Tutorial 3: Quantifying taxonomic diversity

Author

Affiliation

Benjamin Delory

Environmental sciences group, Copernicus institute of sustainable development, Utrecht University

About this tutorial

Welcome to this tutorial on quantifying taxonomic diversity in R!

Throughout this tutorial, you will learn different ways to quantify species diversity in an ecosystem and become aware of some of the challenges associated with quantifying biodiversity. This tutorial focuses on quantifying taxonomic diversity across scales (alpha, beta, and gamma diversity sensu Whittaker 1960). In addition, you will use the knowledge gained in tutorials 1 and 2 to calculate Hill numbers of order q (i.e. the effective number of species for a given diversity order q ) and plot diversity profiles to analyse and compare biodiversity between different sites in a hypothetical metacommunity. Particular attention will be paid to the importance of sampling effort when comparing biodiversity estimates between sites with different levels of biodiversity.

There are R packages that allow you to calculate all the diversity measures listed above, but we will not be using them in this tutorial. We would like you to write your own R code to calculate them. Once again, it’s all about learning by doing.

Let’s get started!

A hypothetical metacommunity

The first step is to create some data to work with. You will do this by sampling wooden meeples (which are small board game pieces) to generate a site by species abundance matrix. Our hypothetical metacommunity consists of six paper bags (which represent different sampling sites), each containing 30 meeples (which represent different individuals living at a site). Meeples of the same colour belong to the same species. We will not give you any information about species diversity and composition at each site. You will have to obtain this information yourself, by repeatedly sampling each site without looking inside the bags.

Our species pool consists of 10 different species, each represented by a different colour. This is a small species pool, but it is enough to meet the objectives of this tutorial.

Sampling procedure

The aim of this exercise is to simulate a survey carried out to characterise the taxonomic diversity of six sites making up our metacommunity. Each site will be sampled multiple times to simulate surveys in a number of quadrats/locations at each site.

You can start generating biodiversity data using the following procedure:

Divide into 6 sub-groups.
Each sub-group takes one bag (each bag is labelled with a number from 1 to 6). The bags have a different composition and contain a different number of species.
One member of each a sub-group carefully mixes the content of the bag and randomly samples 6 individuals from the bag (this represents 20% of the individuals in a bag). Try not to look inside the bag to avoid any sampling bias.
Write down your observations in an Excel sheet (site as rows, species as columns).
Once your observations have been recorded, put your sample back into the bag and give the bag to another person in your group. Then, repeat the previous steps until you have three replicates. In your data sheet, report your observations independently for each replicate (i.e., each row in your data is a replicate within a site).
Once you have all the replicates needed for your bag, switch bags with another group and repeat the previous steps. Do this until you have three replicates for at least four different bags. Note that all the calculations below are based on six bags, but you can also do the same calculations with four bags (you will get different answers).
Make sure that your data are formatted properly.

Exercises

Load R packages

Load tidyverse, knitr, and viridis.

Code

library(tidyverse)
library(knitr)
library(viridis)

Exercise 1: Calculate the average species richness at each site

Use what you learned in previous tutorials to calculate the average number of species at each site. Once again, the functions pivot_longer(), group_by(), and summarise() are essential for this purpose.

Code

kable(head(data, n=10))

Bag	Replicate	Red	Orange	Yellow	Green	Blue	Purple	Pink	White	Grey	Black
1	1	1	0	0	1	2	1	1	0	0	0
1	2	0	1	0	1	1	0	1	1	0	1
1	3	0	1	0	0	0	0	1	0	3	1
2	1	0	0	0	4	0	2	0	0	0	0
2	2	0	0	0	4	0	2	0	0	0	0
2	3	0	0	0	5	0	1	0	0	0	0
3	1	0	1	0	2	1	0	0	1	0	1
3	2	0	2	0	0	0	0	0	1	1	2
3	3	1	1	1	1	1	0	0	1	0	0
4	1	0	2	0	0	1	0	0	0	0	3

Code

table <- data |>            
  pivot_longer(Red:Black,                        
               names_to = "Species",                        
               values_to = "Abundance") |>            
  group_by(Bag, Replicate) |>            
  summarise(Richness = sum(Abundance>0)) |>            
  group_by(Bag) |>            
  summarise(Avg_richness = round(mean(Richness), 1))

Code

kable(table)

Bag	Avg_richness
1	5.0
2	2.0
3	5.0
4	3.3
5	3.7
6	3.0

Exercise 2: Calculate alpha, beta and gamma diversity

Alpha diversity (within-habitat diversity) is the average number of species across sites in our hypothetical metacommunity. Gamma diversity (regional diversity) is the total number of species present in our hypothetical metacommunity. Beta diversity (between-habitat diversity differentiation) is gamma diversity divided by alpha diversity.

