Author: Martina Sánchez-Pinillos

# Ecological resilience to pulse disturbances

#### Martina Sánchez-Pinillos

## 1. Introduction

### 1.1. Ecological resilience and the EDR framework

Ecological resilience is defined as the ability of ecological systems to tolerate disturbances and still maintain the same relationships between state variables (Holling, 1973). Measuring ecological resilience requires considering two important components: the dynamic trends of the system, including its cyclic behaviors, and the random forces representing positive and negative feedback relationships between the components of the system (Holling, 1973). As such, assessing ecological resilience must account for the system dynamic regimes.

An ecological dynamic regime (EDR) is defined as the “fluctuations of an ecological system around some trend or average resulting from the interaction between internal processes and external forces that, in the absence of perturbations, keep the system within the basin of attraction” (Sánchez-Pinillos et al., 2023). The main dynamic trends characterizing an EDR are useful to identify cyclic behaviors and other more complex dynamics, such as transient dynamics. Additionally, the fluctuations and variability of ecological dynamics resulting from the interaction of multiple factors define the shape, size, and characteristics of dynamic regimes and potential domains of attraction (Sánchez-Pinillos et al., 2024).

The EDR framework is a set of algorithms and metrics to characterize and compare ecological dynamic regimes from empirical data so they can be used as the reference to assess ecological resilience accounting for both the dynamic trends of the system and the feedback relationships between its components.

To know more about the relevance of dynamic regimes for assessing ecological resilience, see this publication:

- Sánchez-Pinillos M., Dakos, V., Kéfi, S. 2024. Ecological dynamic regimes: A key concept for assessing ecological resilience. Biological Conservation. https://doi.org/10.1016/j.biocon.2023.110409

Additional information about the EDR framework can be found in this publication:

- 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 illustrate how ecological resilience can be assessed using the EDR framework implemented in `ecoregime`. In particular, this vignette focuses on the quantitative indices proposed in Sánchez-Pinillos et al., 2024 and their geometric rationale, taking ecological dynamic regimes as the reference to evaluate ecological resilience.

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, Fortin M, Messier C, Kneeshaw D (2026).
    #>   "Forecasting ecological trajectories from ecological dynamic regimes
    #>   to improve resilience analysis." _Methods in Ecology and Evolution_.
    #>   <https://doi.org/10.1111/2041-210x.70372>.
    #>
    #>   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)'.

Like other analyses and metrics of the EDR framework, the assessment of ecological resilience relies on a state space constructed from a dissimilarity matrix (\(D_O\)) including pairwise comparisons of all states (or observations) in the ecological trajectories being evaluated (see `vignette("EDR_framework")` for further details). The choice of the coefficient used to generate \(D_O\) is, therefore, a key step that you should make carefully. As we will use species abundance data as example, we will apply the percentage difference (or Bray-Curtis dissimilarity). The percentage difference and other ecologically relevant dissimilarity coefficients can be applied using the function ``vegdist()` in `vegan`.

## 2. Disturbed and undisturbed trajectories

To illustrate the assessment of ecological resilience based on ecological dynamic regimes (EDRs), we will use the data included in `ecoregime` (i.e., `EDR_data`).

In `EDR_data`, you will find three abundance matrices corresponding to three alternative EDRs (EDR1, EDR2, EDR3) composed of undisturbed trajectories.

    abun_EDR1 <- EDR_data$EDR1$abundance
    abun_EDR2 <- EDR_data$EDR2$abundance
    abun_EDR3 <- EDR_data$EDR3$abundance

