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.