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Before starting
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In this practical, we will go through the three main approaches to compute \(\beta\)-diversity.

  • \(\alpha\) diversity versus \(\gamma\) diversity
  • slope of the species-surface curve
  • similarity indices

Species turnover between 3 plots.

Species turnover between 3 plots.


We will use the same dataset for this practical than for the practical 2, i.e. the Barro Colorado Island (BCI) forest-plot inventory in Panama. Refer to the practical 2 for more details regarding this dataset.

1. Pool level

1.1. alpha diversity versus gamma diversity

Lande’s additive partitionning. \[ \beta = \gamma - \overline{\alpha} \]

Calculate the Lande’s index on BCI data.

Solution

## [1] 134.22


Whittaker’s multiplicative partitionning. \[ \beta = \frac{\gamma}{\overline{\alpha}} \]

Calculate the Whittaker’s index.

Solution

## [1] 2.478519


1.2. Species-surface curve

The slope of the species-area curve also gives us a measure of \(\beta\) diversity since it tells us how many new species are added when we consider a bigger area.
With BCI data, we now that each plot has a 1 hectare surface. We can therefore reconstruct a species-surface curve using random sample of a varying number of plots. At each random sample, we calculate the average species number. The approach is similar to the species accumulation curve we saw in the practical 4.

To do so, we will use a for-loop with the \(\alpha\) richness of random plot aggregates of 2, 5, 10, 20, 30 and then 40 plots.
Let us start with a random sample of two plots.

## [1] "Plot4"  "Plot39"
## [1] 185
## [1] 111
## [1] 100
## [1] 121.17

We now repeat this step with several aggregates of plot number (and therefore surface):

Now the plot.

\(\beta\)-diversity estimate corresponds to the exponent \(z\) of the Arrhenius model \(S = cA^z\), where \(S\) is the species number and \(A\) is the area.

To retrieve it, we can calculate a linear model of the log-transformed number of species as a function of the log-transformed area, since we have: \[ log_{10} = log_{10}(c) + zlog_{10}(A) \]

## 
## Call:
## lm(formula = log_S ~ log_area, data = sp_area)
## 
## Coefficients:
## (Intercept)     log_area  
##       2.054        0.186
## log_area 
##    0.186

The slope of 0.186 is the rate at which \(S\) increases against area and therefore our estimate of \(\beta\)-diversity.

2. Pairwise level: similarity indices

Contrarily to the previous sections, we now evaluate \(\beta\)-diversity for each pair of sites.

Each pair of sites can share some species and have unique set of species, like in the following figure:


Example with two plots.

Example with two plots.


with \(a\) the set of species shared by both plots, \(b\) the species unique to plot 1 and \(c\) the species unique to plot 2.

The idea behind this concept is to find an index that identify the sites that share a lot of species and the ones that are distant, regarding their species composition.

2.1. Plot of abc

Let us build a simple virtual matrix to visualize \(abc\) with simple plots.

##       sp1 sp2 sp3 sp4 sp5
## plot1   1   1   1   0   0
## plot2   1   1   1   0   0
## plot3   1   0   0   0   0
## plot4   0   0   0   1   1
## plot5   0   1   1   1   0

We can see that plots 1 and 2 are identical, plot 3 is nested within 1 and 2, plot 4 shares no species with the first three plots. Plot 5 displays an intermediate situation.

Let’s manually calculate the number of shared and unique species for a given pair of virtual plots.

Calculate the number of unique and shared species in the first two virtual plots 1 and 5.

Solution

## plot1 plot5 
##     3     3
## [1] 2
## plot1 plot5 
##     1     1


To compute automatically these numbers, we can use betadiver() function from vegan package. With method = NA, no index is calculated, but instead an object of class betadiver is returned. This is a list of the elements $a, $b and $c.

Each element is a distance matrix of size \(n\times(n-1)/2\) with \(n\) the number of plots.

We can also plot them in a triangular plot showing how each pair of plots is similar, as suggested by (Koleff, Gaston, and Lennon 2003).

##       plot1 plot2 plot3 plot4
## plot2     3                  
## plot3     1     1            
## plot4     0     0     0      
## plot5     2     2     0     1
## [1] 10
## [1] 10
##       plot1 plot2 plot3 plot4
## plot2     0                  
## plot3     0     0            
## plot4     2     2     2      
## plot5     1     1     3     2
##       plot1 plot2 plot3 plot4
## plot2     0                  
## plot3     2     2            
## plot4     3     3     1      
## plot5     1     1     1     1

If we draw the same triangular plot on BCI data, what do we get?

