Author: Martina Sánchez-Pinillos

# Ecological Dynamic Regime framework

#### Martina Sánchez-Pinillos

## 1. Introduction

### 1.1. Ecological Dynamic Regimes and the EDR framework

An Ecological Dynamic Regime (EDR) is defined as the “fluctuations of ecosystem states around some trend or average resulting from an intricate mix of internal processes and external forces that, in the absence of perturbations, keep the system within specific domains of attraction” (Sánchez-Pinillos et al., 2023).

EDRs are composed of multiple ecological trajectories showing similar processes in the development, interaction, and reorganization of some state variables over time. Thus, ecological trajectories are the essential units of EDRs and can be defined as sequences of observations capturing the temporal changes of some state variables. Usually, ecological trajectories are defined based on multiple state variables (e.g., species, functional traits, land uses) that make challenging the characterization and comparison of EDRs.

The EDR framework is a set of algorithms and metrics aiming to characterize and compare EDRs composed of ecological trajectories in multidimensional state spaces. You can find more information in the following publication, including the formal definition of the EDR concept, the methodological details of the framework, and some illustrative examples with artificial and real data sets:

- Sánchez-Pinillos M., Kéfi, S., De Cáceres, M., Dakos, V. 2023. Ecological Dynamic Regimes: Identification, characterization, and comparison. Ecological Monographs. https://doi.org/10.1002/ecm.1589

### 1.2. About this vignette

This vignette aims to give you an overview of the EDR framework and its implementation in `ecoregime` to characterize and compare EDRs.

You can install `ecoregime` directly from CRAN or from my GitHub account (development version):

    # install.packages("ecoregime")
    # devtools::install_github(repo = "MSPinillos/ecoregime", dependencies = T, build_vignettes = T)

Once you have installed `ecoregime` you will have to load it:

    library(ecoregime)

    citation("ecoregime")
    #> To cite 'ecoregime' in publications use:
    #>
    #>   Sánchez-Pinillos M, Kéfi S, De Cáceres M, Dakos V (2023). "Ecological
    #>   dynamic regimes: Identification, characterization, and comparison."
    #>   _Ecological Monographs_, e1589. <https://doi.org/10.1002/ecm.1589>.
    #>
    #>   Sánchez-Pinillos M, Dakos V, Kéfi S (2024). "Ecological dynamic
    #>   regimes: A key concept for assessing ecological resilience."
    #>   _Biological Conservation_, 110409.
    #>   <https://doi.org/10.1016/j.biocon.2023.110409>.
    #>
    #>   Sánchez-Pinillos M (2023). _ecoregime: Analysis of Ecological Dynamic
    #>   Regimes_. <https://doi.org/10.5281/zenodo.7584943>.
    #>
    #> To see these entries in BibTeX format, use 'print(<citation>,
    #> bibtex=TRUE)', 'toBibtex(.)', or set
    #> 'options(citation.bibtex.max=999)'.

## 2. Trajectory data, state and trajectory spaces

If you are familiar with ecological trajectories defined in multivariate spaces, you can probably skip this section. Otherwise, I recommend you to read on.

Let me define a simple example that I will use along this vignette. Imagine that we have 10 sampling units in which we are monitoring the temporal changes in some state variables. For example, those sampling units (A, B, …, J) could be permanent plots or transects in which we have inventoried the abundance of two species (sp1, sp2) at three temporal points (t = 1, t = 2, t = 3).

I will use the Lotka-Volterra model to generate some artificial data and illustrate our example. I will define the parameters of the model to generate trajectories leading to the competitive exclusion of sp2 by sp1 regardless of the initial state. As such, all sampling units will have analogous trajectories that can be considered part of the same EDR.

    # ID of the sampling units and observations
    ID_sampling <- LETTERS[1:10]
    ID_obs <- 1:3

    # Define initial species abundances
    set.seed(123)
    initial <- data.frame(sampling_units = ID_sampling,
                       sp1 = round(runif(10), 2),
                       sp2 = round(runif(10), 2))

    # Define the parameters for the Lotka-Volterra model
    parms <- c(r1 = 1, r2 = 0.1, a11 = 0.02, a21 = 0.03, a22 = 0.02, a12 = 0.01)

    # We can use primer and deSolve to run the simulations
    library(primer)
    #> Loading required package: deSolve
    #> Loading required package: ggplot2
    simulated_abun <- lapply(1:nrow(initial), function(isampling){
      # Initial abundance in the sampling unit i
      initialN <- c(initial$sp1[isampling], initial$sp2[isampling])
      # Simulate community dynamics
      simulated_abun <- data.frame(ode(y = initialN, times = ID_obs, func = lvcomp2, parms = parms))
      # Add the names of the sampling units
      simulated_abun$sampling_unit <- ID_sampling[isampling]
      # Calculate relative abundances
      simulated_abun$sp1 <- simulated_abun$X1/rowSums(simulated_abun[, 2:3])
      simulated_abun$sp2 <- simulated_abun$X2/rowSums(simulated_abun[, 2:3])

      return(simulated_abun[, c("sampling_unit", "time", "sp1", "sp2")])
    })