Additionally, you will find an abundance matrix associated with three disturbed trajectories (identified as 31, 32, and 33), in which the column `"disturbed_states"` indicates whether observations correspond to pre-disturbance states (`disturbed_states = 0`) or disturbed states (`disturbed_states > 0`):

    head(EDR_data$EDR3_disturbed$abundance)
    #>      EDR  traj state disturbed_states   sp1   sp2   sp3   sp4   sp5   sp6   sp7
    #>    <int> <int> <int>            <num> <num> <num> <num> <num> <num> <num> <num>
    #> 1:     3    31     1                0    13    13     3     8     5    58     0
    #> 2:     3    31     2                0    10    18     4    11     3    53     0
    #> 3:     3    31     3                1    10    18     4     3     3    53     0
    #> 4:     3    31     4                2    11    19     5     3     4    58     0
    #> 5:     3    31     5                3     8    26     7     4     3    52     0
    #> 6:     3    31     6                4     6    34     9     6     2    44     0
    #>      sp8   sp9  sp10  sp11  sp12
    #>    <num> <num> <num> <num> <num>
    #> 1:     0     0     0     0     0
    #> 2:     0     0     0     0     0
    #> 3:     0     0     0     0     0
    #> 4:     0     0     0     0     0
    #> 5:     0     0     0     0     0
    #> 6:     0     0     0     0     0

As a preliminary analysis, we can use the function `plot_EDR` to visualize the stability landscape composed of three alternative EDRs and the disturbed trajectories. For that, we need to generate a state dissimilarity matrix comparing all states in the trajectories of the three EDRs and the disturbed trajectories.

    library(data.table)
    #>
    #> Attaching package: 'data.table'
    #> The following object is masked from 'package:base':
    #>
    #>     %notin%
    # Include all abundances in one matrix
    abun_all <- rbindlist(list(
      EDR_data$EDR1$abundance,
      EDR_data$EDR2$abundance,
      EDR_data$EDR3$abundance,
      EDR_data$EDR3_disturbed$abundance),
      fill = TRUE)

    # Compute state dissimilarities
    d_all <- vegan::vegdist(abun_all[, paste0("sp", 1:12)], method = "bray")

    # We can apply mMDS to the dissimilarity matrix
    mds_all <- smacof::mds(d_all, ndim = 3)

    # Visualize the stability landscape
    plot_edr(x = mds_all$conf,
             trajectories = paste0(abun_all$EDR, "_", abun_all$traj), # provide a unique ID to each trajectory
             states = as.integer(abun_all$state),
             traj.colors = c(rep(palette.colors(4, "Table")[c(1:2, 4)], each= 30),
                             rep(palette.colors(4, "Table")[3], 3)),
             xlab = "MDS D1", ylab = "MDS D2",
             main = "Stability landscape and disturbed trajectories")
    legend("bottomleft", c(paste0("EDR", 1:3), "Disturbed trajectories"),
           lty = 1, lwd = 2, col = palette.colors(4, "Table")[c(1,2,4,3)], bty = "n", cex = 0.8)

## 3. Assessing ecological dynamic regimes

### 3.1. Identifying the reference ecological dynamic regime

To assess the ecological resilience of a system, we need to identify its ecological dynamic regime (EDR) so it can be used as the reference to compare the disturbed dynamics. That is, the EDR to which the disturbed trajectories belonged before being disturbed. In the figure of the stability landscape produced for our example data, we can see that the reference EDR should be EDR3, since all disturbed trajectories are located in its proximity. However, when we are working with empirical data, the distinction is often not straightforward.

Given a stability landscape composed of alternative dynamic regimes, we can compute the dynamic dispersion (dDis) of the pre-disturbance portion of the disturbed trajectories in relation to each alternative EDR. The function `dDis()` allows us to quantify the degree of membership of any trajectory within an EDR. You can see more information about `dDis()` in `vignette("EDR_framework")`.

