Calculations using life tables

The first calculation we’ll look at is that for ex – the ‘life expectancy (average years of remaining life) for an organism of age x.

To estimate of the mean life expectancy for an age interval, something needs to be done to correct lx, which is the survivorship to the beginning of the age interval. That correction is to get the average survivorship over the interval, i.e.

Lx = (lx + lx+1)/2

Next we determine the total number of 'animal-years' of survival left in the population following age x, and divide by the total number of animals which accumulated those animal-years of survival.

Method: sum the Lx column from x (the age for which you are determining ex) to the bottom of the life table, and divide the sum by lx. Each unit in Lx is one unit of

animal survival.

Here is an example:

Age x lx Lx Lx ex

  0 1000 900 2500 2.50 1 800 700 1600 2.00 2 600 500 900 1.50 3 400 300 400 1.00 4 200 100 100 .50 5 0 0 0 0

Exam hint: If given ex, could you back-calculate lx?

This life table indicates a monotonic decline in both lx and ex. The monotonic decline in lx is required by the way it is defined. However, real data for human populations (e.g. England, early 20th century) and many others indicates life expectancy rises early in life. How can ex rise at any point, when the lx from which it's derived falls?

This life table indicates a monotonic decline in both lx and ex. The monotonic decline in lx is required by the way it is defined. However, real data for human populations (e.g. England, early 20th century) and many others indicates life expectancy rises early in life. How can ex rise at any point, when the lx from which it's derived falls?

Answer: When sources of mortality are concentrated upon some particular age class(es), surviving that period may well mean that the life expectancy for the average individual just being exposed to that mortality is less than that for one who's made it through. Look back at the figure that showed ex from Roman times to early 20th century England. The life expectancy showed a transient increase after birth in that English data.

An example:

Age x lx Lx Lx ex

  0 1000 600 800 0.8 1 200 150 200 1.0 2 100 50 50 0.5 3 0 0 0 0

In the life table heavy mortality is concentrated in the neonate age class; 80% of all newborns die before their first birthday. Of those who survive to have a first birthday, a heavy, but lower mortality occurs between ages 1 and 2. The result is a higher life expectancy for those who reach their first birthday than for newborns.

Now we’ll do the same calculation for a life table you choose, so that you can see how the calculations are done.

There are a number of other ways of presenting the same data, with differing advantages for each. One of the ways is to look at the number of animals dying in each age class, called dx. This number can be added across age categories to give the total number of deaths. However, it does not indicate how intensely mortality factors are acting.

That information is more easily drawn from the age-specific mortality rate, or qx. That number is the proportion of animals who survived to age (or age class) x, but died before reaching age (or age class) x+1.

qx = (lx - lx+1)/lx

Here’s our original life table, now calculating qx:

Age x lx lx - lx+1 qx

0 1000 200 .200 1 800 200 .250 2 600 200 .333 3 400 200 .500 4 200 200 1.000 5 0 - -

There are some common patterns in qx that make it useful (with limitations). A plot of qx versus age is a direct, visual indication of mortality pressures on individuals of different ages. qx curves for a variety of species are remarkably similar when age scales are adjusted to ‘match’. Here’s the plot of human survivorships as qx:

Here’s a curve of qx for Ovis dalli, a ‘mountain sheep’:

Caughley noted, as have others since, that susceptibility to mortality shows an age-dependent maximum at very early ages and in the post-reproductive period. That seems to be a broad, accurate generality.

qx is the probability that an animal of age x will die before its next 'birthday'. A plot of qx is thus a direct, visual indication of mortality pressures on individuals of different ages.

If a population faces some sort of age-dependent bottleneck, i.e. a time (age) period during which either biotic or abiotic stress is particularly severe, then qx will rise there andindicate that.

The impact of modern medicine in man seems mostly to influence those susceptible ages. Limits to the ability toprolong life are set at present by genetically 'programmed' biological aging (senescence). Genetic engineering may, sometime in the future, alter that limitation.

There is one problem with qx. If you are interested in working with a different time interval, or combining age classes, qx is not additive.

A new variable corrects the problem. The variable is called 'k', and referred to as the killing power. k has the advantages of both dx in additivity across age classes, and of qx in indicating the intensity of mortality in the age classes individually. It gains these advantages by using logs. kx is defined as:

  kx = log10Nx - log10Nx+1

Now consider why this measure is additive. Let’s look across 2 units of time.

kx→x+2 = kx→x+1 + kx+1→x+2 = log10Nx - log10Nx+1

+log10Nx+1 - log10Nx+2

Age x lx lx - lx+1 qx kx

  0 1000 200 .200 .0969 1 800 200 .250 .1249 2 600 200 .333 .1761

3 400 200 .500 .3010 4 200 200 1.000 - 5 0 - -

A book by Williamson (1972), that strongly advocated the use of killing power provided two examples:

K for different life stages of the pine looper:

k values are both higher and more variable for two life stages – eggs and larvae.

As a control strategy, those are the life history stages most vulnerable to chemical or biological control.

What is the pattern of k versus population density if different forms of population regulation occur?

If logistic growth occurs, then 1/N dN/dt decreases in direct proportion to density. That can occur through decreases in the birth rate and/or increases in the death rate. The latter would be evident in a plot of k versus density which has a positive slope.

If changes in both birth and death occurred, only a fraction of the density response would be evident in k, and the slope would be <1. If only death responses occurred the slope would be 1.

Under some circumstances overcompensation could occur,producing a slope greater than 1.

Inverse density dependence seems unlikely, but where groups act to mutually protect their members, or at least influence thelikelihood of mortality, like schools of fish or prairie dog towns, it might occur.

The absence of any density-dependence is, of course, evident in a slope of 0.

Graphs of the various possibilities form the next slide…

There is an advantage in comparing susceptibility tomortality among age classes within a population.

Draw a straight line from the starting population to the age at which all have died (in a ln plot). If this were the plot of survivorship, then rates of mortality would be constant for all age classes, mortality would be uniformly distributed, or a constant proportion would have died in each age class (alternative ways of saying the same thing).

If the real curve is steeper than this line for any age class(es), mortality is relatively concentrated in those age classes, and if shallower, those age classes are relatively less susceptible.

The theoretical type II curve is the base plot. Remember it has uniform proportional mortality across ages.

Hint:

Make sure you understand the relationships among ways of looking at survivorship, and that you can draw curves for each of the ways of looking at survivorship for each of the three basic patterns identified.

Before further life table calculations, we need to consider the other measured variable, births or natality.

Natality

The pattern of births is as important as the pattern of survivorship. We already have one portion in hand: semelparity (in plants – monocarpic) versus iteroparity (in plants – polycarpic).

Monocarpic plants are typically short-lived, e.g. annuals or biennials. There are some interesting exceptions we’ll consider later. Among them are century plants (agave) and some bamboo species. Even in monocarpic plants, there is a distribution of reproductive activity through the season (and, thus, for the population).

Although these data, for temporal distribution of seed production across a season in an annual grass, Poa annua, are species specific, the general pattern in the graph – a rapid rise to peak output, then a similarly rapid fall – is one fairly widely observed pattern.

Age specific patterns of interest are all among iteroparous species. Some species increase their reproductive output until slowed by old age. Think of a maple tree. It grows each year, and, as a result of growth, generally increases its seed output with age.

α Ω

Other species more-or-less rapidly increase reproductive output soon after , then maintain a fairly stable output for an extended period until approaching age . Dogs (or wolves) are a well known example of this pattern.

α Ω