mcbrnet: an R package for simulating metacommunity dynamics in a branching network

Akira Terui May 07, 2021

• Overview
• Installation
• Instruction
  • brnet()
    • Basic usage
    • Quick start
    • Custom setting: visualization
    • Custom setting: environment
    • Custom setting: disturbance
    • Custom setting: asymmetric distance matrix
  • mcsim()
    • Basic usage
    • Quick start
    • Custom setting: combine brnet() and mcsim()
    • Custom setting: detailed parameters
    • Model description
  • adjtodist
    • Basic usage
• References

Overview

The package mcbrnet is composed of two functions: brnet() and mcsim().

• brnet: Function brnet() generates a random branching network with the specified number of patches and probability of branching. The function returns adjacency and distance matrices, hypothetical environmental values at each patch, and the number of patches upstream (akin to the watershed area in river networks). The output may be used in function mcsim() to simulate metacommunity dynamics in a branching network.

• mcsim: Function mcsim() simulates metacommunity dynamics. By default, it produces a square-shaped landscape with randomly distributed habitat patches (x- and y-coordinates are drawn from a uniform distribution). If a distance matrix is given, the function simulates metacommunity dynamics in the specified landscape. Function mcsim() follows a general framework proposed by Thompson et al. (2020). However, it has several unique features that are detailed in the following sections.

Installation

The mcbrnet package can be installed with the following script:

#install.packages("remotes")
remotes::install_github("aterui/mcbrnet")
library(mcbrnet)

Instruction

brnet()

Basic usage

The key arguments are the number of habitat patches (n_patch) and the probability of branching (p_branch), which users must specify. With these parameters, the function generates a branching network through the following steps:

1. Draw the number of branches in the network. An individual branch is defined as a series of connected patches from one confluence (or outlet) to the next confluence upstream (or upstream terminal). The number of branches in a network BR is drawn from a binomial distribution as BR ~ Binomial(N, P_br), where N is the number of patches and P_br is the branching probability.

2. Draw the number of patches in each branch. The number of patches in each branch N_br is drawn from a geometric distribution as N_br ~ Ge(P_br) but conditional on ΣN_br = N.

3. Organize branches into a bifurcating branching network.

Sample script:

net <- brnet(n_patch = 50, p_branch = 0.5)

The function returns:

• adjacency_matrix: adjacency matrix.

• distance_matrix: distance matrix. Distance between patches is measured as the number of steps required to reach from the focal patch to the target patch through the network.

• weighted_distance_matrix: weighted distance matrix. Upstream steps may be weighted (see Custom setting).

• df_patch: a data frame (dplyr::tibble) containing patch attributes.

  • patch_id: patch ID.
  • branch_id: branch ID.
  • environment: environmental value at each patch (see below for details)
  • disturbance: disturbance level (i.e., proportional mortality) at each patch (see below for details)
  • n_patch_upstream: the number of upstream contributing patches (including the focal patch itself; akin to the watershed area in river networks).

Quick start

The following script produce a branching network with n_patch = 50 and p_branch = 0.5. By default, brnet() visualizes the generated network using functions in packages igraph (Csardi and Nepusz 2006) and plotfunctions (van Rij 2020) (plot = FALSE to disable):

net <- brnet(n_patch = 50, p_branch = 0.5)

Randomly generated environmental values color patches and patches’ size is proportional to the number of patches upstream. To view matrices, type the following script:

# adjacency matrix
# showing 10 patches for example
net$adjacency_matrix[1:10, 1:10]

##         patch1 patch2 patch3 patch4 patch5 patch6 patch7 patch8 patch9 patch10
## patch1       0      0      0      0      0      0      0      0      0       0
## patch2       0      0      0      0      0      0      0      0      0       0
## patch3       0      0      0      1      0      0      0      0      0       0
## patch4       0      0      1      0      0      0      0      0      0       0
## patch5       0      0      0      0      0      0      0      0      0       0
## patch6       0      0      0      0      0      0      1      0      0       0
## patch7       0      0      0      0      0      1      0      0      0       0
## patch8       0      0      0      0      0      0      0      0      1       0
## patch9       0      0      0      0      0      0      0      1      0       1
## patch10      0      0      0      0      0      0      0      0      1       0

