Life Tables

Background Definitions

A population is a group of similar individuals of the same kind living in the same area at the same time. Each population has characteristics which are unique to that population, including age distribution, population density, population distribution in time and space, the birth rate, the death rate, and the population growth rate. Populations are interrelated with populations of other organisms. For example, a population of predators affects the mortality rate of a prey population.

Some organisms (populations) have annual life cycles with non-overlapping generations. Examples of these would include many herbaceous plants and many insects. In populations of these organisms, nearly all members of the population are the same age at the same time. Organisms with overlapping generations and/or which are continuously breeding, tend towards a stable age distribution: the ratio of age groups remains the same as long as birth and death rates remain the same. Note that any influence which changes the death rate of a population will also affect the birth rate and age structure of the population. Organisms (populations) of species which live longer can be divided into three ecological periods: pre-reproductive, reproductive, and post-reproductive. In these organisms, the length of time in each stage depends on the overall life history of that species.

In nature, a number of factors determine the rate of increase (r) of a population of a given species. The maximal rate of increase under optimum conditions (the innate capacity for increase) is symbolized by rm. Birth rate (natality) and death rate (mortality) are influencing factors. Note that the birth rate can be less than, equal to, or greater than the death rate.

A life table gives the probability at birth of being alive at age x (designated as lx). At zero age, this is l0, which by definition, equals one (if lx is expressed as a fraction of the total — if lx is expressed in whole numbers, l0 will be equal to the total). For example, in a table where x = 4.5 weeks and lx = 0.87, this means that from a sample of 100 newly-laid eggs, 87 will survive for 4.5 weeks. Some other symbols used in these calculations are:
x = a given age group within the population. This might be expressed in days, weeks, or years depending on the life span of the organism, or may be expressed as stages in the life cycle (such as in insects).
lx = the actual number or the proportion (as a decimal or percentage) of survivors at the beginning of age interval x. Note that since several samples are often averaged together, the lx values may not always be whole numbers.
Lx = the average (X) number of years lived by all individuals in each age category = (lx + lx + 1) ÷ 2.
Tx = total number of time units (years, weeks, months, etc.) left for all individuals to live from age x onward = Σ Lx **FROM THE BOTTOM UP!**.
ex = life expectancy for each interval = Tx ÷ lx.
dx = the number of individuals that die during time interval x (expressed as an actual number or as a proportion of the total). Note that lx+1 = lx – dx.
Σ dx = the overall number of individuals that died.
dxf = the cause of death (not a mathematical quantity).
qx = the mortality rate = dx/lx (often × 100 = percentage or × 1000 = number per 1000). Optionally, qx may also be calculated based on the number surviving at the end of a given time period ÷ the number alive at the beginning of that time period.
Mx = total eggs or young produced per female at age x.
mx = the X number of female births to each age group of “mothers”; the number of eggs or young which are female (in a species with a 1:1 sex ratio, this = Mx/2). Since, for most organisms, one male can fertilize a number of females, the size of the population is more dependent on the number of females present, and the calculations are usually done using only females.
lxmx = the X number of females born to each age group, adjusted for survivorship, or (prob. of reaching age x) × (# of female eggs at age x) = # of births per female.
t = some other time interval.
lt/lx = the proportion of females living from age x to age t.
vx = the reproductive value of each age group = (lt/lx) × mx.
N0 = the number of individuals at time zero; the number of females at the beginning of the experiment
Nt = the number of individuals after time t; the number of females at time t; the number of females after one or more generations
R0 = Σ lxmx (summed over all ages) = the net reproductive rate; the ratio of total female births in two successive generations, the ratio of offspring to parents, or the number of female offspring that will be left during her lifetime by one female. Also, R0 = N1/N0 = Nt+1/Nt, and thus, N1 = N0 × R0 (the population is growing exponentially). The closer R0 is to 1, the slower the population growth, and if R0 is less than one, the population is declining.
R0 = 
N0Σ lxmx
N0		 = Σ lxmx = Nt
N0
T = the mean time from birth of parents to birth of offspring; the average length of time for one generation; average age of parents who had offspring
T = 
(# of offspring) × (age when had)
total # of births (= R0)		 = Σ lxmxx
Σ lxmx		 = Σ lxmxx
R0
If rm = maximal rate of increase and T = time for one generation, then rmT = # of individuals at time T.
By definition, R0 = ermT or ln(R0) = rmT, thus rm = [ln(R0)]/T