    # Species abundances in pre-disturbance states of the disturbed trajectories
    # (Pre-disturbance states are identified by disturbed_states = 0)
    abun_predist <- EDR_data$EDR3_disturbed$abundance[disturbed_states == 0]
    selcols <- names(EDR_data$EDR1$abundance)

    ## EDR1 ------------------------------------------------------------------------

    # Species abundances in EDR1 and the predisturbed states of disturbed trajectories
    abun1_predist <- rbind(EDR_data$EDR1$abundance, abun_predist[, ..selcols])

    # State dissimilarities in EDR1 and the predisturbed states of disturbed trajectories
    d1_predist <- vegan::vegdist(x = abun1_predist[, paste0("sp", 1:12)], method = "bray")

    # dDis of the disturbed trajectories in relation to EDR1
    dDis1 <- sapply(unique(abun_predist$traj), function(ipredist){
      dDis(d = d1_predist, d.type = "dStates",
           trajectories = abun1_predist$traj, states = abun1_predist$state,
           reference = as.character(ipredist))
    })

    ## EDR2 ------------------------------------------------------------------------

    # Species abundances in EDR2 and the predisturbed states of disturbed trajectories
    abun2_predist <- rbind(EDR_data$EDR2$abundance, abun_predist[, ..selcols])

    # State dissimilarities in EDR2 and the predisturbed states of disturbed trajectories
    d2_predist <- vegan::vegdist(x = abun2_predist[, paste0("sp", 1:12)], method = "bray")

    # dDis of the disturbed trajectories in relation to EDR2
    dDis2 <- sapply(unique(abun_predist$traj), function(ipredist){
      dDis(d = d2_predist, d.type = "dStates",
           trajectories = abun2_predist$traj, states = abun2_predist$state,
           reference = as.character(ipredist))
    })

    ## EDR3 ------------------------------------------------------------------------

    # Species abundances in EDR3 and the predisturbed states of disturbed trajectories
    abun3_predist <- rbind(EDR_data$EDR3$abundance, abun_predist[, ..selcols])

    # State dissimilarities in EDR3 and the predisturbed states of disturbed trajectories
    d3_predist <- vegan::vegdist(x = abun3_predist[, paste0("sp", 1:12)], method = "bray")

    # dDis of the disturbed trajectories in relation to EDR3
    dDis3 <- sapply(unique(abun_predist$traj), function(ipredist){
      dDis(d = d3_predist, d.type = "dStates",
           trajectories = abun3_predist$traj, states = abun3_predist$state,
           reference = as.character(ipredist))
    })

    ## Compare dynamic dispersion --------------------------------------------------

    # Compare dDis values for the three EDRs
    dDis_df <- data.frame(EDR1 = dDis1, EDR2 = dDis2, EDR3 = dDis3)

If we compare the values of dDis for each disturbed trajectory (ref. 31, ref. 32, ref. 33) and EDR (EDR1, EDR2, EDR3), we see that the lowest dDis values are associated with EDR3. We can take EDR3 as the reference to assess the ecological resilience of the three disturbed communities.

| EDR1 | EDR2 | EDR3
--- | --- | --- | ---
dDis (ref. 31) | 0.649 | 0.748 | 0.356
dDis (ref. 32) | 0.734 | 0.556 | 0.268
dDis (ref. 33) | 0.711 | 0.436 | 0.412

Now, we can “zoom” in the stability landscape to the reference EDR and the disturbed trajectories. This time, we will use different colors to distinguish (i) the states in the reference EDR, (ii) the pre-disturbance states, and (iii) the disturbed states (one color for each disturbed trajectory).

    # Index of the states in the reference EDR
    iref <- which(abun_all$EDR == 3 & is.na(abun_all$disturbed_states))

    # Index of the states in disturbed trajectories
    idist <- which(!is.na(abun_all$disturbed_states))

    # Index of the states in the reference EDR and disturbed trajectories
    iref_idist <- c(iref, idist)

    # Filter the dissimilarity matrix to select the states of the reference EDR and
    # the disturbed trajectories
    d_all <- as.matrix(d_all)
    d_EDR3_dist <- d_all[iref_idist, iref_idist]

    # Apply mMDS for the new matrix
    mds_EDR3_dist <- smacof::mds(d_EDR3_dist, ndim = 3)

    # Vector of colors
    state.colors <- abun_all[iref_idist,
                             # color for reference states
                             state.colors := ifelse(is.na(disturbed_states), palette.colors(8, "R4")[8],
                                                    # color for pre-disturbance states
                                                    ifelse(disturbed_states == 0, palette.colors(8, "R4")[7],
                                                           # color for disturbed states
                                                           ifelse(traj == 31, palette.colors(8, "R4")[2],
                                                                  ifelse(traj == 32, palette.colors(8, "R4")[3],
                                                                         palette.colors(8, "R4")[4]))))]$state.colors
    state.colors <- state.colors[!is.na(state.colors)]