# distance matrix
# showing 10 patches for example
net$distance_matrix[1:10, 1:10]

##         patch1 patch2 patch3 patch4 patch5 patch6 patch7 patch8 patch9 patch10
## patch1       0      9      3      4      3     12     13      9     10      11
## patch2       9      0     12     13     12      3      4      2      3       4
## patch3       3     12      0      1      2     15     16     12     13      14
## patch4       4     13      1      0      3     16     17     13     14      15
## patch5       3     12      2      3      0     15     16     12     13      14
## patch6      12      3     15     16     15      0      1      5      6       7
## patch7      13      4     16     17     16      1      0      6      7       8
## patch8       9      2     12     13     12      5      6      0      1       2
## patch9      10      3     13     14     13      6      7      1      0       1
## patch10     11      4     14     15     14      7      8      2      1       0

The following script lets you view branch ID, environmental values, and the number of upstream contributing patches for each patch:

net$df_patch

## # A tibble: 50 x 5
##    patch_id branch_id environment disturbance n_patch_upstream
##       <int>     <dbl>       <dbl>       <dbl>            <dbl>
##  1        1         1      -1.33        0.899               50
##  2        2         8      -0.693       0.899                9
##  3        3         6      -1.96        0.895                6
##  4        4         6      -1.93        0.895                5
##  5        5        23      -0.145       0.898                1
##  6        6        21       0.872       0.889                2
##  7        7        21       0.804       0.889                1
##  8        8        13      -0.890       0.889                5
##  9        9        13      -0.938       0.889                4
## 10       10        13      -0.967       0.889                3
## # ... with 40 more rows

Custom setting: visualization

Arguments: patch_label, patch_scaling, patch_size

Users may add patch labels using the argument patch_label:

# patch ID
net <- brnet(n_patch = 50, p_branch = 0.5, patch_label = "patch")

# branch ID
net <- brnet(n_patch = 50, p_branch = 0.5, patch_label = "branch")

# number of upstream contributing patches
net <- brnet(n_patch = 50, p_branch = 0.5, patch_label = "n_upstream")

To remove patch size variation, set patch_scaling = FALSE and specify patch_size:

# number of upstream contributing patches
net <- brnet(n_patch = 50, p_branch = 0.5, patch_scaling = FALSE, patch_size = 8)

Custom setting: environment

Arguments: mean_env_source, sd_env_source, rho, sd_env_lon

Some flexibility exists to simulate environmental values, which are determined through an autoregressive process, as detailed below:

1. Environmental values for upstream terminal patches (i.e., patches with no upstream patch) are drawn from a normal distribution as z₁ ~ Normal(μ_source, σ_source²) (arguments mean_env_source and sd_env_source).

2. Downstream environmental values are determined by an autoregressive process as z_x = ρz_x-1 + ε_x (‘x-1’ means one patch upstream), where ε_x ~ Normal(0, σ²_env) (argument sd_env_lon). At bifurcation patches (or confluence), the environmental value takes a weighted mean of the two contributing patches given the size of these patches s (the number of upstream contributing patches): z_x = ω(ρz_1,x-1 + ε_1,x) + (1 - ω)(ρz_2,x-1 + ε_2,x), where ω = s₁/(s₁ + s₂).

