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

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

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

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

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

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

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

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

13. 104 N = 1,700 S = 17 103 Abundance 102 101 100 Rank in abundance

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

15. 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

16. 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 =

17. Combinatorial Explosion

18. 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:

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

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

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

22. 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

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

24. Generating random samples of the feasible set

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

27. 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

28. Observed abundance Abundance at center of the feasible set

29. Observed abundance 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. 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

35. 0.93 0.88 0.91 0.91 Observed home runs 0.94 0.93 Center of the feasible set 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 SADfor N=5 and S=3

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

40. Generate a random SADfor N=5

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

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

43. 4+3+2

44. 4+3+2

45. 4+3+2

46. 3+3+2+1 4+3+2

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

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

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

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