Survivorship Curves From the information in life tables, various curves may be plotted. A mortality curve is a plot of qx vs. x. These are often “J”-shaped: sometimes there is a high, intitial mortality, but there is typically a period of low mortality followed by a period of highter mortality later in life (more die when they’re older). Survivorship curves may be a plot of lx vs. x, but are more often a plot of log(lx) vs. x. If the log(lx) is used, survivorship curves tend to fall into one of three general types, indicative of higher mortality later in life (I), constant mortality throughout life (II), or higher mortality early in life (III). Mortality and survivorship curves may be used to compare survival of the sexes or of populations existing in different places or at different times.


Problems

Generic Life Table

  1. Complete Table 1 and then calculate T and rm using the formulae above.

Table 1. Life Table and Age-Specific Fecundity Rate of Insects in a Laboratory Environment.
Age in Weeks (x)lxMx/2=mxlxmxlxmxx
0.5
1.5
2.5
3.5		 0.90 . . . Immature Stages . . .
4.5 0.87 20.0
5.5 0.83 23.0
6.5 0.81 15.0
7.5 0.80 12.5
8.5 0.79 12.5
9.5 0.77 14.0
10.5 0.74 12.5
11.5 0.66 14.5
12.5 0.59 11.0
13.5 0.52 9.5
14.5 0.45 2.5
15.5 0.36 2.5
16.5 0.29 2.5
17.5 0.25 4.0
18.5 0.19 1.0
Σ = R0 =   Σ = 
T = Σ lxmxx/Σ lxmx = Σ lxmxx/R0 = 
rm = ln(R0)/T = 
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  1. In your lab notebook, make a graph (actually two graphs in one) of the data in Table 1 as follows:
    1. On the horizontal (x) axis, place the age in weeks (x) using a scale of 0 to 20.
    2. On the vertical (y) axis on the LEFT side of the page, place the probability of survival (lx) using a scale of 0 to 1.
    3. On the vertical axis on the RIGHT side of the page, place the female offspring per female per week (mx) using a scale of 0 to 25.
    4. Graph the data against the appropriate axes.

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Comparing and Interpreting Life Table Data

  1. The comparison of life tables over a period of years/generations can help to reveal the influences of climate, parasites and predators, diseases, food supply, etc. Tables 2 and 3 are each for one generation of an insect called a spruce budworm in an experimental plot consisting of 10 sq. ft. of branch surface on the spruce tree. Note that instar refers to an insect or a stage in an insect’s life in between two molts. Calculate the qx values and other indicated values for Tables 2 and 3, then in your lab notebook, answer the questions which follow.

Table 2. Life Table for the Spruce Budworm in Experimental Plot A.
xlxdxFdx100(dx/lx)
= 100(qx)
eggs 174.00 parasites, predators 18.00
other causes 1.00
TOTAL
instar I (1st larval stage)   155.00 dispersion, etc. 74.40
overwintering stage 80.60 winter 13.70
instar II 66.90 dispersion, etc. 42.20
instars III through VI 24.70 parasites 8.92
disease 0.54
birds 3.39
other causes 10.57
TOTAL (III - VI)
pupae 1.28 parasites, predators 0.23
other causes 0.23
TOTAL
# moths to adult 0.82 sex ratio 50:50 — OK 0.00 0.00
# moths remaining 0.82 abnormal or small 0.00 0.00
# living to reproduce 0.82