Users may change the values of μ_source (default: mean_env_source = 0), σ_source (sd_env_source = 1), ρ (rho = 1), and σ_env (sd_env_lon = 0.1). Increasing the value of sd_env_source leads to greater variation in environmental values at upstream terminals. The argument rho (0 ≤ ρ ≤ 1) determines the strength of longitudinal autocorrelation (the greater the stronger autocorrelation). The argument sd_env_lon (σ_env > 0) determines the strength of longitudinal environmental noise. The following script produces a network with greater environmental variation at upstream terminals (z₁ ~ Normal(0, 3²)), weaker longitudinal autocorrelation (ρ = 0.5), and stronger local noises (σ_env = 0.5).

net <- brnet(n_patch = 50, p_branch = 0.5,
             sd_env_source = 3, rho = 0.5, sd_env_lon = 0.5)

Custom setting: disturbance

Arguments: mean_disturb_source, sd_disturb_source

Some flexibility exists to simulate disturbance levels, as detailed below:

1. Disturbance levels for upstream terminal patches (i.e., patches with no upstream patch) are drawn from a normal distribution as m₁ ~ Normal(μ_m, σ_m²). The argument mean_disturb_source controls the proportional mean of the disturbance level, which will be converted to a logit-scale value in the function (i.e., μ_m = logit(mean_disturb_source)). The argument sd_disturb_source controls the variation in disturbance level among headwaters in a logit scale (i.e., σ_m = sd_disturb_source)

2. Downstream disturbance levels are identical to the adjacent upstream patch except at confluences. At bifurcation patches (or confluence), the disturbance value takes a weighted mean of the two contributing patches given the size of these patches s (the number of upstream contributing patches): m_x = ωm_1,x-1 + (1 - ω)m_2,x-1, where ω = s₁/(s₁ + s₂).

3. The logit-scale values will be back-transformed to the proportional values (return value = inv.logit(m))

Users may change the values of μ_m (default: mean_disturb_source = 0.9) and σ_m (sd_env_source = 0.1).

Custom setting: asymmetric distance matrix

Arguments: asymmetry_factor

Distance between patches is calculated as the number of steps from the origin to the target patch. Specifically, distance from patch x to y, d_xy, is calculated as:

d_xy = step_d + δstep_u

where step_d the number of downstream steps, step_u the number of upstream steps, and δ the asymmetry scaling factor (asymmetry_factor). Users may change δ to control the weight of upstream steps. (1) δ = 1, upstream and dowstream steps have no difference (default), (2) δ > 1, upstream steps have greater costs, (3) δ < 1, downstream steps have greater costs. Changes to asymmetry_factor will be reflected in weighted_distance_matrix.

mcsim()

Basic usage

The key arguments are the number of habitat patches (n_patch) and the number of species in a metacommunity (n_species). The metacommunity dynamics are simulated through (1) local dynamics (population growth and competition among species), (2) immigration, and (3) emigration.

Sample script:

mc <- mcsim(n_patch = 5, n_species = 5)

The function returns:

• df_dynamics a data frame containing simulated metacommunity dynamics*.

  • timestep: time-step.
  • patch_id: patch ID.
  • mean_env: mean environmental condition at each patch.
  • env: environmental condition at patch x and time-step t.
  • carrying_capacity: carrying capacity at each patch.
  • species: species ID.
  • niche_optim: optimal environmental value for species i.
  • r_xt: reproductive number of species i at patch x and time-step t.
  • abundance: abundance of species i at patch x.

• df_species a data frame containing species attributes.

  • species: species ID.
  • mean_abundance: mean abundance (arithmetic) of species i across sites and time-steps.
  • r0: maximum reproductive number of species i.
  • niche_optim: optimal environmental value for species i.
  • sd_niche_width: niche width for species i.
  • p_dispersal: dispersal probability of species i.

• df_patch a data frame containing patch attributes.

  • patch: patch ID.
  • alpha_div: alpha diversity averaged across time-steps.
  • mean_env: mean environmental condition at each patch.
  • carrying_capacity: carrying capacity at each patch.
  • connectivity: structural connectivity at each patch. See below for details.

• df_diversity a data frame containing diversity metrics (α, β, and γ).

• distance_matrix a distance matrix used in the simulation.

• interaction_matrix a species interaction matrix, in which species X (column) influences species Y (row).

*NOTE: The warm-up and burn-in periods will not be included in return values.