Using the data collected earlier, calculate alpha, beta and gamma diversity in our hypothetical metacommunity. You can do this manually, but we also want you to be able to do this in R! The functions pivot_longer(), group_by(), and summarise() are essential for this purpose.

First, import your data into R. Then, use knowledge from tutorials 1 and 2 to write R code to calculate alpha, beta, and gamma diversity in our metacommunity.

Code

gamma <- as.numeric(data |> 
                      pivot_longer(Red:Black,
                                   names_to = "Species",
                                   values_to = "Abundance") |> 
                      group_by(Species) |> 
                      summarise(Abundance = sum(Abundance)) |> 
                      summarise(Gamma = sum(Abundance > 0)))

Gamma diversity: 10

Code

alpha <- round(as.numeric(data |> 
                            pivot_longer(Red:Black,
                                         names_to = "Species",
                                         values_to = "Abundance") |> 
                            group_by(Bag, Species) |> 
                            summarise(Abundance = sum(Abundance)) |> 
                            group_by(Bag) |>
                            summarise(Alpha = sum(Abundance > 0)) |> 
                            summarise(Alpha = mean(Alpha))), 1)

Alpha diversity: 5.7

Code

beta <- round(gamma/alpha, 1)

Beta diversity: 1.8

Exercise 3: Calculate the effective number of species (Hill numbers)

In a community consisting of n species, the Hill number (or effective number of species) of order q can be calculated using Equation 1 ($q \geq 0$, $q\neq1$) and Equation 2 ($q=1$), where p_i is the relative abundance of species i in the community ($p_i \in [0,1]$).

\[ ^qD = (\sum_{i=1}^{n}p_i^q)^{\frac{1}{1-q}} \tag{1}\]

\[ ^1D=exp(-\sum_{i=1}^{n}p_i\log p_i) \tag{2}\]

Use Equation 1 and Equation 2 to calculate the effective number of species of order 0 (species richness), 1 (Hill-Shannon, the exponential of Shannon’s entropy) and 2 (Hill-Simpson, the inverse of Simpson’s concentration index) at each site of our hypothetical metacommunity. Plot your results in a figure with 6 panels (one panel for each site). For each site, plot the effective number of species as a function of q (this is called a diversity profile plot). Plot separate lines (using geom_line()) for each replicate (the group argument in aes() will be useful here).

What do you observe? What can you conclude about the taxonomic diversity at each site?

Solution (code)
Solution (plot)

Your code should include the following steps (use the pipe!):

Create a new variable (called “Total”) by summing the number of individuals of each species sampled for each replicate. Use mutate() to do this.
Transform all absolute species abundance values into relative abundance values by dividing the number of individuals of a species by the total number of individuals (“Total”) sampled for each replicate. You can also use mutate() to do this.
Use pivot_longer() to reorganise the dataset and create a column called “Species” to store all the species names (e.g. Red, Orange, etc.) and a column called “Abundance” to store the relative abundance values of the species in the communities.
Filter the data to eliminate all rows where the relative abundance of a species is zero. You can use filter() to do this.
For each replicate, use Equation 1 and Equation 2 to calculate the effective number of species at q=0 (name this variable “q0”), q=1 (name this variable “q1”), and q=2 (name this variable “q2”). You can do this using group_by() and summarise().
Use pivot_longer() to reorganise the dataset and create a column called “q” to store all diversity order values (0, 1, and 2) and a column called “diversity” to store the effective number of species in each community. In pivot_longer(), you will need to use two extra arguments: names_prefix = "q" (this will make sure to remove the letter “q” from “q0”, “q1”, and “q2”) and names_transform = as.numeric (to convert “0”, “1” and “2” into numeric values).
Finally, transform the contents of the “Bag” column by adding the string “Bag” (do not forget the space!) in front of the bag number (for example, converting 1 to “Bag 1”), and ensure that the Replicate variable is considered a factor. You can perform all these transformations using mutate(), paste() and as.factor().
Finally, use gggplot2 to plot your results in a figure with 6 panels (one panel for each site). For each site (1 site = 1 bag), plot the effective number of species as a function of q (this is called a diversity profile plot). Plot separate lines (using geom_line()) for each replicate (the group argument in aes() will be useful here). Personalise your plot to make it look nice! Be sure to apply the data visualisation principles discussed in the last lecture and tutorial.