    # Compile species abundances of all sampling units in the same data frame
    abundance <- do.call(rbind, simulated_abun)

We obtain an abundance matrix similar to what we could get from field data. For each sampling unit (`sampling_unit`) and observation (`time`), we know the relative abundance of two species (`sp1` and `sp2`).

    head(abundance)
    #>   sampling_unit time       sp1        sp2
    #> 1             A    1 0.2320000 0.76800000
    #> 2             A    2 0.4222631 0.57773693
    #> 3             A    3 0.6354107 0.36458929
    #> 4             B    1 0.6370968 0.36290323
    #> 5             B    2 0.8078814 0.19211857
    #> 6             B    3 0.9066680 0.09333198

Next, we need to define the state space in which ecological trajectories live. The state space is a multidimensional resemblance space defined by the dissimilarities between every pair of states (the observations) in terms of some state variables (the species). That is, we need to generate a matrix (\(D_O\)) representing the state space and containing the dissimilarities between the observations of all sampling units.

The choice of the dissimilarity metric used to generate \(D_O\) depends on the characteristics of your data and the nature of the variables. Your decision should be made carefully since it will affect the analyses in the EDR framework. In our example, we are using species abundances. A common dissimilarity metric adequate for that is the percentage difference (a.k.a. Bray-Curtis index). However, if you have presence/absence data, you could prefer the Jaccard’s distance, and if your state variables are categorical (e.g., functional traits), you should use a multi-trait dissimilarity (e.g., the Gower distance).

In our example, we will use `vegan` to calculate Bray-Curtis dissimilarities and generate \(D_O\).

    # Generate a matrix containing dissimilarities between states
    state_dissim <- vegan::vegdist(abundance[, c("sp1", "sp2")], method = "bray")
    as.matrix(state_dissim)[1:6, 1:6]
    #>           1         2           3           4          5          6
    #> 1 0.0000000 0.1902631 0.403410706 0.405096774 0.57588143 0.67466802
    #> 2 0.1902631 0.0000000 0.213147639 0.214833707 0.38561836 0.48440495
    #> 3 0.4034107 0.2131476 0.000000000 0.001686068 0.17247072 0.27125731
    #> 4 0.4050968 0.2148337 0.001686068 0.000000000 0.17078465 0.26957124
    #> 5 0.5758814 0.3856184 0.172470722 0.170784654 0.00000000 0.09878659
    #> 6 0.6746680 0.4844050 0.271257312 0.269571243 0.09878659 0.00000000

We can visualize the state space using ordination methods. Here, I apply multidimensional scaling (mMDS; `smacof`) to the dissimilarity matrix \(D_O\) and plot the distribution of the ecological states in an ordination space. We will use different colors and the code of the sampling unit to represent the observations of each sampling unit:

    # Multidimensional scaling
    state_mds <- smacof::smacofSym(state_dissim, ndim = 2)
    state_mds <- data.frame(state_mds$conf)

    # Define different colors for each trajectory
    traj.colors <- RColorBrewer::brewer.pal(10, "Paired")

    # Plot the distribution of the states in the state space
    plot(state_mds$D1, state_mds$D2,
         col = rep(traj.colors, each = 3), # use different colors for each sampling unit
         pch = rep(ID_sampling, each = 3), # use different symbols for each sampling unit
         xlab = "Axis 1", ylab = "Axis 2",
         main = "State space")

To see how ecological trajectories are distributed in the state space, we just need to link the states of each sampling unit in chronological order.

    ## -->> This code shows you the process step by step. You could directly use
    ## -->> ecotraj::trajectoryPlot()

    # Plot the distribution of the states in the state space
    plot(state_mds$D1, state_mds$D2,
         col = rep(traj.colors, each = 3), # use different colors for each sampling unit
         pch = rep(ID_sampling, each = 3), # use different symbols for each sampling unit
         xlab = "Axis 1", ylab = "Axis 2",
         main = "State space")

    # Link trajectory states in chronological order
    for (isampling in seq(1, 30, 3)) {
      # From observation 1 to observation 2
      shape::Arrows(state_mds[isampling, 1], state_mds[isampling, 2],
             state_mds[isampling + 1, 1], state_mds[isampling + 1, 2],
             col = traj.colors[1+(isampling-1)/3], arr.adj = 1)
      # From observation 2 to observation 3
      shape::Arrows(state_mds[isampling + 1, 1], state_mds[isampling + 1, 2],
             state_mds[isampling + 2, 1], state_mds[isampling + 2, 2],
             col = traj.colors[1+(isampling-1)/3], arr.adj = 1)
    }

The trajectory space is a multidimensional space defined by the dissimilarities between trajectories (i.e., the whole sequence of states for each sampling unit). Analogously to the state space, you need to generate a dissimilarity matrix (\(D_T\)) and the metric that you use will affect the subsequent analyses. Although there are multiple metrics, there are no formal analyses evaluating their performance in ecological applications. Here, I suggest the directed segment path dissimilarity (De Cáceres et al., 2019, Ecol. Monogr., https://doi.org/10.1002/ecm.1350), but you could use any other dissimilarity metric.