Quick start

The following script simulates metacommunity dynamics with n_patch = 5 and n_species = 5. By default, mcsim() simulates metacommunity dynamics with 200 warm-up (initialization with species introductions: n_warmup), 200 burn-in (burn-in period with no species introductions: n_burnin), and 1000 time-steps for records (n_timestep).

mc <- mcsim(n_patch = 5, n_species = 5)

Users can visualize the simulated dynamics using plot = TRUE, which will show five sample patches and species that are randomly chosen:

mc <- mcsim(n_patch = 5, n_species = 5, plot = TRUE)

A named list of return values:

mc

## $df_dynamics
## # A tibble: 25,000 x 9
##    timestep patch_id mean_env     env carrying_capaci~ species niche_optim  r_xt
##       <dbl>    <dbl>    <dbl>   <dbl>            <dbl>   <dbl>       <dbl> <dbl>
##  1        1        1        0 -0.0494              100       1      -0.968  1.43
##  2        1        1        0 -0.0494              100       2       0.291  2.79
##  3        1        1        0 -0.0494              100       3      -0.562  1.29
##  4        1        1        0 -0.0494              100       4      -0.311  3.07
##  5        1        1        0 -0.0494              100       5       0.108  2.83
##  6        1        2        0 -0.0142              100       1      -0.968  1.35
##  7        1        2        0 -0.0142              100       2       0.291  2.85
##  8        1        2        0 -0.0142              100       3      -0.562  1.11
##  9        1        2        0 -0.0142              100       4      -0.311  2.97
## 10        1        2        0 -0.0142              100       5       0.108  2.86
## # ... with 24,990 more rows, and 1 more variable: abundance <dbl>
## 
## $df_species
## # A tibble: 5 x 6
##   species mean_abundance    r0 niche_optim sd_niche_width p_dispersal
##     <dbl>          <dbl> <dbl>       <dbl>          <dbl>       <dbl>
## 1       1           5.44     4      -0.968          0.752         0.1
## 2       2          61.0      4       0.291          0.696         0.1
## 3       3           0        4      -0.562          0.351         0.1
## 4       4          62.3      4      -0.311          0.536         0.1
## 5       5          60.6      4       0.108          0.807         0.1
## 
## $df_patch
## # A tibble: 5 x 5
##   patch_id alpha_div mean_env carrying_capacity connectivity
##      <dbl>     <dbl>    <dbl>             <dbl>        <dbl>
## 1        1      3.80        0               100       0.307 
## 2        2      3.57        0               100       0.0674
## 3        3      3.68        0               100       0.0971
## 4        4      3.5         0               100       0.0388
## 5        5      3.81        0               100       0.281 
## 
## $df_diversity
## # A tibble: 1 x 3
##   alpha_div beta_div gamma_div
##       <dbl>    <dbl>     <dbl>
## 1      3.67     1.04      3.81
## 
## $df_xy_coord
## # A tibble: 5 x 2
##   x_coord y_coord
##     <dbl>   <dbl>
## 1    5.10   4.46 
## 2    5.69   0.185
## 3    3.59   6.52 
## 4    9.60   1.11 
## 5    6.30   3.48 
## 
## $distance_matrix
##          patch1   patch2   patch3   patch4   patch5
## patch1 0.000000 4.318499 2.546035 5.621133 1.550521
## patch2 4.318499 0.000000 6.670318 4.025012 3.352600
## patch3 2.546035 6.670318 0.000000 8.088696 4.065424
## patch4 5.621133 4.025012 8.088696 0.000000 4.072806
## patch5 1.550521 3.352600 4.065424 4.072806 0.000000
## 
## $weighted_distance_matrix
## NULL
## 
## $interaction_matrix
##     sp1 sp2 sp3 sp4 sp5
## sp1   1   0   0   0   0
## sp2   0   1   0   0   0
## sp3   0   0   1   0   0
## sp4   0   0   0   1   0
## sp5   0   0   0   0   1

Custom setting: combine brnet() and mcsim()

