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

7. The Feasible Set: A New Understanding of Constraints on Ecological Patterns of Abundance

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

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

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

11. Rank-abundance curve (RAC) Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0

12. Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0 Predicting the SAD Observed Predicted

13. Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0 N = 1,700 S = 17

14. Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0 How many forms of the SAD for a given N and S?

15. 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 order e.g. 3+2+1

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

17. Combinatorial Explosion NSShapes of the SAD 100010> 886 trillion 1000100> 302 trillion trillion

18. Random integer partitions Goal: Random partitions for N = 5, S = 3: 5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1 Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York.

19. SAD feasible sets are dominated by hollow curves Frequency log 2 (abundance)

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

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

22. DATA Ethan 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 DatasetNumber of sites Christmas Bird Count1992 North American Breeding Bird Survey2769 Gentry’s Forest Transect222 Forest Inventory & Analysis10356 Mammal Community Database103 TOTAL15442

23. DatasetNumber of sites Indoor Fungal Communities128 Terrestrial metagenomes Chu Arctic Soils, Lauber 88 Soils 128 Aquatic metagenomes Catlin Arctic Waters, Hydrothermal Vents 252 TOTAL METAGENOMES512 GRAND TOTAL15954 Microbial metagenomic datasets obtained from MG-RAST metagenomics.anl.gov

24. 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

25. Datasettotal sitesanalyzable sites Christmas Bird Count1992129 (6.5%) North American Breeding Bird Survey 27691586 (57%) Gentry’s Forest Transect222182 (82%) Forest Inventory & Analysis103567359 (71%) Mammal Community Database10342 (41%) Indoor Fungal Communities128124 (97%) Terrestrial metagenomes 12892 (72%) Aquatic metagenomes 25248 (19%) TOTAL159509562 (60%)

26. The center of the feasible set ln(abundance) Rank in abundance N=1000, S=40

27. R 2 = 0.93 10 0 10 1 10 2 10 2 10 1 10 0 Observed abundance Abundance at center of the feasible set North American Breeding Bird Survey (1583 sites)

28. Abundance at center of the feasible set Observed abundance

29. Abundance at center of the feasible set

30. DOI: 10.1111/ele.12154

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

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

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

34. 0.91 Observed Urban population sizes among nations (1960-2009, rescaled) Oil related CO2 emission among nations (1980-2009, rescaled) 0.92 Center of the feasible set

35. Observed home runs 0.930.88 0.91 0.940.93 http://mlb.mlb.com

36. General Conclusions ●The integer partitioning approach needs improvement

37. CHAPTER 2: Efficient algorithms for sampling feasible sets

38. Generate a random SAD for N=5 and S=3 5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1

39. Combinatorial Explosion NSSAD shapes 100010> 886 trillion 10001,...,1000> 2.4x10 31 Probability of generating a random partition of 1000 having 10 parts: < 10 -17

40. Generate a random SAD for N=5 1) 5 2) 4+1 3) 3+2 4) 3+1+1 5) 2+2+1 6) 2+1+1+1 7) 1+1+1+1+1

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

42. 4+3+24+3+2

43. 4+3+24+3+2

44. 4+3+24+3+2

45. 4+3+24+3+2

46. 3+3+2+13+3+2+1 4+3+24+3+2

47. 1.Generate a random partition of N with S as the largest part 2.Conjugate the partition A recipe for random SADs N = total abundance S = species richness

48. Generate a random partition of N with S as the largest part Divide & Conquer 5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1 Multiplicity Top down Bottom up

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

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

51. 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

52. 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

53. Abundance class frequency 012345 Intraspecific spatial abundance distribution (SSAD) N = abundance of a species S = number of subplots

54. SSAD N = total abundance S = no. subplots S non-negative abundances that sum to N without respect to order (weak) Integer partition N = positive integer S = number of parts S non-negative integers that sum to N without respect to order = Intraspecific spatial abundance distribution (SSAD)

55. Abundance class Frequency Abundance class

56. Frequency 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 = 1K Subplots = 100 Community abundance =1K Species = 50 SSAD Abundance class

57. 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

58. 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

59. Acknowledgements For collecting, managing and providing datasets: North American Breeding Bird Survey Christmas Bird Count Gentry’s Forest Transect Data Forest Inventory and Analysis dataset Microbial metagenomic datasets accessed from MG-RAST Mammal Community Database My committee: Morgan Ernest, David Koons, Jeannette Norton, Jacob Parnell Past: Mike Pfrender, Paul Cliften Colleagues: Justin Kitzes, James O’Dwyer, Bill Burnside, Jay Lennon, Paul Stone and the Stone Crew Faculty and Staff of the Biology Dept: esp. Brian Joy, Kami McNeil Funding: W. L. Eccles Graduate Research Fellow 2008-2011 James A. and Patty MacMahon Scholarship Joseph E. Greaves Scholarship in Biology Dissertation Fellowship CAREER grant from NSF to Ethan White ( DEB-0953694 ) Research grant from Amazon Web Services American Museum of Natural History Theodore Roosevelt Memorial Grant

60. Weecology I you guys

62. Sampling the SAD feasible Set Density Evenness Density Sample size = 300Sample size = 500Sample size = 700

63. Future Directions in Feasible Sets

64. Evenness and diversity metrics

66. The ubiquitous hollow-curve

67. New feasible sets: integer composition: all ordered ways that S positive integers can sum to N

68. New feasible sets: integer composition: all ordered ways that S positive integers can sum to N