Code

#Organise data to create the plot

data_plot <- data |> 
          mutate(Total = Red+Orange+Yellow+Green+Blue+Purple+Pink+White+Grey+Black) |> 
          mutate(Red=Red/Total,
                 Orange=Orange/Total,
                 Yellow=Yellow/Total,
                 Green=Green/Total,
                 Blue=Blue/Total,
                 Purple=Purple/Total,
                 Pink=Pink/Total,
                 White=White/Total,
                 Grey=Grey/Total,
                 Black=Black/Total) |> 
          pivot_longer(Red:Black,
                       names_to = "Species",
                       values_to = "Abundance") |> 
          filter(Abundance!=0) |> 
          group_by(Bag, Replicate) |> 
          summarise(q0=sum(Abundance^0)^(1/(1-0)),
                    q1=exp(-sum(Abundance*log(Abundance))),
                    q2=sum(Abundance^2)^(1/(1-2))) |> 
          pivot_longer(q0:q2,
                       names_to = "q",
                       names_prefix = "q",
                       names_transform = as.numeric,
                       values_to = "diversity") |>
          mutate(Bag=paste("Bag ", Bag, sep=""),
                 Replicate=as.factor(Replicate)) 

#Plot data

plot <- ggplot(data_plot,
               aes(x = q,
                   y = diversity,
                   group = Replicate,
                   colour=Replicate))+
            geom_line()+
            geom_point(shape=16)+
            facet_wrap(~Bag)+
            theme_bw()+
            xlab("Diversity order (q)")+
            ylab("Effective number of species (D)")+
            scale_colour_viridis(discrete=TRUE)+
            theme(axis.title.x = element_text(margin = margin(t=10)),
                  axis.title.y = element_text(margin = margin(r=10)),
                  axis.text = element_text(colour="black"),
                  panel.grid.minor = element_blank())+
            scale_x_continuous(breaks=c(0,1,2),
                               limits = c(0,2))+
            scale_y_continuous(breaks=seq(from = 0,
                                          to = 10,
                                          by = 2),
                               limits=c(0,10))

Code

plot

Suggested further reading

Our tutorial can’t cover everything, and there’s still a lot to learn about quantifying biodiversity! If you are interested, take a look at these excellent papers.

Chao A, Chiu C-H, Jost L. 2014. Unifying Species Diversity, Phylogenetic Diversity, Functional Diversity, and Related Similarity and Differentiation Measures Through Hill Numbers. Annual review of ecology, evolution, and systematics 45: 297–324.

Hill MO. 1973. Diversity and evenness: A unifying notation and its consequences. Ecology 54: 427–432.

Roswell M, Dushoff J, Winfree R. 2021. A conceptual guide to measuring species diversity. Oikos 130: 321–338.

Whittaker RH. 1960. Vegetation of the siskiyou mountains, Oregon and California. Ecological monographs 30: 279–338.