Return values of brnet() are compatible with mcsim(). For example, df_patch$environment, df_patch$n_patch_upstream, and df_patch$distance_matrix may be used to inform parameters of mcsim():

patch <- 100
net <- brnet(n_patch = patch, p_branch = 0.5, plot = F)
mc <- mcsim(n_patch = patch, n_species = 5,
            mean_env = net$df_patch$environment,
            carrying_capacity = net$df_patch$n_patch_upstream*10,
            distance_matrix = net$distance_matrix,
            plot = T)

Custom setting: detailed parameters

Users may use the following arguments to custom metacommunity simulations regarding (1) species attributes, (2) competition, (3) patch attributes, and (4) landscape structure.

Species attributes

Arguments: r0, niche_optim OR min_optim and max_optim, sd_niche_width OR min_niche_width and max_niche_width, niche_cost, p_dispersal

Species attributes are determined based on the maximum reproductive rate r0, optimal environmental value niche_optim (or min_optim and max_optim for random generation of niche_optim), niche width sd_niche_width (or min_niche_width and max_niche_width for random generation of sd_niche_width) and dispersal probability p_dispersal (see Model description for details).

For optimal environmental values (niche optimum), the function by default assigns random values to species as: μ_i ~ Uniform(min_μ, max_μ), where users can set values of min_μ and max_μ using min_optim and max_optim arguments (default: min_optim = -1 and max_optim = 1). Alternatively, users may specify species niche optimums using the argument niche_optim (scalar or vector). If a single value or a vector of niche_optim is provided, the function ignores min_optim and max_optim arguments.

Similarly, the function by default assigns random values of σ_niche to species as: σ_niche,i ~ Uniform(min_σ, max_σ), where users can set values of min_σ and max_σ using min_niche_width and max_niche_width arguments (default: min_niche_width = 0.1 and max_niche_width = 1). If a single value or a vector of sd_niche_width is provided, the function ignores min_niche_width and max_niche_width arguments.

The argument niche_cost determines the cost of having wider niche. Smaller values imply greater costs of wider niche (i.e., decreased maximum reproductive rate; default: niche_cost = 1). To disable (no cost of wide niche), set niche_cost = Inf.

For other parameters, users may specify species attributes by giving a scalar (assume identical among species) or a vector of values whose length must be one or equal to n_species. Default values are r0 = 4, sd_niche_width = 1, and p_dispersal = 0.1.

Competition

Arguments: interaction_type, alpha OR min_alpha and max_alpha

The argument interaction_type determines whether interaction coefficient alpha is a constant or random variable. If interaction_type = "constant", then the interaction coefficients α_ij (i != j) for any pairs of species will be set as a constant alpha (i.e., off-diagonal elements of the interaction matrix). If interaction_type = "random", α_ij will be drawn from a uniform distribution as α_ij ~ Uniform(min_α, max_α), where users can specify min_α and max_α using arguments min_alpha and max_alpha. The argument alpha is ignored under the scenario of random interaction strength (i.e., interaction_type = "random"). Note that the diagonal elements of the interaction matrix (α_ii) are always 1.0 regardless of interaction_type, as alpha is the strength of interspecific competition relative to that of intraspecific competition (see Model description). By default, interaction_type = "constant" and alpha = 0.

Patch attributes

Arguments: carrying_capacity, mean_env, sd_env, spatial_auto_cor, phi

The arguments carrying_capacity (default: carrying_capacity = 100) and mean_env (default: mean_env = 0) determines mean attributes of habitat patches, which can be a scalar (assume identical among patches) or a vector (length must be equal to n_patch).

The arguments sd_env (default: sd_env = 0.1), spatial_auto_cor (default: spatial_auto_cor = FALSE) and phi (default: phi = 1) determine spatio-temporal dynamics of environmental values. sd_env determines the magnitude of temporal environmental fluctuations. If spatial_auto_cor = TRUE, the function models spatial autocorrelation of temporal environmental fluctuation based on a multi-variate normal distribution. The degree of spatial autocorrelation would be determined by phi, the parameter describing the strength of distance decay in spatial autocorrelation (see Model description).

