These are the only RADs possible for the community you think you have

Whittling down RADsKen Locey

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

Q. How many possible RADs for a community of 45 individuals?

A. 89, 134

Q. How many possible ways to add integers to reach a sum of 45?

A. 89, 134

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

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

Community State Variables = Partition State Variables

A community

• Community size =• Community richness =

• Most abundant species =• Singleton species =

An integer partition

Integer valuePartition lengthLargest integer in partition1’s in the partition

N = 30

# partitions (i.e. RADs) = 5604

N = 30, S = 4

# partitions (i.e. RADs) = 297

N = 30, S = 4, max = 10

# partitions (i.e. RADs) = 23 from 5604 99.6% decrease

N = 45, S = 5, max <= 25, min = 1

# RADs = 458

N = 45, S = 5, max <= 25, min = 1

# RADs = 458 from 89,134 99.5% decrease

N = 45, S = 5, max = 25, min = 1

# RADs = 30 from 89,134 99.97% decrease

Having community size…

• A community of 100 individuals will have 190,569,292 possible RAD’s

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?

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

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

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