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

Macroecology …characterizing and explaining patterns of abundance, distribution, and diversity. The Feasible Set : A New Understanding of Constraints on Ecological Patterns of Abundance. CHAPTER 2: Efficient algorithms for sampling feasible sets.

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

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

  2. The Feasible Set: A New Understanding of Constraintson Ecological Patterns of Abundance

  3. CHAPTER 2: Efficient algorithms for sampling feasible sets CHAPTER 1: How species richness and total abundance constrain the distribution of abundance

  4. Rank-abundance curve (RAC) Species abundance distribution (SAD) Frequency distribution frequency Abundance Rank in abundance Abundance class

  5. The ubiquitous hollow-curve Frequency distribution frequency Abundance class

  6. Rank-abundance curve (RAC) 104 103 Abundance 102 101 100 Rank in abundance

  7. Predicting the SAD 104 Observed Predicted 103 Abundance 102 101 100 Rank in abundance

  8. 104 N = 1,700 S = 17 103 Abundance 102 101 100 Rank in abundance

  9. How many forms of the SAD for a given N and S? 104 103 Abundance 102 101 100 Rank in abundance

  10. Integer Partitioning Integer partition: A positive integer expressedas anunorderedsum of positive integers e.g. 6 = 3+2+1 = 1+2+3 = 2+1+3 Written in non-increasing order e.g. 3+2+1

  11. Rank-abundance curves are integer partitions Rank-abundance curve Integer partition N = total abundance S = species richness Sunlabeled abundances that sum to N N = positive integer S = number of parts Sunordered +integers that sum to N =

  12. Combinatorial Explosion

  13. Random integer partitions Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York. Goal: Random partitions for N = 5, S = 3:

  14. SAD feasible sets are dominated by hollow curves Frequency log2(abundance)

  15. The SAD feasible set N=1000, S=40 ln(abundance) Rank in abundance

  16. Question: Can we explain the SAD based solely on how N and S constrain observable variation?

  17. DATAEthan P. White, Katherine M. Thibault, and Xiao Xiao2012. Characterizing species abundance distributions across taxa and ecosystems using a simple maximum entropy model. Ecology 93:1772–1778

  18. Microbial metagenomic datasets obtained from MG-RASTmetagenomics.anl.gov

  19. Generating random samples of the feasible set

  20. The center ofthe feasible set N=1000, S=40 ln(abundance) Rank in abundance

  21. North American Breeding Bird Survey (1583 sites) 102 101 100 R2 = 0.93 Observed abundance 100 101 102 Abundance at center of the feasible set

  22. Observed abundance Abundance at center of the feasible set

  23. Observed abundance Abundance at center of the feasible set

  24. DOI: 10.1111/ele.12154

  25. Public code and data repository https://github.com/weecology/feasiblesets

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

  27. General Conclusions Extending the feasible set approach: • Spatial abundance distribution • Species area relationship • Distributions of wealth and abundance The ubiquitous hollow curve

  28. Urban population sizes among nations (1960-2009, rescaled) Oil related CO2 emission among nations (1980-2009, rescaled) 0.91 0.92 Observed Center of the feasible set

  29. 0.93 0.88 0.91 0.91 Observed home runs 0.94 0.93 Center of the feasible set http://mlb.mlb.com

  30. General Conclusions • The integer partitioning approach needs improvement

  31. CHAPTER 2: Efficient algorithms for sampling feasible sets

  32. Generate a random SADfor N=5 and S=3

  33. Combinatorial Explosion Probability of generating a random partition of 1000 having 10 parts:< 10-17

  34. Generate a random SADfor N=5

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

  36. Task: Generate random partitions of N=9 having S=4 parts 4+3+2

  37. 4+3+2

  38. 4+3+2

  39. 4+3+2

  40. 3+3+2+1 4+3+2

  41. A recipe for random SADsN = total abundanceS = species richness • Generate a random partition of N with S as the largest part • Conjugate the partition

  42. Generate a random partition of N with S as the largest part Divide & Conquer Top down Multiplicity Bottom up

  43. Un(bias) Density Skewness of partitions in a random sample

  44. Speed N = 50 N = 100 Sage/algorithm N = 150 N = 200 Number of parts (S)

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