    # Generate a matrix containing dissimilarities between trajectories
    traj_dissim <- ecotraj::trajectoryDistances(state_dissim,
                                                sites = rep(ID_sampling, each = 3),
                                                surveys = rep(ID_obs, 10), distance.type = "DSPD")

    as.matrix(traj_dissim)[1:6, 1:6]
    #>           A          B         C          D         E         F
    #> A 0.0000000 0.30509564 0.1084426 0.27711311 0.5425424 0.2600516
    #> B 0.3050956 0.00000000 0.1409823 0.02576483 0.1520544 0.6244592
    #> C 0.1084426 0.14098231 0.0000000 0.11299979 0.3784291 0.4231264
    #> D 0.2771131 0.02576483 0.1129998 0.00000000 0.1756016 0.6031298
    #> E 0.5425424 0.15205442 0.3784291 0.17560156 0.0000000 0.8146293
    #> F 0.2600516 0.62445921 0.4231264 0.60312977 0.8146293 0.0000000

In contrast to the state space, in the trajectory space, each trajectory is represented by one point:

    # Multidimensional scaling
    traj_mds <- smacof::smacofSym(traj_dissim, ndim = 2)
    traj_mds <- data.frame(traj_mds$conf)

    # Plot the distribution of the trajectories in the trajectory space
    plot(traj_mds$D1, traj_mds$D2,
         col = traj.colors, # use different colors for each sampling unit
         pch = ID_sampling, # use different symbols for each sampling unit
         xlab = "Axis 1", ylab = "Axis 2",
         main = "Trajectory space")

## 3. Defining EDRs

### 3.1. Identifying EDRs from clustering analyses

Identifying EDRs involves finding groups of ecological trajectories showing dynamic patterns more similar between each other than with any other ecological trajectory in the trajectory space. In this sense, it is possible to use the trajectory dissimilarity matrix (\(D_T\)) to identify EDRs through clustering. However, applying clustering analyses to trajectory data is not always straightforward. Depending on the characteristics of your data, your goals, and the system that you are evaluating, the choice of the clustering algorithm may have important consequences.

Currently, this is out of the scope of `ecoregime` but you can find more details in the paper introducing the EDR framework (Sánchez-Pinillos et al., 2023)

### 3.2. Defining EDRs based on ecological properties

Clustering analyses are not the only way to define EDRs. As I stated before, the ecological trajectories of our simulated data can be considered part of the same EDR for having similar dynamic patterns. Thus, you could define groups of trajectories that you want to evaluate for sharing similar aspects in the state variables. For example, you could consider groups of trajectories with the same number of observations in which the state variables change following similar patterns.

## 4. Representative trajectories

Regardless of the method used to define EDRs, you can use `ecoregime` to summarize their dynamics. In our example, it is easy to identify a clear pattern of competitive exclusion of sp2 by sp1. We can simply plot the changes in species abundances over time to see this pattern:

    # Plot species abundances over time
    plot(abundance$time, abundance$sp1, type = "n", ylim = c(-0.01, 1),
         xlab = "Observation", ylab = "Species abundance",
         main = "Temporal variation in species composition")

    for (i in seq_along(ID_sampling)) {
      points(abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$time,
             abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$sp1,
             col = traj.colors[i], pch = ID_sampling[i])
      lines(abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$time,
             abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$sp1,
             col = "black", pch = ID_sampling[i])

      points(abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$time,
             abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$sp2,
             col = traj.colors[i], pch = ID_sampling[i])
      lines(abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$time,
             abundance[which(abundance$sampling_unit == ID_sampling[i]), ]$sp2,
             col = "red", pch = ID_sampling[i], lty = 2)
    }

    legend("bottomleft", legend = c("sp1", "sp2"), col = 1:2, lty = 1:2, bg = "white")

Or we can test the correlations between the first axis of the state space and the species abundances along the trajectories.

    # Correlation of the first axis of the state space with the abundance of sp1
    cor(state_mds$D1, abundance$sp1)
    #> [1] 1

    # Correlation of the first axis of the state space with the abundance of sp2
    cor(state_mds$D1, abundance$sp2)
    #> [1] -1

These analyses are fine for extremely simple systems as the one simulated in our toy data. However, when we want to apply these or similar approaches to empirical data, finding clear dynamical patterns becomes more difficult. In real ecosystems, we usually have to deal with a great heterogeneity, multiple variables interacting between each other, and noise. As a consequence, we need robust tools able to summarize a great variety of trajectories in multidimensional spaces. That is precisely what RETRA-EDR pursues.

### 4.1. Identifying representative trajectories with RETRA-EDR

RETRA-EDR (REpresentative TRAjectories in Ecological Dynamic Regimes) is an algorithm that aims to identify representative trajectories in EDRs based on the distribution and density of their ecological trajectories (Sánchez-Pinillos et al., 2023). Whereas many algorithms with similar goals where developed for moving objects in low-dimensional spaces, RETRA-EDR can be applied to high-dimensional spaces, capturing the complexity of empirical data.