Landscape structure

Arguments: xy_coord OR distance_matrix (+ weighted_distance_matrix), landscape_size, theta

These arguments define landscape structure. By default, the function produces a square-shaped landscape (landscape_size = 10 on a side) in which habitat patches are distributed randomly through a Poisson point process (i.e., x- and y-coordinates of patches are drawn from a uniform distribution). The parameter θ describes the shape of distance decay in species dispersal (see Model description) and determines patches’ structural connectivity (default: theta = 1). Users can define their landscape by providing either xy_coord or distance_matrix (landscape_size will be ignored if either of these arguments is provided). If xy_coord is provided (2-column data frame denoting x- and y-coordinates of patches, respectively; NULL by default), the function calculates the distance between patches based on coordinates. Alternatively, users may provide distance_matrix (the object must be matrix), which describes the distance between habitat patches. The argument distance_matrix overrides xy_coord.

weighted_distance_matrix is the supplemental argument to account for asymmetry in dispersal. For example, users may set an asymmetric distance matrix (distance between patches differs by direction, e.g., upstream to downstream v.s. downstream to upstream). This argument is enabled only if both weighted (weighted_distance_matrix) and unweighted distance matrix (distance_matrix) are given. If both arguments are given, weighted_distance_matrix will be used to calculate dispersal matrix while distance_matrix being used to calculate the spatial autocorrelation in temporal environmental variation (see Dispersal and Local community dynamics in Model description).

Others

Arguments: n_warmup, n_burnin, n_timestep

The argument n_warmup is the period during which species introductions occur (default: n_warmup = 200). The initial number of individuals introduced follows a Poisson distribution with a mean of 0.5 and independent across space and time. This random introduction events occur multiple times over the n_warmup period, in which propagule_interval determines the timestep interval of the random introductions (default: propagule_interval = ceiling(n_warmup / 10)).

The argument n_burnin is the period that will be discarded as burn-in to remove the influence of initial values (default: n_burnin = 200). During the burn-in period, species introductions do not occur.

The argument n_timestep is the simulation peiord that is recorded in the return df_dynamics (default: n_timestep = 1000). As a result, with the default setting, the function simulates 1400 timesteps (n_warmup + n_burnin + n_timestep = 1400) but returns only the last 1000 timesteps as the resulting metacommunity dynamics. All the derived statistics (e.g., diversity metrics in df_diversity and df_patch) will be calculated based on the results during n_timestep.

Model description

The metacommunity dynamics are described as a function of local community dynamics and dispersal (Thompson et al. 2020). Specifically, the realized number of individuals N_ix(t + 1) (species i at patch x and time t + 1) is given as:

where n_ix(t) is the expected number of individuals given the local community dynamics at time t, I_ix(t) the expected number of immigrants to patch x, and E_ix(t) the expected number of emigrants from patch x.

Local community dynamics

Local community dynamics are simulated based on the Beverton-Holt model:

where r_ix(t) is the reproductive rate of species i given the environmental condition at patch x and time t, r_0,i the maximum reproductive rate of species i (argument r0), K_x the carrying capacity at patch x (argument carrying_capacity), and α_ij the interaction coefficient with species j (argument alpha). Note that α_ij is the strength of interspecific competition relative to that of intraspecific competition (intraspecific competition is greater than interspecific competition if α_ij < 1; α_ii is set to be 1.0). The density-independent reproductive rate r_ix(t) is affected by environments and determined by a Gaussian function:

where μ_i is the optimal environmental value for species i (argument niche_optim), z_x(t) the environmental value at patch x and time t, and σ_niche the niche width of species i (argument sd_niche_width). The cost of having wider niche is expressed by multiplying c (Chaianunporn and Hovestadt, 2015):