    # Plot the reference EDR and disturbed trajectories
    plot_edr(x = mds_EDR3_dist$conf,
             trajectories = paste0(abun_all$EDR[iref_idist], "_", abun_all$traj[iref_idist]),
             states = as.integer(abun_all$state[iref_idist]),
             type = "states", # it allows assigning different colors to the states
             state.colors = state.colors,
             xlab = "MDS D1", ylab = "MDS D2",
             main = "Reference EDR and disturbed trajectories")
    legend("topleft", c("Reference EDR", "Pre-disturbed states",
                        "Disturbed states T31", "Disturbed states T32",
                        "Disturbed states T33"),
           lty = 1, lwd = 2, col = palette.colors(8, "R4")[c(8, 7, 2:4)],
           cex = 0.9, bty = "n")

In this figure, we already observe contrasting post-disturbance dynamics. T32 seems to remain within the boundaries of the EDR, T33 temporarily visits the boundaries before returning towards the reference EDR, and T31 is pushed out of the EDR. In Section 4, we will learn to quantify these patterns.

### 3.2. Define the main dynamic trends through representative trajectories

Once we have identified the EDR of reference, we can compute the representative trajectories using the function `retra_edr()`. See `vignette("EDR_framework")` for more information.

    # State dissimilarities for EDR3 (considering only the undisturbed trajectories)
    d_EDR3 <- vegan::vegdist(EDR_data$EDR3$abundance[, paste0("sp", 1:12)])

    # Representative trajectories
    retra <- retra_edr(d = d_EDR3,
                       trajectories = EDR_data$EDR3$abundance$traj,
                       states = EDR_data$EDR3$abundance$state, minSegs = 5)

Although there are five representative trajectories, we will select “T4” as the reference for being the longest and covering all regions of the EDR relatively well.

    # Summarize retra
    summary(retra)
    #>    ID Size     Length   Avg_link   Sum_link Avg_density Max_density Avg_depth
    #> T1 T1    8 0.56912543 0.12435644 0.24871287    7.250000           8  4.750000
    #> T2 T2    6 0.39900498 0.10718905 0.21437811    7.666667           8  4.666667
    #> T3 T3    4 0.08970149 0.03482587 0.03482587    6.500000           7  6.500000
    #> T4 T4   16 1.11654181 0.08961127 0.53766760    6.625000           8  5.125000
    #> T5 T5    6 0.35468137 0.08723881 0.17447761    7.333333          10  5.333333
    #>    Max_depth
    #> T1         6
    #> T2         6
    #> T3         7
    #> T4         7
    #> T5         7

    # Define T4 as the unique representative trajectory and generate an object of class 'RETRA'
    retra_ref <- define_retra(data = retra$T4$Segments, d = d_EDR3,
                              trajectories = EDR_data$EDR3$abundance$traj,
                              states = EDR_data$EDR3$abundance$state,
                              retra = retra)

    # Plot EDR3 and its representative trajectories
    plot(retra, d = d_EDR3,
         trajectories = EDR_data$EDR3$abundance$traj,
         states = EDR_data$EDR3$abundance$state, select_RT = "T4",
         xlab = "MDS D1", ylab = "MDS D2",
         main = "Representative trajectories in EDR3")
    legend("topleft", c("Representative trajectory 'T4'",
                        "Other representative trajectories",
                        "Individual trajectories in EDR3"),
           lty = 1, col = c("red", "black", "grey"), bty = "n")

## 4. Metrics of ecological resilience to pulse disturbances

When an ecological system is disturbed, its dynamics can deviate from the reference EDR and its representative trajectories. Depending on the immediate impact of the disturbance and the changes in the state variables following the release of the disturbance, post-disturbance trajectories can be characterized through four complementary indices: resistance (Rt), amplitude (A), recovery (Rc), and net change (NC).

