fullsecder {demogR}

fullsecder

Description

Calculates the second derivatives of the dominant eigenvalue of a square matrix, A, with respect to all non-zero elements of A.

Usage

fullsecder(A)

Arguments

A

a demographic projection matrix

Details

Produces a matrix the columns of which are re-shaped matrices of second derivatives of the dominant eigenvalue of the projection matrix A with respect to each non-zero element in A.

See Caswell (1996, 2001) for details on second derivatives of the dominant eigenvalue.

Value

A square matrix. If A is a Leslie matrix of rank k, then the maximum rank of the resulting matrix is 2k-1 (since there are at most k non-zero fertilities and k-1 survival probabilities in a k x k Leslie matrix).

References

Caswell, H. 1996. Second derivatives of population growth rate: Calculation and applications. Ecology 77 (3):870-879.

Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.

See Also

secder, elassens, eigen.analysis, stoch.sens

Examples

## eigenvalue second derivatives of the US projection matrix from 1967
data(goodman)
ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx))
mx <- goodman$usa.bx/goodman$usa.nKx
usa <- leslie.matrix(lx=ult$nLx,mx=mx)

fs <- fullsecder(usa)

## plot the survival cross-second derivatives of lambda with respect
## to infant survival

plot( seq(0,40,by=5), fs["21",10:18], type="l", 
	 xlab="Age (j)", 
	 ylab=expression(paste(partialdiff^2 , lambda, "/", 
	 partialdiff, P[1] , partialdiff, P[j])))
abline(h=0,lty=3)