1. Rarefaction curves
Quantifying biodiversity: procedures and pitfalls in the measurements and comparisons of species richness

2. Our value of interest
All organisms are organized in a tree structure depicting groups
An interesting value about those groups is taxon-richness (measuring diversity of a population).
How do we represent this function?

3. Species richness Count the number of different species you find.
A simple measurement of diversity, used for
Comparing sites
Conservation studies
Learning evolutionary patterns
Animal diets
Yet, this intuitive function can be abused quite easily

4. Species Richness Example
Heat map for the number of tree species in north America
now and predicted for 2100

5. Taxon sampling curves
We want to compare the number of tree species between 2 forest plots of 1𝑘 𝑚 2 each.
Approach 1 – individual-based
Randomly check individual trees within each plot
Approach 2 – sample-based
Define quadrats in each plot, and record all within a number of random quadrats.

6. Taxon sampling curves (2)
Do we want to check specific sets, or do a mean over them?
Accumulation curves
total number of species revealed during the process of data collection as units are pooled together
Rarefaction curves
the mean of repeatedly re-sampling the pool of N individuals

7. Accumulation / Rarefaction curves
The belly represents spatial and temporal (patchiness or heterogeneity) relations

8. Rarefaction curve definition
𝑁 – number of individuals
𝐺 – number of groups (species)
𝑁 𝑖 - number of individuals under group 𝑖
𝑀 𝑗 - number of groups consisting of j individuals, such that
𝑗 𝑀 𝑗 =𝑁 ; 𝑀 𝑗 =𝐺
𝑋 𝑛 - number of groups present under a total of 𝑛≤𝑁 individuals.

9. Rarefaction curve definition (2)
𝑓 𝑛 =𝐸 𝑋 𝑛 =𝐺− 𝑁 𝑛 −1 𝑖=1 𝐺 𝑁− 𝑁 𝑖 𝑛 Where 𝑓 0 =0 – no groups with 0 individuals 𝑓 1 =1 – only 1 individual 𝑓 𝑁 =𝐺 – all examples

10. Confidence intervals
As long as there is a variance in sampled values, we will never gain an exact result.
Therefore we need a method to measure how “close” does an average on our samples gets to the exact value.
Usually we search for an interval [𝑎,𝑏] such that
𝑃 𝑎≤𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 𝑎𝑣𝑔.≤𝑏 =0.95

11. Confidence interval Example
Lets measure the amount of Caffeine in a 100 gram Energy bar.
Assume that the measuring method has an error with 𝜎=0.5
Take 25 samples ( 𝑋 1 ,…, 𝑋 25 ). We got 𝑋 =3𝑚𝑔
We can compute 𝑉𝑎𝑟 𝑋 =
Assuming a normal distribution we get:
𝑃 −1.96≤ 𝑋 −𝜇 𝑉𝑎𝑟 𝑋 ≤1.96 =0.95
⇒𝑃 𝜇∈ 𝑋 −0.196, 𝑋 =0.95

12. Rarefaction example
The speedy convergence of the pallets suggest a more uniform distribution
Yet note the smaller asymptote (owls don’t catch all that traps catch)
Small mammals in the diet of Barn owls
Rita G. Rocha et al – Zoologia

13. Databases pitfall 1
Comparing differences in abundance on raw species richness as a function of samples can produce misleading results (sample size, sampling method)

14. Category-subcategory ratios
Instead of normalizing, many databases save average species count per individual
Pitfalls

15. Species-genus ratio
With a low species – genus ratio, we can deduct a strong intrageneric competition (Islands for instance usually have a low species – genus ratio)
A pitfall of the ratio :
A simple example
10 genus
100 species per genus
After a 100 sampled individuals – we most likely would have found all genus, therefore a smaller sample might cause a low, unreliable species-genus ratio

16. A different perspective of the pitfall
If we try using the Genus-Species ratio (reverse) we can clearly see the non- linear problem arising (find more species in each previous genus with every genus)

17. Species richness vs species density
Most ecology studies standardize datasets via area or sampling effort instead of number of individuals
So a comparison with such a standardization actually compare species density (need to re-scale to individual and use the species mean density).
For conservations and large areas problems we use species density
For evolutionary predictions or theoretical predictions in ecology, we use species richness

18. Examples of rarefaction

19. The high fidelity of the cetacean stranding record: insights into measuring diversity by integrating taphonomy and macroecology ND Pyenson – The royal society publishing

20. Limited role of functional differentiation in early diversification of animals M.L Knope – Nature

21. Stranded dolphin stomach contents represent the free-ranging population's diet Glenn Dunshea et al – The royal society publishing