### 4.1. Resistance

The resistance index (Rt) quantifies the impact of the disturbance on the system based on the immediate changes in the state variables. That is, it is a measure of how similar the disturbed (1) and the pre-disturbance (0) states are (Sánchez-Pinillos et al., 2019). As such, the resistance index does not depend on the position of the system within the EDR.

\[Rt = 1 - d_{pre, dist}\]

where \(d_{pre,dist}\) is the dissimilarity between the pre-disturbance (0) and the disturbed (1) states.

Given the three disturbed trajectories in EDR3 (i.e., `EDR3_disturbed`), we can compute the resistance index using the function `resistance()`:

    # To calculate resistance, we need a state dissimilarity matrix for the disturbed trajectories
    d_disturbed <- vegan::vegdist(EDR_data$EDR3_disturbed$abundance[, paste0("sp", 1:12)],
                           method = "bray")

    # Compute resistance
    # Note that the disturbed states are identified by disturbed_states = 1
    Rt <- resistance(d = d_disturbed,
                     trajectories = EDR_data$EDR3_disturbed$abundance$traj,
                     states = EDR_data$EDR3_disturbed$abundance$state,
                     disturbed_trajectories = unique(EDR_data$EDR3_disturbed$abundance$traj),
                     disturbed_states = EDR_data$EDR3_disturbed$abundance[disturbed_states == 1]$state)

The three hypothetical systems show a relatively high resistance (close to 1) to the disturbance:

Disturbed trajectories | Rt
--- | ---
31 | 0.958
32 | 0.812
33 | 0.693

### 4.2. Amplitude

The amplitude (A) quantifies how much the system is deviated from its trajectory during the disturbance assuming that, in the absence of disturbances, the system would keep a constant distance to a representative trajectory taken as the reference.

The amplitude can be calculated in absolute terms as the difference of the dissimilarity between the disturbed state (1) and the representative trajectory (\(d_{dist,RT}\)) and the dissimilarity between the pre-disturbance state (0) and the representative trajectory (\(d_{pre,RT}\)):

\[A_{abs} = d_{dist,RT} - d_{pre,RT}\]

Alternatively, the amplitude can be calculated in relation to the impact of the disturbance (\(d_{pre,dist}\)). In this case, the amplitude quantifies the ability of the system to remain close to the representative trajectory in relation to the changes in the state variables provoked by the disturbance.

\[A_{rel} = \frac{d_{dist,RT} - d_{pre,RT}}{d_{pre,dist}}\]

In any case, positive amplitude values indicate that the system is deviated towards the boundaries of the EDR, whereas negative values indicate the deviation towards the representative trajectory taken as the reference.

Amplitude can be calculated using the function `amplitude()`:

    # We need a state dissimilarity matrix containing the states of the disturbed
    # trajectories and the representative trajectory taken as the reference
    abun <- rbind(EDR_data$EDR3$abundance, EDR_data$EDR3_disturbed$abundance, fill = T)
    d <- vegan::vegdist(abun[, paste0("sp", 1:12)], method = "bray")

    # Compute amplitude
    A <- amplitude(d = d,
                   trajectories = abun$traj,
                   states = abun$state,
                   disturbed_trajectories = abun[disturbed_states == 1]$traj,
                   disturbed_states = abun[disturbed_states == 1]$state,
                   reference = retra_ref, method = "nearest_state")

The three considered systems show positive amplitude values, indicating that the disturbance lead them towards the boundaries of the EDR. Whereas the absolute amplitude of trajectory 31 (\(A_{abs}(31) = 0.027\)) is smaller than the absolute amplitude of trajectory 32 (\(A_{rel}(32) = 0.119\)), both have similar relative values (\(A_{rel}(31) = 0.653\); \(A_{rel}(32) = 0.631\)). This result indicates that despite the small deviation of trajectory 31 from the representative trajectory during the disturbance, such deviation is disproportionately high in relation to the impact of the disturbance on the state variables.