I will not explain the details of RETRA-EDR in this vignette (you can find them in Sánchez-Pinillos et al., 2023), but it is important that you know its main steps to understand the output returned by `retra_edr()`:

1. The ecological trajectories forming the EDR are split into segments. Then, a segment space analogous to the trajectory space is generated from a matrix containing segment dissimilarities.
2. The segment space is divided into regions with a minimum number of trajectory segments (`minSegs`). For that, RETRA-EDR recursively divides the space into halves following the axes of the segment space and generating a kd-tree structure. The greater the density of segments in a particular region, the larger the number of partitions required until obtaining regions with `minSegs` segments. The number of partitions is known as the depth of the kd-tree (`kdTree_depth`) and the number of segments in the final region is the `Density`.
3. For each region with at least `minSegs` segments, RETRA-EDR extracts the medoid as the representative segment.
4. All representative segments are joined forming a network of representative trajectories.
5. RETRA-EDR returns a set of attributes characterizing the representative trajectories identified in the EDR, including the value of `minSegs`; the sequence of representative segments (`Segments`) and the number of states (`Size`) forming each representative trajectory; the total length of the trajectory (`Length`); the links (`Link`) generated to join representative segments and the dissimilarity between the connected states (`Distance`); the number of segments in each region represented by each representative segment (`Density`), and the number of partitions of the segment space until obtaining a region with `minSegs` segments or less (`kdTree_depth`).

Let’s apply `retra_edr()` to our data. As the EDR only contains 10 trajectories, we will assign a value of `minSegs` relatively low.

    # Use set.seed to obtain reproducible results of the segment space in RETRA-EDR
    set.seed(123)

    # Apply RETRA-EDR
    repr_traj <- retra_edr(d = state_dissim, trajectories = rep(ID_sampling, each = 3),
                       states = rep(ID_obs, 10), minSegs = 2)

RETRA-EDR has returned two representative trajectories. We can summarize their attributes using `summary()`:

    summary(repr_traj)
    #>    ID Size    Length   Avg_link   Sum_link Avg_density Max_density Avg_depth
    #> T1 T1    6 0.3577606 0.02996245 0.05992491           3           3  3.666667
    #> T2 T2    7 0.6048257 0.03925918 0.07851836           3           3  3.500000
    #>    Max_depth
    #> T1         4
    #> T2         4

The output of `retra_edr()` is an object of class `RETRA`. We can extract specific attributes as we do with lists.

`Segments` is probably the most important attribute since it informs about the states that form the representative trajectories.

- T1 is composed of six states of three different trajectories (B, I, and E).
- T2 is composed of the three states of the sampling unit C and converges towards T1 through a link between the last state of C and the second state of I.

    lapply(repr_traj, "[[", "Segments")
    #> $T1
    #> [1] "B[1-2]" "I[2-3]" "E[2-3]"
    #>
    #> $T2
    #> [1] "C[1-2]" "C[2-3]" "I[2-3]" "E[2-3]"

### 4.2. Visualizing representative trajectories in the state space

We can also visualize the representative trajectories in the state space using the function `plot()`.

    # Plot the representative trajectories of an EDR
    plot(repr_traj, # <-- This is a RETRA object returned by retra_edr()
         # data to generate the state space
         d = state_mds, trajectories = rep(ID_sampling, each = 3), states = rep(ID_obs, 10),
         # use the colors previously used for individual trajectories.
         # (make them more transparent to highlight representative trajectories)
         traj.colors = alpha(traj.colors, 0.3),
         # display representative trajectories in blue
         RT.colors = "blue",
         # select T2 to be displayed with a different color (black)
         select_RT = "T2", sel.color = "black",
         # Identify artificial links using dashed lines (default) and a different color (red)
         link.lty = 2, link.color = "red",
         # We can use other arguments in plot()
         main = "Representative trajectories")

    # Add a legend
    legend("bottomright", c("T1", "T2", "Link"),
           col = c("blue", "black", "red"), lty = c(1, 1, 2))

### 4.3. Defining representative trajectories based on trajectory features

RETRA-EDR returns the longest possible trajectory according to the procedure defined before. However, you could be interested in only a fraction of a representative trajectory, split representative trajectories based on some criteria (e.g. long artificial links), or maybe, define your own representative trajectory.

Let’s say that we want to split T1 and T2 from the last segment (`"E[2-3]"`). For that, we can use the function `define_retra()`.

There are two ways of defining our “new” trajectories. The first one consists in providing all the details in a data frame:

    # Generate a data frame indicating the states forming the new trajectories
    new_traj_df <- data.frame(
      # name of the new trajectories (as many times as the number of states)
      RT = c(rep("T1.1", 4), rep("T2.1", 5), rep("T1.2", 2)),
      # name of the trajectories (sampling units)
      RT_traj = c(rep("B", 2), rep("I", 2), # for the first trajectory (T1.1)
                  rep("C", 3), rep("I", 2), # for the second trajectory (T2.1)
                  rep("E", 2)), # for the third trajectory (T1.2)
      # states in each sampling unit
      RT_states = c(1:2, 2:3, # for the first trajectory (T1.1)
                    1:3, 2:3, # for the second trajectory (T2.1)
                    2:3), # for the third trajectory (T1.2)
      # representative trajectories obtained in retra_edr()
      RT_retra = c(rep("T1", 4), rep("T2", 5),
                   rep("T1", 2)) # The last segment belong to both (T1, T2), choose one
    )

    new_traj_df
    #>      RT RT_traj RT_states RT_retra
    #> 1  T1.1       B         1       T1
    #> 2  T1.1       B         2       T1
    #> 3  T1.1       I         2       T1
    #> 4  T1.1       I         3       T1
    #> 5  T2.1       C         1       T2
    #> 6  T2.1       C         2       T2
    #> 7  T2.1       C         3       T2
    #> 8  T2.1       I         2       T2
    #> 9  T2.1       I         3       T2
    #> 10 T1.2       E         2       T1
    #> 11 T1.2       E         3       T1

The second one consists in providing a list with the sequences of segments defining each trajectory. This is particularly useful when we want to select a subset in the output of RETRA-EDR:

    # List including the sequence of segments for each new trajectory
    new_traj_ls <- list(
      # First part of T1 excluding the last segment
      repr_traj$T1$Segments[1:(length(repr_traj$T1$Segments) - 1)],
      # First part of T2 excluding the last segment
      repr_traj$T2$Segments[1:(length(repr_traj$T2$Segments) - 1)],
      # Last segment of T1 and T2: segment composed of states 2 and 3 of the sampling unit E ("E[2-3]")
      "E[2-3]"
    )

    new_traj_ls
    #> [[1]]
    #> [1] "B[1-2]" "I[2-3]"
    #>
    #> [[2]]
    #> [1] "C[1-2]" "C[2-3]" "I[2-3]"
    #>
    #> [[3]]
    #> [1] "E[2-3]"

In any case, `define_retra()` will return the same output

    # Define representative trajectories using a data frame
    new_repr_traj <- define_retra(data = new_traj_df,
                 # Information of the state space
                 d = state_dissim, trajectories = rep(ID_sampling, each = 3),
                 states = rep(ID_obs, 10),
                 # RETRA object returned by retra_edr()
                 retra = repr_traj)

    # Define representative trajectories using a list with sequences of segments
    new_repr_traj_ls <- define_retra(data = new_traj_ls,
                 # Information of the state space
                 d = state_dissim, trajectories = rep(ID_sampling, each = 3),
                 states = rep(ID_obs, 10),
                 # RETRA object returned by retra_edr()
                 retra = repr_traj)

    if (all.equal(new_repr_traj, new_repr_traj_ls)) {
      print("Yes, both are equal!")
    }
    #> [1] "Yes, both are equal!"

`define_retra()` returns an object of class `RETRA` that you can use to plot the new trajectories:

    plot(new_repr_traj, # <-- This is the RETRA object returned by define_retra()
         # data to generate the state space
         d = state_mds, trajectories = rep(ID_sampling, each = 3), states = rep(ID_obs, 10),
         # display individual trajectories in light blue
         traj.colors = "lightblue",
         # display representative trajectories in dark blue
         RT.colors = "darkblue",
         # select T1.2 to be displayed in a different color (red)
         select_RT = "T1.2", sel.color = "red",
         # Identify artificial links using dashed lines (default), but use the same
         # color than the representative trajectories (default)
         link.lty = 2, link.color = NULL,
         # We can use other arguments in plot()
         main = "Defined representative trajectories")

    # Add a legend
    legend("bottomright", c("T1.1, T2.1", "T1.2", "Link"),
           col = c("darkblue", "red", "darkblue"), lty = c(1, 1, 2))

### 4.4. Variation of ecological properties along representative trajectories

The definition of representative trajectories is based on the real states of the sampling units. Thus, it is possible to assess the variation of some ecological properties along the representative trajectories, and therefore, across the EDR.

Let’s see how species diversity varies along the representative trajectories returned by `retra_edr()`. First, we need to calculate an index of species diversity (e.g., Shannon index) for the states in each representative trajectory and set the order of the states in the trajectory:

    # Set an ID in the abundance matrix
    abundance$ID <- paste0(abundance$sampling_unit, abundance$time)

    # Calculate the Shannon index in all trajectory states
    abundance$Shannon <- vegan::diversity(abundance[, c("sp1", "sp2")], index = "shannon")

    # Identify the states forming both representative trajectories
    traj_states <- lapply(repr_traj, function(iRT){
      segments <- iRT$Segments
      seg_components <- strsplit(gsub("\\]", "", gsub("\\[", "-", segments)), "-")
      traj_states <- vapply(seg_components, function(iseg){
        c(paste0(iseg[1], iseg[2]), paste0(iseg[1], iseg[3]))
      }, character(2))
      traj_states <- unique(as.character(traj_states))
      traj_states <- data.frame(ID = traj_states, RT_states = 1:length(traj_states))
    })

    # Associate the states of the representative trajectories with their corresponding
    # value of the Shannnon index
    RT_diversity <- lapply(traj_states, function(iRT){
      data <- merge(iRT, abundance, by = "ID", all.x = T)
      data <- data[order(data$RT_states), ]
    })

