eigen.analysis: eigen.analysis
In demogR: Analysis of Age-Structured Demographic Models

Description Usage Arguments Details Value References See Also Examples

Calculates the asymptotic growth rate and related quantities from a demographic projection matrix.

1	eigen.analysis(A)

`A`	demographic projection matrix

Calculates the asymptotic growth rate (lambda) of a population described by demographic projection matrix A. The asymptotic growth rate of the population is given by the dominant eigenvalue of the projection matrix. By the Perron-Frobenius Theorem, this eigenvalue is guaranteed to be real, positive and strictly greater than all the other eigenvalues if the matrix A is non-negative, irreducible, and primitive (for details see Caswell (2001)).

Also calculates the damping ratio (rho), eigenvalue sensitivities, eigenvalue elasticities, the stable age distribution (for the communicating parts of the life cycle), and scaled reproductive values.

The damping ratio is the ratio of the dominant eigenvalue and the absolute value of the second eigenvalue. rho is a measure of the rate of convergence to the stable age-distribution. A population characterized by damping ratio rho will converge asymptotically to the stable age distribution exponentially with rate at least as fast as log(rho). Clearly, a population already at or very near the stable age distribution will converge faster, but rho provides an upper bound.

The eigenvalue sensitivities are the partial derivatives of lambda with respect to a perturbation in matrix element a_{ij}. The sensitivities measure the selection gradient on the life-cycle (Lande 1982). The eigenvalue elasticities are scaled to be proportional sensitivities of lambda to a perturbation in a_{ij}. Elasticities have a number of desirable properties including, their sum across all life-cycle transitions is unity and the sum of the elasticities of all incoming arcs to a life-cycle stage must equal the sum of all outgoing arcs (van Groenendael et al 1994).

The stable age distribution is normalized to represent the proportion in each of the communicating age classes. If the population is characterized by post-reproductive survival (and hence age classes that do not communicate with the rest of the life cycle graph), then other methods should be used to calculate to stable distribution. For example, from classic stable population theory, we know that the stable age distribution of the population c(x) is given by the relationship:

c(x) = b l(x) exp(-r*x)

where b is the gross birth rate, l(x) is survivorship to age x and r is the rate of increase of the population (=log(lambda)). See Coale (1972) or Preston et al. (2001) for details.

The age-specific reproductive values are normalized so that the reproductive value of the first age class is unity. Problems associated with post-reproductive survival are irrelevant for reproductive value since the reproductive value of post-reproductive individuals is, by definition, zero.

A list with six components:

`lambda1`	the asymptotic growth rate (dominant eigenvalue) of A
`rho`	damping ratio of A
`sensitivities`	eigenvalue sensitivities of A
`elasticities`	eigenvalue elasticities of A
`stable.age`	stable age distribution of A
`repro.value`	reproductive values of A

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

Coale, A.J. 1972. The growth and structure of human populations: A mathematical investigation. Princeton: Princeton University Press.

Lande, R. A. 1982. A quantitative genetic theory of life history evolution. Ecology 63:607-615.

van Groenendael, J., H. De Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: Evaluating life history pathways in population projection matrices. Ecology 75 (8):2410-2415.

leslie.matrix, secder

data(goodman)
ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx))
ven.mx <- with(goodman, ven.bx/ven.nKx)
A <- leslie.matrix(lx=ven$nLx,mx=ven.mx)

usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx))
usa.mx <- with(goodman, usa.bx/usa.nKx)
B <- leslie.matrix(lx=usa$nLx,mx=usa.mx)

eav <- eigen.analysis(A)
eau <- eigen.analysis(B)

## compare rates of increase

eav$lambda1
eau$lambda1

## compare stable age distributions for Venezuela and the USA
op <- par(no.readonly = TRUE)
age <- seq(0,50,by=5)
par(mfrow=c(1,2))
barplot(eav$stable.age, names.arg=age, 
	     horiz=TRUE, 
	     col="grey", 
	     ylab="Age")
barplot(eau$stable.age, names.arg=age[-11], 
	     horiz=TRUE, 
	     col="blue", 
	     ylab="Age", 
	     xlim=c(0,0.2))

par(op)

## Compare reproductive values for Venezuela and the USA

plot(age, eav$repro.value, type="l", xlab="Age", ylab="Reproductive Value")
lines(age[-11], eau$repro.value, lty=2)
legend("topright", c("Venezuela (1965)", "USA (1967)"), lty=c(1,2))