In franzikoch/Feedbackloops: Feedback loop analysis for competition networks

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(Feedbackloops)

Converting empirical data on overgrowth competition between bryozoan colonies into Jacobian community matrices requires the following steps:

1. Read in the raw data files (species-contact matrix and abundance table).
2. Convert the species-contact matrix to a tabular format that contains one row for each pair of competitors
3. Calculation of biomass loss rates and interaction strengths using the table created in the previous step and the abundances of each species (measured in colonies per species).
4. Assembling the Jacobian matrices by putting each interaction strength into its correct position within the matrix.

Read in raw data files

For each Jacobian, two data files are needed:

1. The species-contact-matrix csv file contains the wins/losses/draws/total interactions for each pair of interacting species. The first row contains species names and is used to name the columns of the table. Because the species contact matrix contains a 2 x 2 block of cells for each interaction, each species name is always repeated twice (both in the rows and in the columns).
2. The abundance csv file contains a table with two columns: species name and corresponding number of colonies per species.

The function read_data(path_to_species_contact_matrix, path_to_abundance_table) is used to read in the raw data files (only works if they have the exact format specified above!). The function returns a list with two elements: The first is a data.frame containing the species contact matrix in the correct format to be processed further and the second is another data.frame containing the abundance table. The function removes species that appear in the abundance table but not in the species contact matrix. Also, it makes sure that species in the abundance table are ordered in the same way as in the species-contact-matrix (Important because Dave ordered his matrices according to competitive rank.)

To read datasets from csv files use:

path_competition <- "Competition_Signy_1.csv"
path_abundance <- "Abundance_Signy_1.csv"

contact_matrix <- read_contact_matrix(path_competition)
abundance <- read_abundance(path_abundance, contact_matrix)

The competition table/species-contact matrix should look like this:

head(contact_matrix)

and the abundance table like this:

head(abundance)

Create a table of pairwise interactions, calculate biomass loss rates and interaction strengths

To make calculations easier, the function interaction_strengths(competition_table, abundance_table, cost_list) first converts the competition matrix to a data.frame with one row for each pairwise interaction. In this format, the biomass loss rate of species $i$ due to competition with species $j$ can easily be calculated with the following formula:

$F_{ij} = c_W W_{ij} + c_L L_{ij} + c_D D_{ij}$

where $W_{ij}$, $L_{ij}$ and $D_{ij}$ represent the number of wins, losses and ties for species $i$ in confrontation with species $j$ and $c_W$, $c_L$ and $c_D$ are the so-called cost-values. Cost-values are the fixed amount of lost biomass assigned to each type of competitive outcome. In the case of intraspecific interactions (species $i$ == species $j$), draws need to be counted twice, as such an interaction affects two colonies but only one confrontation is recorded in the species-contact matrix. For each row in the interaction table, two biomass loss rates ($F_{ij}$ and $F_{ji}$) are calculated.

To convert a given biomass loss rate $F_{ij}$ into interaction strengths (as in the element $a_{ij}$ of the Jacobian matrix), it is simply divided by the abundance $B$ of species $j$: $a_{ij} = \frac{F_{ij}}{B_j}$ (this follows from the formulation as Lotka-Volterra equation, see Methods in Koch \& Neutel (2023) for details).

The interaction table returned by interaction_strengths() contains biomass loss rates and interaction strengths for all those interactions. The cost values must be supplied in the cost list in the following order: $c_W$, $c_L$, $c_D$.

interaction_table <- interaction_strengths(contact_matrix, abundance, c(-0.1, -0.9, -0.2))
head(interaction_table)

Assemble the Jacobian matrix

We already calculated all elements of the Jacobian in the previous step. The function assemble_jacobian() is responsible for putting them in the correct order. The species are ordered according to their position in the abundance table, so that the Jacobian will have the same order as the original species contact matrix. Note, that the order of species does not affect the eigenvalues / stability of the network.

assemble_jacobian() takes the following arguments:

• the interaction_table which was returned by interaction_strengths()
• species_list a list of species. The Jacobian is ordered according to this list. Use the first column of the abundance table!
• ij_col name of the column that contains the effects of species $i$ on species $j$
• ji_col name of the column that contains the effects of species $j$ on species $i$

Note: Choosing column names is important once we start randomising matrices, which adds new columns to the interaction table.

Jacobian = assemble_jacobian(interaction_table, abundance$species, "a_ij", "a_ji")
print(Jacobian)

franzikoch/Feedbackloops documentation built on July 1, 2023, 12:42 p.m.