    RT_diversity$T2
    #>   ID RT_states sampling_unit time       sp1        sp2    Shannon
    #> 1 C1         1             C    1 0.3761468 0.62385321 0.66214624
    #> 2 C2         2             C    2 0.5930319 0.40696810 0.67573602
    #> 3 C3         3             C    3 0.7754031 0.22459689 0.53266482
    #> 6 I2         4             I    2 0.8009390 0.19906100 0.49909794
    #> 7 I3         5             I    3 0.9044378 0.09556219 0.31522122
    #> 4 E2         6             E    2 0.9574203 0.04257971 0.17605768
    #> 5 E3         7             E    3 0.9809725 0.01902749 0.09422978

Then, we can evaluate the variation of species diversity along the representative trajectories graphically:

    # Plot the variation of species diversity in T1
    plot(x = RT_diversity$T1$RT_states + 1, y = RT_diversity$T1$Shannon,
         type = "l", col = "blue", xlim = c(1, 7), ylim = c(0, 0.7),
         xlab = "RT state", ylab = "Shannon index",
         main = "Variation of species diversity")
    # Add the variation of species diversity in T2
    lines(x = RT_diversity$T2$RT_states, y = RT_diversity$T2$Shannon,
          col = "black")
    # Add a legend
    legend("topright", c("T1", "T2"), col = c("blue", "black"), lty = 1)

## 5. Distribution and heterogeneity of ecological trajectories in EDRs

Representative trajectories aim to summarize the main dynamic patterns of ecological systems in a given EDR. As a complement, you can compute additional metrics to understand the internal structure of the EDR.

### 5.1. Dynamic dispersion (dDis)

Dynamic dispersion (dDis) is calculated as the average dissimilarity between the trajectories in an EDR and another trajectory taken as a reference (Sánchez-Pinillos et al., 2023).

If the trajectory taken as the reference is one of the representative trajectories (e.g. T2), dDis can be used as an indicator of its fitness to the data and as an overall metric of the dispersion of the trajectories in the EDR.

First, you need to prepare the data. Even if the states of the representative trajectories are already in your abundance and dissimilarity matrices, you will need to duplicate them to indicate that this is a different trajectory. We can directly use the data frame that we produced to calculate species diversity.

    # Change the trajectory identifier and the number of the state
    abundance_T2 <- RT_diversity$T2
    abundance_T2$sampling_unit <- "T2"
    abundance_T2$time <- abundance_T2$RT_states

    # Add representative trajectories' data to the abundance matrix
    abundance_T2 <- rbind(abundance, abundance_T2[, names(abundance)])

    # Calculate state dissimilarities including the representative trajectory
    state_dissim_T2 <- vegan::vegdist(abundance_T2[, c("sp1", "sp2")], method = "bray")

Now, we can compute dDis using `dDis()`:

    # Compute dDis taking T2 as reference
    dDis_T2 <- dDis(d = state_dissim_T2, d.type = "dStates",
             trajectories = abundance_T2$sampling_unit, states = abundance_T2$time,
             reference = "T2")
    dDis_T2
    #> dDis (ref. T2)
    #>      0.1463935

Instead of using the representative trajectory as the reference, we could use any other trajectory of the EDR to quantify how “immersed” it is in the EDR. Let’s compare dDis for trajectories of the sampling units I, which is part of both representative trajectories, and F, which was displayed relatively far from the rest of trajectories in the state and trajectory spaces.

In this case, we will use the trajectory dissimilarity matrix generated for the original data:

    # dDis: reference C
    dDis_I <- dDis(d = traj_dissim, d.type = "dTraj",
             trajectories = ID_sampling,
             reference = "I")
    # dDis: reference F
    dDis_F <- dDis(d = traj_dissim, d.type = "dTraj",
             trajectories = ID_sampling,
             reference = "F")

    dDis_I
    #> dDis (ref. I)
    #>     0.1847822
    dDis_F
    #> dDis (ref. F)
    #>     0.5809146

As expected, the dispersion when trajectory F is taken as the reference is larger than for trajectory I, indicating that F is more isolated than I within the EDR.

Alternatively, you can weight each trajectory depending on some criteria. Currently, `dDis()` is able to compute dDis weighting ecological trajectories by their size (`w.type = "size"`) and length (`w.type = "length"`). However, if you want to use different criteria, you can use a set of weights of your choice (`w.type = "precomputed"`) and indicate them through the `w.values` argument. For example, imagine that you want to weight each trajectory by the initial abundance of sp2:

    # Define w values
    initial_sp2 <- abundance[which(abundance$time == 1), ]$sp2

    # Identify the index of the reference trajectories
    ind_I <- which(ID_sampling == "I")
    ind_F <- which(ID_sampling == "F")

    # Compute dDis with weighted trajectories:
    # Considering I as reference
    dDis_I_w <- dDis(d = traj_dissim, d.type = "dTraj",
             trajectories = ID_sampling, reference = "I",
             w.type = "precomputed", w.values = initial_sp2[-ind_I])
    # Considering F as reference
    dDis_F_w <- dDis(d = traj_dissim, d.type = "dTraj",
             trajectories = ID_sampling, reference = "F",
             w.type = "precomputed", w.values = initial_sp2[-ind_F])
    dDis_I_w
    #> dDis (ref. I)
    #>     0.2517555
    dDis_F_w
    #> dDis (ref. F)
    #>     0.4742493

In comparison with the previous results, dDis has increased and decreased for I and F, respectively, as a result of the greater abundance of sp2 in F than in I. This result could be interpreted as a lower isolation of F in relation to the trajectories initially dominated by sp2.

