getStability: Get the mathematical stability of a matrix.

In dswdejonge/fwstability: Infer Food-Web Stability

Get the mathematical stability of a matrix.

Description

This function can find the initial and asymptotic stability of a Jacobian matrix.

Usage

getStability(
  JM,
  method = "eigenvalue",
  MR = NULL,
  dead_names = NULL,
  iters = 100
)

Arguments

JM	(required) A square named Jacobian matrix with numeric values representing the effect of one compartment (rows) on another compartment (columns).
method	(required) Either "eigenvalue" (default), "scalar", or "initial". The method "eigenvalue" finds asymptotic stability as the maximum real part of the eigenvalues calculated from the Jacobian matrix. The "scalar" method finds asymptotic stability as the scalar of natural mortality rates needed to acquire a stable matrix. The "initial" method finds initial stability as equation 4 from Arnoldi et al. (2016); initial stability is half the maximum real part of the eigenvalues derived from the matrix obtained by addition of the Jacobian matrix to its transpose.
MR	(required if method is "scalar") A named numeric vector containing mortality of the faunal compartments (per unit time, t-1). The values and names must be in the same order as the Jacobian matrix, and the values for dead compartments should be set to NA.
dead_names	(optional if method is "scalar") Character vector with all names of detritus and nutrient compartments (everything that is not fauna).
iters	(required if method is "scalar") Default is 100. The max number of iterations to find stability with the iterative scalar method. A greater number of iters will result in a higher resolution of stability value.

Details

Initial stability is the initial reaction of a system to a small perturbation. Initial stability is found by equation 4 from Arnoldi et al. (2016), with the difference that th function in this package does not change the sign of the value. Therefore, a negative initial stability means the system is reactive i.e. the perturbations are amplified.

Asymptotic stability is the long term reaction of a system to a small perturbation. Asymptotic stability can be found either as the maximum value of the real part of its eigenvalues (requires a quantified diagonal) (May 1972) or as the scalar of natural mortality rates that results in a stable matrix (requires mortality rate estimates) (De Ruiter et al. 1995).

The interpretation of the "eigenvalue" method relies on the quantification of the diagonal in the Jacobian matrix. If there has been no thought on the quantification of the diagonal, the "eigenvalue" method might not be informative. The diagonal can be quantified with upperbound values for self-dampening by setting the argument diagonal to "model" in the function getJacobian

The "scalar" method for asymptotic stability relies on the estimation of natural mortality rates (unit per time, t-1). A critical matrix, i.e. a matrix that is on the stability threshold, is calculated by setting the diagonal to the given mortality values and subtracting the maximum real part of the eigenvalues from each diagonal value. A stepsize is then determined by estimating the scalars needed to acquire the critical matrix from the given mortality values, and dividing the largest scalar by 100. Subsequently, the provided natural mortality is iteratively scaled with the determined stepsize until the matrix becomes stable, i.e. the maximum real part of the eigenvalues is negative. If dead compartments exist, their diagonal values are not scaled, because dead compartments do not have natural mortality rates. Therefore, the diagonal values of dead compartments in the Jacobian matrix should be quantified correctly, or set to zero (assuming there is no intracompartmental detritus feedback).

Mortality rates per unit time can for example be calculated as the mortality flux (biomass per surface area per unit time) divided by the biomass per surface area, or as the inverse of the natural lifespan of the species. Mortality fluxes can be found with the function getMortalityRates if you have the relevant data; the function assumes natural mortality equals production (AE x GE x Consumption) minus predation (flux to all other faunal compartments).

Value

This function returns a numeric value. For the "eigenvalue" method a negative value indicates a stable matrix. For the "scalar" method the value represents the fraction self-dampening effect needed for system stability.

References

• Arnoldi, J.F., Loreau, M., Haegeman, B., 2016. Resilience, reactivity and variability: A mathematical comparison of ecological stability measures. J. Theor. Biol. 389, 47–59. https://doi.org/10.1016/j.jtbi.2015.10.012

• de Ruiter, P.C., Neutel, A.M., Moore, J.C., 1995. Energetics, Patterns of Interaction Strengths, and Stability in Real Ecosystems. Science (80-. ). 269, 1257–1260. https://doi.org/10.1126/science.269.5228.1257

• May, R.M., 1972. Will a large network be stable? Nature 238, 37–38. https://doi.org/10.1038/238413a0

Examples

# Use example food-web model of soil
model <- fwmodels::LovinkhoeveCP
# Calculate the Jacobian Matrix
JM <- getJacobian(model)
# Find asymptotic stability using the eigenvalue method
getStability(JM)
# Find asymptotic stability using the scalar method
getStability(JM, method = "scalar", MR = model$MR, dead_names = model$dead$names)
# Find initial stability
getStability(JM, method = "initial")

dswdejonge/fwstability documentation built on Dec. 7, 2022, 7:24 p.m.