--- title: "Tutorial 3: Quantifying taxonomic diversity" author: - name: Benjamin Delory orcid: 0000-0002-1190-8060 email: b.m.m.delory@uu.nl affiliations: - name: Environmental sciences group, Copernicus institute of sustainable development, Utrecht University --- ## About this tutorial Welcome to this tutorial on quantifying taxonomic diversity in R! Throughout this tutorial, you will **learn different ways to quantify species diversity in an ecosystem and become aware of some of the challenges associated with quantifying biodiversity**. This tutorial focuses on quantifying taxonomic diversity across scales (alpha, beta, and gamma diversity sensu [Whittaker 1960](https://esajournals.onlinelibrary.wiley.com/doi/abs/10.2307/1943563)). In addition, you will use the knowledge gained in tutorials 1 and 2 to calculate [Hill numbers](https://www.jstor.org/stable/1934352?seq=1) of order *q* (i.e. the effective number of species for a given diversity order *q* ) and plot diversity profiles to analyse and compare biodiversity between different sites in a hypothetical metacommunity. Particular attention will be paid to the importance of sampling effort when comparing biodiversity estimates between sites with different levels of biodiversity. There are R packages that allow you to calculate all the diversity measures listed above, but we will not be using them in this tutorial. We would like you to write your own R code to calculate them. Once again, it's all about learning by doing. Let's get started! ## A hypothetical metacommunity The first step is to create some data to work with. You will do this by sampling wooden meeples (which are small board game pieces) to generate a site by species abundance matrix. Our hypothetical metacommunity consists of six paper bags (which represent different sampling sites), each containing 30 meeples (which represent different individuals living at a site). Meeples of the same colour belong to the same species. We will not give you any information about species diversity and composition at each site. You will have to obtain this information yourself, by repeatedly sampling each site without looking inside the bags. Our species pool consists of 10 different species, each represented by a different colour. This is a small species pool, but it is enough to meet the objectives of this tutorial. ![](Colors_meeples.png){fig-align="center" width="850"} ## Sampling procedure The aim of this exercise is to simulate a survey carried out to characterise the taxonomic diversity of six sites making up our metacommunity. Each site will be sampled multiple times to simulate surveys in a number of quadrats/locations at each site. You can start generating biodiversity data using the following procedure: 1. Divide into 6 sub-groups. 2. Each sub-group takes one bag (each bag is labelled with a number from 1 to 6). The bags have a different composition and contain a different number of species. 3. One member of each a sub-group carefully mixes the content of the bag and randomly samples 6 individuals from the bag (this represents 20% of the individuals in a bag). Try not to look inside the bag to avoid any sampling bias. 4. Write down your observations in an Excel sheet (site as rows, species as columns). 5. Once your observations have been recorded, put your sample back into the bag and give the bag to another person in your group. Then, repeat the previous steps until you have three replicates. In your data sheet, report your observations independently for each replicate (i.e., each row in your data is a replicate within a site). 6. Once you have all the replicates needed for your bag, switch bags with another group and repeat the previous steps. Do this until you have three replicates for at least four different bags. Note that all the calculations below are based on six bags, but you can also do the same calculations with four bags (you will get different answers). 7. Make sure that your data are formatted properly. ## Exercises ### Load R packages Load *tidyverse*, *knitr*, and *viridis*. ```{r} #| eval: true #| echo: true #| warning: false #| message: false library(tidyverse) library(knitr) library(viridis) ``` ### Exercise 1: Calculate the average species richness at each site Use what you learned in previous tutorials to calculate the average number of species at each site. Once again, the functions `pivot_longer()`, `group_by()`, and `summarise()` are essential for this purpose. ::: panel-tabset ## Example dataset ```{r} #| eval: true #| echo: false #| warning: false #| message: false #Simulate some data Bag1 <- rep(3,10) Bag2 <- c(0,0,0,15,0,15,0,0,0,0) Bag3 <- c(2,5,1,4,4,3,0,6,2,3) Bag4 <- c(0,7,0,0,11,0,0,5,0,7) Bag5 <- c(2,1,18,1,2,1,0,2,2,1) Bag6 <- c(6,0,0,0,0,0,17,0,5,2) data <- data.frame(Bag = rep(1:6, each=3), Replicate = rep(1:3, 6), Red=rep(NA, 18), Orange=rep(NA, 18), Yellow=rep(NA, 18), Green=rep(NA, 18), Blue=rep(NA, 18), Purple=rep(NA, 18), Pink=rep(NA, 18), White=rep(NA, 18), Grey=rep(NA, 18), Black=rep(NA, 18)) set.seed(123) for (i in 1:nrow(data)){ if (i <= 3){bag <- Bag1} if (i > 3 & i<=6){bag <- Bag2} if (i > 6 & i<=9){bag <- Bag3} if (i > 9 & i<=12){bag <- Bag4} if (i > 12 & i<=15){bag <- Bag5} if (i > 15 & i<=18){bag <- Bag6} s <- sample(c(rep("red", bag[1]), rep("orange", bag[2]), rep("yellow", bag[3]), rep("green", bag[4]), rep("blue", bag[5]), rep("purple", bag[6]), rep("pink", bag[7]), rep("white", bag[8]), rep("grey", bag[9]), rep("black", bag[10])), 6, replace=FALSE) data$Red[i] <- sum(s=="red") data$Orange[i] <- sum(s=="orange") data$Yellow[i] <- sum(s=="yellow") data$Green[i] <- sum(s=="green") data$Blue[i] <- sum(s=="blue") data$Purple[i] <- sum(s=="purple") data$Pink[i] <- sum(s=="pink") data$White[i] <- sum(s=="white") data$Grey[i] <- sum(s=="grey") data$Black[i] <- sum(s=="black")} ``` ```{r} #| eval: true #| echo: true #| warning: false #| message: false kable(head(data, n=10)) ``` ## Solution (code) ```{r} #| eval: true #| echo: true #| warning: false #| message: false table <- data |> pivot_longer(Red:Black, names_to = "Species", values_to = "Abundance") |> group_by(Bag, Replicate) |> summarise(Richness = sum(Abundance>0)) |> group_by(Bag) |> summarise(Avg_richness = round(mean(Richness), 1)) ``` ## Solution (table) ```{r} #| eval: true #| echo: true #| warning: false #| message: false kable(table) ``` ::: ### Exercise 2: Calculate alpha, beta and gamma diversity **Alpha diversity** (within-habitat diversity) is the average number of species across sites in our hypothetical metacommunity. **Gamma diversity** (regional diversity) is the total number of species present in our hypothetical metacommunity. **Beta diversity** (between-habitat diversity differentiation) is gamma diversity divided by alpha diversity. Using the data collected earlier, calculate alpha, beta and gamma diversity in our hypothetical metacommunity. You can do this manually, but we also want you to be able to do this in R! The functions `pivot_longer()`, `group_by()`, and `summarise()` are essential for this purpose. First, import your data into R. Then, use knowledge from tutorials 1 and 2 to write R code to calculate alpha, beta, and gamma diversity in our metacommunity. ::: panel-tabset ## Solution (gamma) ```{r} #| eval: true #| echo: true #| warning: false #| message: false gamma <- as.numeric(data |> pivot_longer(Red:Black, names_to = "Species", values_to = "Abundance") |> group_by(Species) |> summarise(Abundance = sum(Abundance)) |> summarise(Gamma = sum(Abundance > 0))) ``` Gamma diversity: `r gamma` ## Solution (alpha) ```{r} #| eval: true #| echo: true #| warning: false #| message: false alpha <- round(as.numeric(data |> pivot_longer(Red:Black, names_to = "Species", values_to = "Abundance") |> group_by(Bag, Species) |> summarise(Abundance = sum(Abundance)) |> group_by(Bag) |> summarise(Alpha = sum(Abundance > 0)) |> summarise(Alpha = mean(Alpha))), 1) ``` Alpha diversity: `r alpha` ## Solution (beta) ```{r} #| eval: true #| echo: true #| warning: false #| message: false beta <- round(gamma/alpha, 1) ``` Beta diversity: `r beta` ::: ### Exercise 3: Calculate the effective number of species (Hill numbers) In a community consisting of *n* species, the Hill number (or effective number of species) of order *q* can be calculated using @eq-hill-1 ($q \geq 0$, $q\neq1$) and @eq-hill-2 ($q=1$), where *p~i~* is the relative abundance of species *i* in the community ($p_i \in [0,1]$). $$ ^qD = (\sum_{i=1}^{n}p_i^q)^{\frac{1}{1-q}} $$ {#eq-hill-1} $$ ^1D=exp(-\sum_{i=1}^{n}p_i\log p_i) $$ {#eq-hill-2} Use @eq-hill-1 and @eq-hill-2 to calculate the effective number of species of order 0 (species richness), 1 (Hill-Shannon, the exponential of Shannon's entropy) and 2 (Hill-Simpson, the inverse of Simpson's concentration index) at each site of our hypothetical metacommunity. Plot your results in a figure with 6 panels (one panel for each site). For each site, plot the effective number of species as a function of *q* (this is called a diversity profile plot). Plot separate lines (using `geom_line()`) for each replicate (the `group` argument in `aes()` will be useful here). What do you observe? What can you conclude about the taxonomic diversity at each site? ::: panel-tabset ## Solution (code) Your code should include the following steps (use the pipe!): 1. Create a new variable (called "Total") by summing the number of individuals of each species sampled for each replicate. Use `mutate()` to do this. 2. Transform all absolute species abundance values into relative abundance values by dividing the number of individuals of a species by the total number of individuals ("Total") sampled for each replicate. You can also use `mutate()` to do this. 3. Use `pivot_longer()` to reorganise the dataset and create a column called "Species" to store all the species names (e.g. Red, Orange, etc.) and a column called "Abundance" to store the relative abundance values of the species in the communities. 