Disturbed trajectories | Reference | Aabs | Arel
--- | --- | --- | ---
31 | T4.1 | 0.027 | 0.653
32 | T4.1 | 0.119 | 0.631
33 | T4.1 | 0.268 | 0.871

### 4.3. Recovery

The recovery index (Rc) quantifies the ability of the system to reorganize itself after a disturbance and evolve towards the representative trajectory representing its dominant dynamic trends.

The recovery can be calculated in absolute terms as the difference of the dissimilarity between the disturbed state (1) and the representative trajectory (\(d_{dist,RT}\)) and the dissimilarity between one of the post-disturbance states (> 1) and the representative trajectory (\(d_{post,RT}\)):

\[Rc_{abs} = d_{dist,RT} - d_{post,RT}\]

Alternatively, the recovery can be calculated in relation to the changes in the state variables that the system must perform to return towards the representative trajectory from the disturbed state (\(d_{dist, post}\)). In this case, the index penalizes the systems that require major restructuring of the state variables to return towards the representative trajectory (positive recovery) and gives less negative values to the systems minimizing the escape from the dynamic regime through major changes in its state variables.

\[Rc_{rel} = \frac{d_{dist,RT} - d_{post,RT}}{d_{dist, post}}\]

Both absolute and relative recovery indices are positive when the system evolves towards the representative trajectory. Otherwise, negative values indicate that the system evolves towards the boundaries of the EDR.

Recovery can be calculated using the function `recovery()` considering all states after the disturbed state (> 1):

    # Compute recovery using the same data used for amplitude
    Rc <- recovery(d = d,
                   trajectories = abun$traj,
                   states = abun$state,
                   disturbed_trajectories = abun[disturbed_states == 1]$traj,
                   disturbed_states = abun[disturbed_states == 1]$state,
                   reference = retra_ref, method = "nearest_state")

We can plot the variation of recovery over time considering all post-disturbance states:

    # Number of states after the disturbed state
    Rc <- data.table::data.table(Rc)
    Rc[, ID_post := 1:(.N), by = disturbed_trajectories]

    par(mfrow = c(1, 2))
    # Plot absolute recovery over time
    plot(x = range(Rc$ID_post), y = range(Rc$Rc_abs), type = "n",
         xlab = "Nb. states after disturbance", ylab = "Absolute recovery",
         main = "Variation of absolute recovery")
    for (i in unique(Rc$disturbed_trajectories)) {
      lines(Rc[disturbed_trajectories == i, c("ID_post", "Rc_abs")],
            col = which(unique(Rc$disturbed_trajectories) %in% i) + 1)
    }
    legend("bottomleft", legend = unique(Rc$disturbed_trajectories), lty = 1,
           col = seq_along(unique(Rc$disturbed_trajectories)) + 1, bty = "n")

    # Plot relative recovery over time
    plot(x = range(Rc$ID_post), y = range(Rc$Rc_rel), type = "n",
         xlab = "Nb. states after disturbance", ylab = "Relative recovery",
         main = "Variation of relative recovery")
    for (i in unique(Rc$disturbed_trajectories)) {
      lines(Rc[disturbed_trajectories == i, c("ID_post", "Rc_rel")],
            col = which(unique(Rc$disturbed_trajectories) %in% i) + 1)
    }
    legend("topright", legend = unique(Rc$disturbed_trajectories), lty = 1,
           col = seq_along(unique(Rc$disturbed_trajectories)) + 1, bty = "n")

### 4.4. Net change

Net change (NC) quantifies how much the system is deviated from its trajectory after the release of the disturbance assuming that, in the absence of disturbances, the system would keep a constant distance to a representative trajectory taken as the reference.

