popdemo (version 0.1-2)

tf: Calculate transfer function

Description

Calculate the transfer function of a given matrix and perturbation structure.

Usage

tf(A, z, b, c, exp=-1)

Arguments

A

a matrix of any dimension.

z

a numeric vector representing the set of real values over which to analyse the transfer function.

b,c

a numeric column vectors of class matrix that determines the perturbation structure (see details).

exp

the exponent to raise the resolvent to. Normally, exp=-1 but for some usages an exponent of larger magnitude may be needed.

Value

• a vector of perturbation magnitudes whose INVERSE will achieve the dominant eigenvalues given by z for the triple A, b, c.

Details

tf is primarily designed to be called by tfa, tfsens, inertia.tfa and inertia.tfsens for evaluating transfer functions and sensitivities of population projection matrix models. However, it can also function as a standalone means of calculating a transfer function. Currently tf can only work with rank-one, single-parameter perturbations. For tf to work properly, the dimensions of A, b and c must match. If A is an n by m matrix, then b must be a column vector of length m and c must be a column vector of length n. The perturbation structure is determined by b%*%t(c). Therefore, the rows to be perturbed are determined by b and the columns to be perturbed are determined by c. The specific values in b and c will determine the relative perturbation magnitude. So for example, if only entry [3,2] of a 3 by 3 matrix is to be perturbed, then b = matrix(c(0,0,1), ncol=1) and c = matrix(c(0,1,0), ncol=1). If entries [3,2] and [3,3] are to be perturbed with the magnitude of perturbation to [3,2] half that of [3,3] then b = matrix(c(0,0,1), ncol=1) and c = matrix(c(0,0.5,1), ncol=1). See Hodgson et al. (2006) for more information on perturbation structures.

References

Townley & Hodgson (2004) J. Appl. Ecol., 41, 1155-1161. Hodgson et al. (2006) J. Theor. Biol., 70, 214-224.

See Also

tfa, tfsens

Examples

# Create a 3x3 matrix
    A <- matrix(c(0,1,2,0.5,0.1,0,0,0.6,0.6), byrow=TRUE, ncol=3)
    A

    # Calculate the transfer function given a range of z and a perturbation structure
    evals<-eigen(A)$values
    lmax<-which.max(Re(evals))
    lambda<-Re(evals[lmax])
    z <- seq(lambda-0.04, lambda+0.04, 0.01)
    z <- z[ z != lambda]
    b <- matrix(c(0,0,1), ncol=1)
    c <- matrix(c(0,0,1), ncol=1)
    p<-1/tf(A, z=z, b=b, c=c)
    p

    ## Plot p and z
    plot(z~p,type="l", xlab="p",ylab=expression(lambda[max]))