Life Table Calculations
Authors
Gareth J. Russell, Department of Ecology, Evolution and Environmental biology, Columbia University, New York, NY, USA.
Introduction
This Eco-Tool uses matrix alegbra to calculate a variety of statistics based on a table of survivorship and maternity values by age class. These terms are used sensu Caswell (1989, 2001): Survivorship l(x) is the proportion of individuals born that survive to age x. By definition, l(0) = 1, and can only decrease or remain the same for greater values of x. Maternity m(x) is the rate of production of offspring per individual per unit time at age x (usually expressed as female offspring per female in sexually reproucing species).
Age (x) is a continuous variable, but the functions l(x) and m(x) that depend on age may be continuous or discontinuous. In particular, births often occur in discrete ‘pulses.’ In either case, matrix population models require that we first convert the data into a form based on discrete age classes i = 1, 2, 3, …, etc. Age class i = 1 summarizes data for 0 < x ≤ 1, age class i = 2 summarizes data for 1 < x ≤ 2, and so on. The age class representation of suvivorship is survival, Pi, and the age class representation of maternity is fecundity, Fi.
The procedure for converting survivorship and maternity data into survival and fecundity age class data depends on whether births occur in a continuous (“flow”) or discrete (“pulse”) manner. The algorithms used here are described in detail in Caswell 1989; in particular, we use equations 2.12, 2.22 and 2.23 in the birth-flow case, and equations 2.27 or 2.28 and 2.31 or 2.32 in the birth-pulse case, depending on whether the breeding census is just before or just after the birth pulse.
After calculating survival and fecundity, the Eco-Tool constructs a Leslie matrix, and from this obtains an estimate of the long-term population growth rate λ in the form of the dominant eigenvalue of the Leslie matrix, as well the stable age structure from the corresponding eignenvector (normalized to sum to one, or alternatively 100%).
The next step — if requested — is to examine the relative sensitivity of the population growth rate λ to small changes in the non-zero values of survival and fecundity. Sensitivity analysis is very useful in a conservation context because it indicates which combinations of life history trait and age class have the biggest influence on a population's potential long-term overall growth. In this case each non-zero entry is, by itself, increased by 10% and decreased by 10%. Each time a new value of λ is calculated, and the sensitivity measure is the difference in the two λ values. Note that sensitivities calculated in this way may only be understood in the context of the other sensitivities from the same dataset — the choice of how much to perturb each life history parameter is arbitrary, and may differ from analysis to analysis, even of the same dataset (although the relative results should remain approximately the same).
The final step — if requested — is to project the overall population size based on a vector of age-class population numbers provided in the original dataset. This is accomplished by iterative pre-multiplication of the population vector by the Leslie Matrix (see Caswell 1989, Section 2.4).
References
Caswell, H. (1989) Matrix population models. Sinauer, MA.
Caswell, H. (2001) Matrix population models, 2nd ed. Sinauer, MA.