primer (version 0.1)

pimmlawton: A function to analysis Jacobian matrices of foodwebs

Description

Used primarily to repeat simulations and analyses of Pimm and Lawton (1977), given a Jacobian matrix. Analyses include eigenanalysis, but also measuring average interaction strength (May 1972), average intraspecific negative density dependence, and the strength of the omnivory interaction, if present.

Usage

pimmlawton(mat, N = 1, omni.i = NA, omni.j = NA, omega = NULL)

Arguments

mat

a numerical matrix; the "maximum" Jacobian matrix. See details below.

N

a scalar for the number of randomizations

omni.i

if omnivory is present, the row/col index for the prey.

omni.j

if omnivory is present, the row/col index for the predator.

omega

if not NULL, a scalar $0

Value

• Returns a data frame, where each row corresponds to a single random Jacobian matrix, with the following columns.
• DomEig
  the real part of the dominant eigenvalue
• Im
  the imaginary part of the dominant eigenvalue
• IntraDD
  average magnitude, over all species, of the intraspecific negative density dependence; the square root of the sum of the squared diagonal elements of the random Jacobian matrix
• I
  average interaction strength (after May 1972); the square root of, sum of the squared off diagonal elements divided by the number of off diagonal elements.
• I.omni
  average interaction strength for the omnivory interaction; the square root of, sum of the squared omnivory elements effect of predator on prey and prey on predator divided by two.

Details

This function simulates a constrained randomization of a Jacobian food web matrix. The matrix it uses mat is of a special form, which assumes that all non-zero values are drawn from a uniform distribution between zero and a value of some specified magnitude, either positive or negative.

References

R.M. May. Stability and Complexity in Model Ecosystems, volume 6 of Monographs in Population Biology. Princeton University Press, 1973.

K. McCann, A. Hastings, and G.R. Huxel. Weak trophic interactions and the balance of nature. Nature, 395:794--798, 1998.

S.L. Pimm and J.H. Lawton. Number of trophic levels in ecological communities. Nature, 268:329--331, 1977.

M.H.H. Stevens. A Primer of Ecology with R. Use R! Series. Springer. 2009.

Examples

### A relevant style of matrix for the first food chain in Pimm and Lawton (1977). Note each non-zero element is the appropriate sign, and the maximum magnitude specified by Pimm and Lawton (1977)
Aq = matrix(c(
  -1,   -10,   0,     0,
  0.1, 0,   -10,  0,
  0,    0.1,   0,    -10,
  0,      0,    0.1,   0),
  nrow=4, byrow=TRUE)

pimmlawton(Aq, N=1)

out <- pimmlawton(Aq, N=2000)
out <- subset(out, -1/DomEig <150)
hist(-1/out$DomEig, main="Frequencies of Return Time")