Weighting ecological trajectories is equivalent to modifying the location of the trajectory taken as the reference. Therefore, I do not recommend you to weight trajectories if the representative trajectory is the reference of dDis because it could make difficult the interpretation of the results.

### 5.2. Dynamic beta diversity (dBD)

Dynamic beta diversity (dBD) was proposed by De Cáceres et al. (2019; Ecol. Monogr., https://doi.org/10.1002/ecm.1350) to quantify the overall variability of a group of trajectories. Conceptually, it measures the average distance to the centroid of the trajectories. Therefore, it is quite similar to dDis when it is calculated taking a representative trajectory as the reference.

dBD is implemented in the function `dBD()`. As you can imagine, dBD in our EDR is very small because the dynamical patterns of all trajectories are very similar.

    # Calculate dBD
    dBD(d = state_dissim, d.type = "dStates",
        trajectories = rep(ID_sampling, each = 3), states = rep(ID_obs, 10))
    #>        dBD
    #> 0.06252172

### 5.3. Dynamic evenness (dEve)

Dynamic evenness (dEve) quantifies the regularity with which an EDR is filled by ecological trajectories (Sánchez-Pinillos et al., 2023). This metric informs about the existence of groups of trajectories within the EDR.

You can compute dEve with the function `dEve()`:

    # Calculate dEve
    dEve(d = traj_dissim, d.type = "dTraj", trajectories = ID_sampling)
    #>      dEve
    #> 0.5827095

Like dDis, dEve can also be calculated using different weights for the individual trajectories of the EDR. In that case, dEve is equivalent to the functional evenness index proposed by Villéger et al. (2003; Ecology, https://doi.org/10.1890/07-1206.1).

For example, weighting ecological trajectories by the initial abundance of sp2 is equivalent to reducing the dissimilarity of the trajectories dominated by sp2 and dEve increases.

    # Calculate dEve weighting trajectories by the initial abundance of sp2
    dEve(d = traj_dissim, d.type = "dTraj", trajectories = ID_sampling,
             w.type = "precomputed", w.values = initial_sp2)
    #>      dEve
    #> 0.6296034

## 6. Comparing EDRs

Representative trajectories and the metrics of dynamic dispersion, beta diversity, and evenness will allow you to characterize EDRs based on their dynamic patterns and the distribution of ecological trajectories. Therefore, you can compare two or more EDRs based on these features, saying “this EDR is dominated by these dynamics, whereas the dynamics in this other EDR are different” or “the trajectories in this EDR are more diverse than in this other EDR”. But what if you want to quantify how different two EDRs are? This is a common situation. Perhaps, you want to compare EDRs of ecosystems under different environmental conditions, for example, forest communities dominated by the same species that grow under different climatic conditions. Perhaps, you want to use simulated data to assess the effects of the introduction of certain species in the EDRs.

Of course, comparing EDR characteristics will provide you with very valuable information, but if you want “a number”, you will have to calculate the dissimilarities between EDRs. You can do it using `ecoregime`.

Let’s generate two more EDRs to be compared with the previous one.

We will use the initial states of our first EDR to define the abundances of sp1 and sp2, but we will introduce a third species (sp3).

    # ID of the sampling units for EDR2
    ID_sampling2 <- paste0("inv_", LETTERS[1:10])

    # Define initial species abundances for sp3
    set.seed(321)
    initial$sp3 <- round(runif(10, 0, 0.5), 2)

    # Define the parameters for the Lotka-Volterra model
    parms2 <- c(r1 = 1, r2 = 0.1, a11 = 0.02, a21 = 0.03, a22 = 0.02, a12 = 0.01,
               r3 = 1.5, a33 = 0.02, a31 = 0.01, a32 = 0.01, a13 = 0.1, a23 = 0.1)

    # We can use primer to run the simulations
    simulated_abun2 <- lapply(1:nrow(initial), function(isampling){
      # Initial abundance in the sampling unit i
      initialN <- c(initial$sp1[isampling], initial$sp2[isampling], initial$sp3[isampling])
      # Simulate community dynamics
      simulated_abun <- data.frame(ode(y = initialN, times = ID_obs, func = lvcomp3, parms = parms2))
      # Add the names of the sampling units
      simulated_abun$sampling_unit <- ID_sampling2[isampling]
      # Calculate relative abundances
      simulated_abun$sp1 <- simulated_abun$X1/rowSums(simulated_abun[, c("X1", "X2", "X3")])
      simulated_abun$sp2 <- simulated_abun$X2/rowSums(simulated_abun[, c("X1", "X2", "X3")])
      simulated_abun$sp3 <- simulated_abun$X3/rowSums(simulated_abun[, c("X1", "X2", "X3")])

      return(simulated_abun[, c("sampling_unit", "time", "sp1", "sp2", "sp3")])
    })