##       Plot1 Plot2 Plot3
## Plot1     0    64    64
## Plot2    64     0    62
## Plot3    64    62     0
##       Plot1 Plot2 Plot3
## Plot1     0    20    26
## Plot2    20     0    28
## Plot3    26    28     0

Low heterogeneity is found among BCI communities, all plots are being clumped in one part of the triangular space. We could say that BCI plots are similarly dissimilar.

2.2. Pairwise distances

Many distances and metrics exist, the three following ones being the most important. With \(a\) the number of species shared by two sites, \(b\) the number of species only in the first site and \(c\) the number of species only in the second one, we have:

  • Sorensen (similarity) \[ Sorensen = \frac{2a}{2a + b + c} \]

Therefore, \(\beta\)-diversity being a metric based on dissimilarity, we get:

\[ \beta_{sor} = 1- \frac{2a}{2a + b + c} \]

  • Jaccard (similarity) \[ Jaccard = \frac{a}{a + b + c} \]

and similarly:

\[ \beta_{jac} = 1- \frac{a}{a + b + c} \]

  • Simpson (dissimilarity)

\[ \beta_{sim} = \frac{min(b, c)}{a + min(b, c)} \]

Calculate the Sorensen similarity between the first two plots of BCI.

Solution

## Plot1 Plot2 
##    93    84
## [1] 64
## Plot1 Plot2 
##    29    20
## [1] 0.7231638


betadiver() function from vegan allows us to compute these indices.

##           Plot1     Plot2
## Plot1 0.0000000 0.7231638
## Plot2 0.7231638 0.0000000
## [1] 0.7231638
##           Plot1     Plot2
## Plot1 0.0000000 0.2380952
## Plot2 0.2380952 0.0000000
## [1] 0.2380952
##           Plot1     Plot2
## Plot1 0.0000000 0.5663717
## Plot2 0.5663717 0.0000000
## [1] 0.5663717


Note
The above betadiver() function computes the original indices. These can quantify either similarity or dissimilarity between plots. When classifying plots between each other, dissimilarity is often used. Dissimilarity simply equals 1 - similarity.

vegdist() returns the dissimilarity between plots.

##           Plot1     Plot2     Plot3
## Plot1 0.0000000 0.2768362 0.3005464
## Plot2 0.2768362 0.0000000 0.2873563
## Plot3 0.3005464 0.2873563 0.0000000
##           Plot1     Plot2     Plot3
## Plot1 1.0000000 0.2768362 0.3005464
## Plot2 0.2768362 1.0000000 0.2873563
## Plot3 0.3005464 0.2873563 1.0000000

vegdist() can also be applied on abundances data, with the use of Bray-Curtis index:

\[ \beta_{ij}=1-\frac{2C_{ij}}{S_i + S_j} \]

with \(C_{ij}\) the sum of the lesser individuals for only common species in both \(i\) and \(j\), \(S_i\) the number of individuals in plot \(i\) and \(S_j\) the number of individuals in plot \(j\).

##           Plot1     Plot2     Plot3
## Plot1 0.0000000 0.2706682 0.3501647
## Plot2 0.2706682 0.0000000 0.2873051
## Plot3 0.3501647 0.2873051 0.0000000

The overall pattern is similar but including abundances can lead to a better understanding of the changing of species composition across plots.

2.3. Partitionning of \(\beta\) diversity

\(\beta\)-diversity can be divided into a nestedness and a turnover component. The nestedness component quantifies the part of \(\beta\)-diversity that results from the mere difference in species numbers. The turnover component quantifies the part that actually results from differences in species identities (Baselga, 2010). The Simpson index is a common measure of the turnover component because it is independent of differences in the number of species in the plots. Jaccard and Sorensen quantify the total beta diversity. Therefore:

\[ \beta_{nes} = \beta_{sor} - \beta_{sim} \]

or

\[ \beta_{nes} = \beta_{jac} - \beta_{sim} \]

##            Plot1      Plot2      Plot3
## Plot1 0.00000000 0.03874092 0.01165756
## Plot2 0.03874092 0.00000000 0.02545156
## Plot3 0.01165756 0.02545156 0.00000000
## [1] 0.3399075
## [1] 0.3090427
## [1] 0.03086482

Note
Other package exist to calculate \(\beta\)-diversity like betarpart. It notably allows to compute the three components of \(\beta\)-diversity (with turnover and nestedness) in a faster way.

Bonus: highliting extreme points on abc plots

Bonus From such a distance matrix, we can target the closest and most distances plots, regarding their species composition.

## [1] 0.4941176
## [1] 723
## [1] 0.7914439
## [1] 954

We can also highlight these extreme pairs in an “abc” triangular plot.

## [1] "a= 42 ; b= 39 ; c= 47"
## [1] "a= 74 ; b= 14 ; c= 25"

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