secder {demogR}

secder

Description

Calculates the second derivatives of the dominant eigenvalue of the demographic projection matrix for all non-zero transitions with respect to one specified transition.

Usage

secder(A, k, l)

Arguments

A

demographic projection matrix

k

row index for the specified transition

l

column index for the specified transition

Details

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

Value

A square matrix of the same rank as A where each element s_{ij} is the second derivative of the dominant eigenvalue of A, \partial^2 \lambda/\partial a_{ij} \partial a_{kl}.

Note

The eigenvalue second derivatives are essential for calculating both perturbation analyses of the eigenvalue elasticities and stochastic sensitivities. secder is used in functions to calculate both these quantities.

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

fullsecder, elassens, eigen.analysis, stoch.sens

Examples

## eigenvalue second derivatives of the US projection matrix from 1967
## with respect to infant survival
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)

sd21 <- secder(usa,2,1)