    # Compile species abundances of all sampling units in the same data frame
    abundance2 <- do.call(rbind, simulated_abun2)

The third EDR will be composed of trajectories dominated by sp2 and a new species (sp4) able to coexist in equilibrium with sp2:

    # ID of the sampling units for EDR3
    ID_sampling3 <- LETTERS[11:20]

    # Define initial species abundances
    set.seed(3)
    initial3 <- data.frame(sampling_units = ID_sampling3,
                       sp4 = round(runif(10), 2))

    # Define the parameters for the Lotka-Volterra model
    parms3 <- c(r1 = 1, r2 = 0.1, a11 = 0.2, a21 = 0.1, a22 = 0.02, a12 = 0.01)

    # We can use primer to run the simulations
    simulated_abun3 <- lapply(1:nrow(initial), function(isampling){
      # Initial abundance in the sampling unit i
      initialN <- c(initial3$sp4[isampling], initial$sp2[isampling])
      # Simulate community dynamics
      simulated_abun <- data.frame(ode(y = initialN, times = ID_obs, func = lvcomp2, parms = parms3))
      # Add the names of the sampling units
      simulated_abun$sampling_unit <- ID_sampling3[isampling]
      # Calculate relative abundances
      simulated_abun$sp4 <- simulated_abun$X1/rowSums(simulated_abun[, c("X1", "X2")])
      simulated_abun$sp2 <- simulated_abun$X2/rowSums(simulated_abun[, c("X1", "X2")])

      return(simulated_abun[, c("sampling_unit", "time", "sp4", "sp2")])
    })

    # Compile species abundances of all sampling units in the same data frame
    abundance3 <- do.call(rbind, simulated_abun3)

Once we have species abundances for all EDRs, we need to calculate trajectory dissimilarities:

    # Bind all abundance matrices
    abundance_allEDR <- data.table::rbindlist(list(abundance, abundance2, abundance3), fill = T)
    abundance_allEDR[is.na(abundance_allEDR)] <- 0

    # Calculate state dissimilarities including states in the three EDRs
    state_dissim_allEDR <- vegan::vegdist(abundance_allEDR[, paste0("sp", 1:4)],
                                          method = "bray")

    # Calculate trajectory dissimilarities including trajectories in the three EDRs
    traj_dissim_allEDR <- ecotraj::trajectoryDistances(state_dissim_allEDR,
                                                       sites = abundance_allEDR$sampling_unit,
                                                       surveys = abundance_allEDR$time)

If we plot the trajectories of all EDRs in a common state space, we can see that EDR2 is closer to EDR1 than EDR3. This is not surprising since all the species in EDR1 are present in EDR2. In contrast, EDR3 only has one species in common with the other EDRs.

    # Multidimensional scaling
    st_mds_all <- smacof::smacofSym(state_dissim_allEDR,
                                      ndim = nrow(as.matrix(state_dissim_allEDR))-1)
    st_mds_all <- data.frame(st_mds_all$conf)

    # Plot ecological trajectories in the state space
    state.colorsEDRs <- rep(RColorBrewer::brewer.pal(3, "Set1"), each = 30)
    # Set an empty plot
    plot(st_mds_all$D1, st_mds_all$D2, type = "n",
         xlab = "Axis 1", ylab = "Axis 2",
         main = "EDRs in the state space")
    # Add arrows
    for (isampling in seq(1, 90, 3)) {
      # From observation 1 to observation 2
      shape::Arrows(st_mds_all[isampling, 1], st_mds_all[isampling, 2],
             st_mds_all[isampling + 1, 1], st_mds_all[isampling + 1, 2],
             col = state.colorsEDRs[isampling], arr.adj = 1)
      # From observation 2 to observation 3
      shape::Arrows(st_mds_all[isampling + 1, 1], st_mds_all[isampling + 1, 2],
             st_mds_all[isampling + 2, 1], st_mds_all[isampling + 2, 2],
             col = state.colorsEDRs[isampling], arr.adj = 1)
    }

    # Add a legend
    legend("bottomleft", paste0("EDR", 1:3), col = unique(state.colorsEDRs), lty = 1)

Let’s see how this is reflected in a dissimilarity matrix generated with `dist_edr()`:

    # Compute the dissimilarities between EDRs
    EDR_dissim <- dist_edr(d = traj_dissim_allEDR, d.type = "dTraj",
                           edr = rep(c("EDR1", "EDR2", "EDR3"), each = 10),
                           metric = "dDR")
    round(EDR_dissim, 2)
    #>      EDR1 EDR2 EDR3
    #> EDR1 0.00 0.25 0.73
    #> EDR2 0.39 0.00 0.82
    #> EDR3 0.67 0.71 0.00

As expected, the dissimilarity values between EDR1 and EDR2 are smaller than the dissimilarities between EDR3 and both, EDR1 and EDR2.

## Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 891477 (RESET project).