# Calculate the intrinsic stochastic invariability of a community from its community matrix. `stoch_var` calculates the intrinsic stochastic ( $I_S$ ) invariability - a theoretical equivalent of the univariate measure of invariability (Arnoldi et al. 2016). ```r stoch_var(B) ``` ## Arguments - `B`: a matrix, containing the interactions between the species or functional groups in the community. Can be calculated with `extractB` from the fitted MARSS object. ## Returns A numeric, the stochastic variability. The larger its value, the more stable the system, as its rate of return to equilibrium is higher. ## Details $$ I_S = \frac{1}{\lVert B^{-1} \rVert} $$ where $\lVert B^{-1} \rVert$ is the spectral norm of the inverse of the matrix $B$. The spectral norm is computed as the dominant eigenvalue of the matrix $$ B \otimes I + I \otimes B $$ where $I$ is the identity matrix. ## Examples ```r library(MARSS) # smaller dataset for example: # 3 functional groups and two insecticide concentrations besides control data_df <- subset( aquacomm_fgps, treat %in% c(0.0, 0.9, 6) & time >= 1 & time <= 28, select = c(time, treat, repli, herb, carn, detr) ) # estimate z-score transformation and replace zeros with NA data_df[, c("herb", "carn", "detr")] <- lapply(data_df[, c("herb", "carn", "detr")], MARSS::zscore) data_df[, c("herb", "carn", "detr")] <- lapply(data_df[, c("herb", "carn", "detr")], function(x) replace(x, x == 0, NA)) # reshape data from wide to long format data_z_ldf <- reshape( data_df, varying = list(c("herb", "carn", "detr")), v.names = "abund_z", timevar = "fgp", times = c("herb", "carn", "detr"), direction = "long", idvar = c("time", "treat", "repli") ) data_z_ldf <- data_z_ldf[order(data_z_ldf$time, data_z_ldf$treat, data_z_ldf$fgp), ] # summarize mean and sd data_z_summldf <- aggregate(abund_z ~ time + treat + fgp, data_z_ldf, function(x) c(mean = mean(x, na.rm = TRUE), sd = sd(x, na.rm = TRUE))) data_z_summldf <- do.call(data.frame, data_z_summldf) names(data_z_summldf)[4:5] <- c("abundz_mu", "abundz_sd") # split dataframe per functional groups # into list to apply the MARSS model more easily split_data_z <- split(data_z_summldf[, c("time", "fgp", "abundz_mu")], data_z_summldf$treat) reshape_to_wide <- function(df) { df_wide <- reshape(df, idvar = "fgp", timevar = "time", direction = "wide") rownames(df_wide) <- df_wide$fgp df_wide <- df_wide[, -1] # Remove the 'fgp' column as.matrix(df_wide) } data_z_summls <- lapply(split_data_z, reshape_to_wide) # fit MARSS models data.marssls <- list( MARSS( data_z_summls[[1]], model = list( B = "unconstrained", U = "zero", A = "zero", Z = "identity", Q = "diagonal and equal", R = matrix(0, 3, 3), tinitx = 1 ), method = "BFGS" ), MARSS( data_z_summls[[2]], model = list( B = "unconstrained", U = "zero", A = "zero", Z = "identity", Q = "diagonal and equal", R = matrix(0, 3, 3), tinitx = 1 ), method = "BFGS" ), MARSS( data_z_summls[[3]], model = list( B = "unconstrained", U = "zero", A = "zero", Z = "identity", Q = "diagonal and equal", R = matrix(0, 3, 3), tinitx = 1 ), method = "BFGS" ) ) # identify experiments names(data.marssls) <- paste0("Conc. = ", c("0", "0.9", "44"), " micro g/L") # extract community matrices (B) data.Bls <- data.marssls |> lapply(extractB, states_names = c("Herbivores", "Carnivores", "Detrivores")) # calculate intrinsic stochastic variability for each of the B matrices purrr::map(data.Bls, stoch_var) ``` ## References Arnoldi, J.-F., Loreau, M., & Haegeman, B. (2016). Resilience, reactivity and variability: A mathematical comparison of ecological stability measures. Journal of Theoretical Biology, 389, 47–59. tools:::Rd_expr_doi("10.1016/j.jtbi.2015.10.012")

stoch_var function