Like amplitude and recovery, net change can be calculated in absolute or relative terms. The absolute net change is expressed as the difference of the dissimilarity between one of the post-disturbance states (> 1) and the representative trajectory (\(d_{post,RT}\)) and the dissimilarity between the pre-disturbance state (0) and the representative trajectory (\(d_{pre,RT}\)):

\[NC_{abs} = d_{post,RT} - d_{pre,RT}\]

The relative net change is calculated in relation to the changes produced in the state variables between the pre-disturbance and the post-disturbance states (\(d_{pre,post}\)). In this way, the index penalizes the systems that deviated from the expected trajectory despite being very similar to the pre-disturbance state.

\[NC_{rel} = \frac{d_{post,RT} - d_{pre,RT}}{d_{pre,post}}\]

As in the amplitude index, positive values indicate that the system is deviated towards the boundaries of the EDR and negative values indicate the deviation towards the representative trajectory taken as the reference.

The function `net_change()` included in `ecoregime` can be used to calculate the net change of all post-disturbance states in relation to the pre-disturbance state:

    # Compute net change using the same data used for amplitude
    NC <- net_change(d = d,
                     trajectories = abun$traj,
                     states = abun$state,
                     disturbed_trajectories = abun[disturbed_states == 1]$traj,
                     disturbed_states = abun[disturbed_states == 1]$state,
                     reference = retra_ref, method = "nearest_state")

We can plot the variation of net change over time considering all post-disturbance states:

    # ID post-disturbance states
    NC <- data.table::data.table(NC)
    NC[, ID_post := 1:(.N), by = disturbed_trajectories]

    par(mfrow = c(1, 2))
    # Plot absolute net change over time
    plot(x = range(NC$ID_post), y = range(NC$NC_abs), type = "n",
         xlab = "Nb. states after disturbance", ylab = "Absolute net change",
         main = "Variation of absolute net change")
    for (i in unique(NC$disturbed_trajectories)) {
      lines(NC[disturbed_trajectories == i, c("ID_post", "NC_abs")],
            col = which(unique(NC$disturbed_trajectories) %in% i) + 1)
    }
    legend("topleft", legend = unique(NC$disturbed_trajectories), lty = 1,
           col = seq_along(unique(NC$disturbed_trajectories)) + 1, bty = "n")

    # Plot relative net change over time
    plot(x = range(NC$ID_post), y = range(NC$NC_rel), type = "n",
         xlab = "Nb. states after disturbance", ylab = "Relative net change",
         main = "Variation of relative net change")
    for (i in unique(NC$disturbed_trajectories)) {
      lines(NC[disturbed_trajectories == i, c("ID_post", "NC_rel")],
            col = which(unique(NC$disturbed_trajectories) %in% i) + 1)
    }
    legend("bottomleft", legend = unique(NC$disturbed_trajectories), lty = 1,
           col = seq_along(unique(NC$disturbed_trajectories)) + 1, bty = "n")

## 5. Assessing the ecological resilience of systems affected by pulse disturbances

Altogether, the metrics of amplitude, recovery, and net change define the geometry of disturbed trajectories in relation to the reference EDR. Additionally, the resistance index helps interpret the response of the system considering the immediate impact of the disturbance.

As an example, we will take the three disturbed trajectories used in the previous section. For simplicity, we will only focus on resistance and the absolute values of amplitude, recovery, and net change and we will consider the 14th post-disturbance state (last post-disturbance state of Trajectory 33) of all trajectories to calculate recovery and net change.

    # Merge the results for resistance, amplitude, recovery, and net change
    results <- Reduce(function(x, y) merge(x, y, all = T),
                      list(Rt, A, Rc, NC))
    results <- results[which(results$ID_post == 14),
                       c("disturbed_trajectories", "Rt", "A_abs", "Rc_abs", "NC_abs")]

Disturbed trajectories | Rt | Aabs | Rcabs | NCabs
--- | --- | --- | --- | ---
31 | 0.958 | 0.027 | -0.474 | 0.502
32 | 0.812 | 0.119 | 0.099 | 0.020
33 | 0.693 | 0.268 | 0.283 | -0.016

