1. These are the only RADs possible for the community you think you have Whittling down RADs Ken Locey

2. Q. How many possible RAD’s for a community of 45 individuals?

3. Q. How many possible RADs for a community of 45 individuals? A. 89, 134

4. Q. How many possible ways to add integers to reach a sum of 45? A. 89, 134

5. Integer partitioning: number of unordered ways to add integers to sum to another integer 4: 1,1,1,1 4 2,2 1,3 5: 1,1,1,1,1 5 2,2,1 1,3,1 2,3 4,1 2,1,1,1 3: 1,1,1 3 2,1

6. Partitioning Community of size N Community of size N Can have 1 to N species The species can take values of 1 to N Abundances of species must sum to N Integer of size N Can have 1 to N summands The summands can take values 1 to N Values of integers must sum to N

7. Community State Variables = Partition State Variables A community Community size = Community richness = Most abundant species = Singleton species = An integer partition Integer value Partition length Largest integer in partition 1’s in the partition

8. N = 30 # partitions (i.e. RADs) = 5604

9. N = 30, S = 4 # partitions (i.e. RADs) = 297

10. N = 30, S = 4, max = 10 # partitions (i.e. RADs) = 23 from 5604 99.6% decrease

11. N = 45, S = 5, max <= 25, min = 1 # RADs = 458

12. N = 45, S = 5, max <= 25, min = 1 # RADs = 458 from 89,134 99.5% decrease

13. N = 45, S = 5, max = 25, min = 1 # RADs = 30 from 89,134 99.97% decrease

14. Having community size… A community of 100 individuals will have 190,569,292 possible RAD’s

15. Having community size… A community of 100 individuals will have 190,569,292 possible RAD’s If we know: S= 8, max <= 50, at least 1 singleton: 487,182 possibilities – 99.7% decrease – Imagine if we knew a little more?

16. Point # RADs for a community can be large but the number can be found Knowing community size and 3 more pieces of information can whittle down the possibilities by more than 99.5%. If you want to predict the distribution of abundance, you do not need to know much – What you know should be accurate

17. Challenges Computer algorithms fast enough to yield answers in a reasonable time – Storage is not the problem Identifying which features whittle-down the possibilities the most. – Most informative state variables – Graphical examination

18. Within these RADs are the only RADs possible …for the community you think you have