in abundance
Abun
danc
e104
103
102
101
100
N = 1,700S = 17
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Rank in abundance
Abun
danc
e104
103
102
101
100
How many forms of the SAD for a given N and S?
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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
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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=
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Combinatorial Explosion
N S Shapes of the SAD
1000 10 > 886 trillion
1000 100 > 302 trillion trillion
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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.
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SAD feasible sets are dominated by hollow curves
Fre
quen
cy
log2(abundance)
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The SAD feasible setln
(abu
ndan
ce)
Rank in abundance
N=1000, S=40
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Question: Can we explain the SAD based solely on how N and S constrain
observable variation?
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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
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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
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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
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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%)
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The center of the feasible setln
(abu
ndan
ce)
Rank in abundance
N=1000, S=40
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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)
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Abundance at center of the feasible set
Obs
erve
d ab
unda
nce
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Obs
erve
d ab
unda
nce
Abundance at center of the feasible set
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DOI: 10.1111/ele.12154
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Public code and data repository
https://github.com/weecology/feasiblesets
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General Conclusions
Feasible set: A primary way to account for how variables constrain ecological patterns…before attributing a pattern to a process
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General Conclusions
Extending the feasible set approach:○ Spatial abundance distribution○ Species area relationship○ Distributions of wealth and abundance
The ubiquitous hollow curve
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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
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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
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General Conclusions
● The integer partitioning approach needs improvement
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CHAPTER 2: Efficient algorithms for sampling feasible sets
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Generate a random SADfor N=5 and S=3
54+13+23+1+12+2+12+1+1+11+1+1+1+1
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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
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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
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Task: Generate random partitions of N=9 having S=4 parts
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4+3+2
Task: Generate random partitions of N=9 having S=4 parts
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4+3+2
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4+3+2
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4+3+2
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3+3+2+14+3+2
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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
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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
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Un(bias)
Skewness of partitions in a random sample
De
nsi
ty
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Speed
Number of parts (S)
Sag
e/al
gorit
hm
N = 50 N = 100
N = 150 N = 200
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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
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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
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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
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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)
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Intraspecific spatial abundance distribution (SSAD)
Abundance class
Freq
uenc
y
Abundance class Abundance class
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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
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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
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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
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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
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Weecology
I you guys
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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
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Future Directionsin Feasible Sets
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Evenness and diversity metrics
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Evenness and diversity metrics
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The ubiquitous hollow-curve
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New feasible sets:
integer composition: all ordered ways that S positive integers can sum to N
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New feasible sets:
integer composition: all ordered ways that S positive integers can sum to N