pimmlawton: Analysis of Jacobian Lotka-Volterra Food Web Matrices
In primer: Functions and Data for the Book, a Primer of Ecology with R

Description Usage Arguments Details Value Author(s) References Examples

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.

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

`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 $0<x<1$ indicating the relative weight of omniovory (after McCann et al. (1998))

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.

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.

Hank Stevens <HStevens@muohio.edu>

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

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

Pimm, S.L and Lawton, J.H. (1997) Number of trophic levels in ecological communities. Nature, 268, 329–331.

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

### 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")