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Estimate the dynamic stability threshold of an ecological network

Source: R/calc_stability_threshold.R
calc_stability_threshold.Rd

This function estimates a stability threshold for an ecological network using maximum eigenvalue simulations derived from the calc_QSS() function. The goal is to approximate the dynamical stability of a system without requiring knowledge of the actual interaction strengths or the real Jacobian matrix.

Usage

calc_stability_threshold(
  g,
  nsim = 1000,
  target_prob = 0.05,
  sd_start = -1,
  sd_step = -0.1,
  sd_min = -20,
  ncores = 0,
  verbose = TRUE
)

Arguments

g

An igraph ecological network.

nsim

Integer. Number of QSS simulations per self-damping value.

target_prob

Numeric. Target proportion of negative eigenvalues (default = 0.05).

sd_start

Numeric. Starting value of selfDamping (default = -1).

sd_step

Numeric. Increment used to decrease selfDamping (default = -0.1).

sd_min

Numeric. Minimum (most negative) value allowed during search.

ncores

Logical. If > 0 run QSS simulations in parallel.

verbose

Logical. Print diagnostic messages.

Returns

A list with: thresholdInterpolated selfDamping value where the target proportion is reached. sd_lowerLower bracketing self-damping value. sd_upperUpper bracketing self-damping value. prob_lowerProportion of negative eigenvalues at sd_lower. prob_upperProportion of negative eigenvalues at sd_upper. sd_valuesAll tested self-damping values. probsCorresponding proportions of negative eigenvalues. qss_interpDistribution of maximum eigenvalues at the interpolated threshold value.

References

Allesina, S. & Pascual, M. (2008). Network structure, predator–prey modules, and stability in large food webs. Theoretical Ecology, 1, 55–64.

Grilli, J., Rogers, T. & Allesina, S. (2016). Modularity and stability in ecological communities. Nature Communications, 7, 12031.

Monteiro, A. B. & Del Bianco Faria, L. (2017). Causal relationships between population stability and food-web topology. Functional Ecology, 31, 1294–1300.

Borrelli, J. J. (2015). Selection against instability: stable subgraphs are most frequent in empirical food webs. Oikos, 124, 1583–1588.

Details

Conceptual Background

Real ecosystems persist over ecological timescales, which implies that their underlying community matrices (Jacobians) must be locally stable—that is, they produce a negative real part of the maximum eigenvalue. Because interaction strengths and self-regulation parameters are rarely known, stability must be inferred indirectly.

The calc_QSS() function provides such an indirect method by generating random Jacobian matrices whose sign structure is determined by the interaction network. Interaction magnitudes are drawn from a uniform distribution whose limits are defined by the parameters negative, positive, and selfDamping (Assumed with default values in this function).

The QSS value is the proportion of simulated matrices that are locally stable (i.e., have negative maximum eigenvalues). A higher QSS implies that the network topology is more compatible with stable dynamics.

Purpose of this Function

Because the true amount of self-regulation in a real system is unknown, this function searches for the value of selfDamping for which a target proportion of negative eigenvalues (typically 5%) is produced. This "stability threshold" represents the minimal level of self-regulation required for the network to behave as a persistently stable system.

This approach follows the idea that:

  • Real food webs must be dynamically stable.

  • Stability requires some level of intraspecific self-regulation.

  • The amount of self-regulation needed depends on network structure.

By interpolating the crossing point where the QSS proportion reaches a target value (e.g., 0.05), we obtain an estimate of stability that can be compared across networks—in the same spirit as structural stability, modularity–stability relationships, and feasibility analysis.

For details, see References below.

Parameters

Examples

if (FALSE) { # \dontrun{
g <- generate_niche(40, 0.1)
result <- calc_stability_threshold(g, nsim = 500)
plot_stability_curve(result)

# calculate the mean of the negative max eigenvalues at the threshold
mean(result$qss_raw_interp[result$qss_raw_interp < 0])

} # }