Based on the values of the indices, we can figure out the shape of the disturbed trajectories. For example, Trajectory 31 was very resistant to the immediate impact of the disturbance (in terms of changes in the state variables) (\(Rt = 0.958\)) and the amplitude provoked by the disturbance was very small (\(A = 0.027\)). However, unlike Trajectory 32 and Trajectory 33, the recovery was negative (\(Rc = -0.474\)) and the net change positive and relatively high (\(NC = 0.502\)). Thus, despite the relatively low impact of the disturbance, the values of recovery and net change indicate that Trajectory 31 goes away from the reference EDR and the system potentially changes to an alternative dynamic regime.

Both Trajectory 32 and Trajectory 33 showed positive values of recovery and net change values close to zero. In both cases, amplitude and recovery have similar values, respectively (Trajectory 32: \(A = 0.119\), \(Rc = 0.099\); Trajectory 33: \(A = 0.268\), \(Rc = 0.283\)), indicating that these systems are able to reorganize and remain within the EDR.

Trajectory 32 was relatively resistant to the impact of the disturbance (\(Rt = 0.812\)) and showed a relatively low deviation from the representative trajectory (\(A = 0.119\)). In contrast, Trajectory 33 was more severely impacted by the disturbance (\(Rt = 0.693\); \(A = 0.268\)) but showed a high recovery capacity (\(Rc = 0.283\)). Thus, while both communities could be considered resilient, they showed different strategies. Whereas Trajectory 32 represents a resistant dynamic within the EDR, Trajectory 33 represents a more variable system whose dynamics visit the borders of the EDR to eventually return towards the representative trajectory.

To support our evaluation of the ecological resilience of disturbed trajectories, we can calculate the dynamic dispersion (dDis) of the post-disturbance portion of the disturbed trajectories in relation to the trajectories in the EDR.

    # Species abundances in post-disturbance states of the disturbed trajectories
    # The states after the release of the disturbance are identified by disturbed_states > 1
    abun_post <- EDR_data$EDR3_disturbed$abundance[disturbed_states > 1]
    selcols <- names(EDR_data$EDR3$abundance)

    # Species abundances in EDR3 and the post-disturbance states of disturbed trajectories
    abun3_post <- rbind(EDR_data$EDR3$abundance, abun_post[, ..selcols])

    # State dissimilarities in EDR3 and the post-disturbance states of disturbed trajectories
    d3_post <- as.matrix(vegan::vegdist(x = abun3_post[, paste0("sp", 1:12)], method = "bray"))

    # dDis of each disturbed trajectory
    dDis_dist <- sapply(unique(abun_post$traj), function(idist){
      rm_disturbed <- unique(abun_post$traj[which(abun_post$traj != idist)])
      irm <- which(abun3_post$traj %in% rm_disturbed)
      dDis(d = d3_post[-irm, -irm], d.type = "dStates",
           trajectories = abun3_post$traj[-irm],
           states = abun3_post$state[-irm],
           reference = as.character(idist))
    })

The values of dDis confirm the resilience of Trajectory 32 and show that, despite being able to recover after the disturbance, the dynamic represented by Trajectory 33 is anomalous in relation to the other dynamics in the EDR.

Disturbed trajectories | dDis
--- | ---
31 | 0.463
32 | 0.257
33 | 0.474

Finally, we can illustrate the responses of the three disturbed communities by representing their trajectories in relation to the reference EDR and its representative trajectory:

    # Plot EDR3 and its representative trajectories
    plot(retra_ref, d = d,
         trajectories = abun$traj,
         states = abun$state,
         traj.colors = c(rep("grey", 30), 2:4),
         xlab = "MDS D1", ylab = "MDS D2",
         main = "Comparison of disturbed trajectories and EDR3")
    legend("topleft", c("Representative trajectory",
                        "Individual trajectories in EDR3",
                        "Trajectory 31",
                        "Trajectory 32",
                        "Trajectory 33"),
           lty = 1, col = c("black", "grey", 2:4), bty = "n")

## 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).