Smaller values of ν (argument niche_cost) imply greater costs of wider niche (i.e., decreased maximum reproductive rate). There is no cost of wider niche if ν approaches infinity. The environmental value z_x(t), which may vary spatially and temporarily, is assumed to follow a multivariate normal distribution:

μ_z is the vector of mean environmental conditions of patches (argument mean_env) and Ω_z is the variance-covariance matrix. If spatial_auto_cor = FALSE, the off-diagonal elements of the matrix are set to be zero while diagonal elements are σ_z² (σ_z; argument sd_env). If spatial_auto_cor = TRUE, spatial autocorrelation is considered by describing the off-diagonal elements as:

where Ω_xy denotes the temporal covariance of environmental conditions between patch x and y, which is assumed to decay exponentially with increasing distance between the patches d_xy (randomly generated or specified by argument distance_matrix). The parameter φ (argument phi) determines distance decay (larger values lead to sharper declines).

Dispersal

The expected number of emigrants at time t E_ix(t) is the product of dispersal probability P_dispersal (argument p_dispersal) and n_ix(t). The immigration probability at patch x, ξ_ix, is calculated given the structural connectivity of patch x, in which the model assumes the exponential decay of successful immigration with the increasing separation distance between habitat patches:

where d_xy is the separation distance between patch x and y (randomly generated or specified by argument distance_matrix or weighted_distance_matrix). The parameter θ (argument theta) dictates the dispersal distance of species (θ^-1 corresponds to the expected dispersal distance) and is assumed to be constant across species. The expected number of immigrants is calculated as:

adjtodist

Basic usage

This function converts an adjacency matrix into a distance matrix. Sample script:

net <- brnet(n_patch = 10, p_branch = 0) # linear network

x <- net$adjacency_matrix

# adjacency matrix
print(x)

##         patch1 patch2 patch3 patch4 patch5 patch6 patch7 patch8 patch9 patch10
## patch1       0      1      0      0      0      0      0      0      0       0
## patch2       1      0      1      0      0      0      0      0      0       0
## patch3       0      1      0      1      0      0      0      0      0       0
## patch4       0      0      1      0      1      0      0      0      0       0
## patch5       0      0      0      1      0      1      0      0      0       0
## patch6       0      0      0      0      1      0      1      0      0       0
## patch7       0      0      0      0      0      1      0      1      0       0
## patch8       0      0      0      0      0      0      1      0      1       0
## patch9       0      0      0      0      0      0      0      1      0       1
## patch10      0      0      0      0      0      0      0      0      1       0

# covert an adjacency matrix into a distance matrix
adjtodist(x)

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
##  [1,]    0    1    2    3    4    5    6    7    8     9
##  [2,]    1    0    1    2    3    4    5    6    7     8
##  [3,]    2    1    0    1    2    3    4    5    6     7
##  [4,]    3    2    1    0    1    2    3    4    5     6
##  [5,]    4    3    2    1    0    1    2    3    4     5
##  [6,]    5    4    3    2    1    0    1    2    3     4
##  [7,]    6    5    4    3    2    1    0    1    2     3
##  [8,]    7    6    5    4    3    2    1    0    1     2
##  [9,]    8    7    6    5    4    3    2    1    0     1
## [10,]    9    8    7    6    5    4    3    2    1     0

References

• Chaianunporn T and Hovestadt T. (2015) Evolutionary responses to climate change in parasitic systems. Global Change Biology 21: 2905-2916.
• Csardi G, Nepusz T: The igraph software package for complex network research, InterJournal, Complex Systems 1695. 2006. http://igraph.org
• Jacolien van Rij (2020). plotfunctions: Various Functions to Facilitate Visualization of Data and Analysis. R package version 1.4. https://CRAN.R-project.org/package=plotfunctions
• Thompson, P.L., Guzman, L.M., De Meester, L., Horváth, Z., Ptacnik, R., Vanschoenwinkel, B., Viana, D.S. and Chase, J.M. (2020), A process‐based metacommunity framework linking local and regional scale community ecology. Ecol Lett, 23: 1314-1329. doi:10.1111/ele.13568