4. Filter the data to eliminate all rows where the relative abundance of a species is zero. You can use `filter()` to do this. 5. For each replicate, use @eq-hill-1 and @eq-hill-2 to calculate the effective number of species at q=0 (name this variable "q0"), q=1 (name this variable "q1"), and q=2 (name this variable "q2"). You can do this using `group_by()` and `summarise()`. 6. Use `pivot_longer()` to reorganise the dataset and create a column called "q" to store all diversity order values (0, 1, and 2) and a column called "diversity" to store the effective number of species in each community. In `pivot_longer()`, you will need to use two extra arguments: `names_prefix = "q"` (this will make sure to remove the letter "q" from "q0", "q1", and "q2") and `names_transform = as.numeric` (to convert "0", "1" and "2" into numeric values). 7. Finally, transform the contents of the "Bag" column by adding the string "Bag" (do not forget the space!) in front of the bag number (for example, converting 1 to "Bag 1"), and ensure that the Replicate variable is considered a factor. You can perform all these transformations using `mutate()`, `paste()` and `as.factor()`. 8. Finally, use *gggplot2* to plot your results in a figure with 6 panels (one panel for each site). For each site (1 site = 1 bag), plot the effective number of species as a function of *q* (this is called a diversity profile plot). Plot separate lines (using `geom_line()`) for each replicate (the `group` argument in `aes()` will be useful here). Personalise your plot to make it look nice! Be sure to apply the data visualisation principles discussed in the last lecture and tutorial. ```{r} #| eval: true #| echo: true #| warning: false #| message: false #Organise data to create the plot data_plot <- data |> mutate(Total = Red+Orange+Yellow+Green+Blue+Purple+Pink+White+Grey+Black) |> mutate(Red=Red/Total, Orange=Orange/Total, Yellow=Yellow/Total, Green=Green/Total, Blue=Blue/Total, Purple=Purple/Total, Pink=Pink/Total, White=White/Total, Grey=Grey/Total, Black=Black/Total) |> pivot_longer(Red:Black, names_to = "Species", values_to = "Abundance") |> filter(Abundance!=0) |> group_by(Bag, Replicate) |> summarise(q0=sum(Abundance^0)^(1/(1-0)), q1=exp(-sum(Abundance*log(Abundance))), q2=sum(Abundance^2)^(1/(1-2))) |> pivot_longer(q0:q2, names_to = "q", names_prefix = "q", names_transform = as.numeric, values_to = "diversity") |> mutate(Bag=paste("Bag ", Bag, sep=""), Replicate=as.factor(Replicate)) #Plot data plot <- ggplot(data_plot, aes(x = q, y = diversity, group = Replicate, colour=Replicate))+ geom_line()+ geom_point(shape=16)+ facet_wrap(~Bag)+ theme_bw()+ xlab("Diversity order (q)")+ ylab("Effective number of species (D)")+ scale_colour_viridis(discrete=TRUE)+ theme(axis.title.x = element_text(margin = margin(t=10)), axis.title.y = element_text(margin = margin(r=10)), axis.text = element_text(colour="black"), panel.grid.minor = element_blank())+ scale_x_continuous(breaks=c(0,1,2), limits = c(0,2))+ scale_y_continuous(breaks=seq(from = 0, to = 10, by = 2), limits=c(0,10)) ``` ## Solution (plot) ```{r} #| eval: true #| echo: true #| warning: false #| message: false #| fig-align: center plot ``` ::: ## Suggested further reading Our tutorial can't cover everything, and there's still a lot to learn about quantifying biodiversity! If you are interested, take a look at these excellent papers. **Chao A, Chiu C-H, Jost L**. **2014**. [Unifying Species Diversity, Phylogenetic Diversity, Functional Diversity, and Related Similarity and Differentiation Measures Through Hill Numbers.](https://www.annualreviews.org/content/journals/10.1146/annurev-ecolsys-120213-091540) *Annual review of ecology, evolution, and systematics* **45**: 297--324. **Hill MO**. **1973**. [Diversity and evenness: A unifying notation and its consequences](https://www.jstor.org/stable/1934352?seq=1). *Ecology* **54**: 427--432. **Roswell M, Dushoff J, Winfree R**. **2021**. [A conceptual guide to measuring species diversity](https://nsojournals.onlinelibrary.wiley.com/doi/full/10.1111/oik.07202). *Oikos* **130**: 321--338. **Whittaker RH**. **1960**. [Vegetation of the siskiyou mountains, Oregon and California](https://www.jstor.org/stable/1943563). *Ecological monographs* **30**: 279--338.