generation totals 174.00 173.18
Since 0.82 moths lived to reproduce and 50% are females, that means 0.41 are
females. The average female lays an average of 150 eggs, thus the expected number
of eggs would be 0.41 × 150 = 61.5. The actual eggs counted = 575, so:
migration:  61.5 – 575
= –513.5†		 (–513/62)×100
= –834.96%*
The index of population trend is:  expected = (61.5/174) × 100 = 35.34% increase§
observed = (575/174) × 100 = 330.46% increase‡
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Since there are equal numbers of males and females, no females are left without mates. Note that in Table 3, the ratio is 54:46, and thus 8 females out of 100 are left without mates. Note that 8/100 × 1.29 = 0.1032, so it’s as though only 1.29 – 0.10 = 1.19 are remaining to potentially reproduce.
†Expected number of eggs – actual, observed number of eggs = difference (in this case negative because the observed number was higher). This would be explained by migration of moths into or out of the study location, resulting in more or less eggs than expected.
*Difference ÷ expected × 100 = % of expected number.
§E = Expected # eggs ÷ starting # eggs × 100 gives the expected % increase in the population.
‡O = Actual # eggs ÷ starting # eggs × 100 gives the actual, observed % increase in the population.


Table 3. Life Table for the Spruce Budworm in Experimental Plot B.
xlxdxFdx100(dx/lx)
= 100(qx)
eggs 2176.00 parasites, predators 175.00
other causes 21.00
TOTAL
instar I (1st larval stage)   1980.00 dispersion, etc. 1148.00
overwintering stage 832.00 winter 141.00
instar II 691.00 dispersion, etc. 484.00
instars III through VI 207.00 parasites 2.90
disease 0.30
birds 1.70
starvation 165.30
DDT 8.30
other causes 10.57
TOTAL (III - VI)
pupae 1.80 parasites, predators 0.24
other causes 0.27
TOTAL
# moths to adult 1.29 sex ratio 54:46 — (abnormal) 0.10
# moths remaining 0.19 abnormal, small, weak 0.57
# living to reproduce 0.62

generation totals 2176.00 2175.38
Since 0.62 moths lived to reproduce and 46% are females, that means are
females. The average female lays an average of 150 eggs, thus the expected number
of eggs would be  × 150 = . The actual eggs counted = 246, so:
migration: 
– 246 =
 (
÷)
×100 =
The index of population trend is:  expected = (÷2176) × 100 = % increase
observed = (246/2176) × 100 = % increase
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    1. On the basis of the initial egg population, in which plot would you expect the most eggs in the next generation? Did this happen?   Show Me!
    2. What was the major mortality factor for each age group/interval? Are the tables alike in this respect? Where are the weakest points in the insect’s life cycle?   Show Me!
    3. What explanation can you give for the difference between the expected and actual numbers of eggs and index of population trend for the next generation?   Show Me!

Virtual Cemetary

  1. Life tables can also be applied to humans. Insurance companies do it all the time.
    1. Take a walk through this cemetary (note that while the numbers in the previous problems are static, you will be transported to a different cemetary each time you click the “Reload” button). For each person, record in your lab notebook the year of death and age at death. (The last guy doesn’t count — he’s just a Medieval superstition.)
    2. Sort the people into two categories based on whether they died before
    3. Within those categories, group the people into five-year sub-categories based on age at death (i.e. 0 - 4, 5 - 9, 10 - 14, etc.)
Boo!
    1. Complete the following life table for these people.
    2. The total number of people in each of the two categories (before and after is l1 for each category.
    3. Fill in all the dx values (number of people dying at each age).
    4. From those, calculate the lx values by subtracting (l2 = l1 – d1, etc).
    5. Calculate each qx value (qx = 100 × dx/lx).
    6. Calculate each Lx value (Lx = [lx + lx + 1] ÷ 2). Note that L100 = [l100 + 0] ÷ 2.
    7. Starting at the bottom of the life table and working your way to the top, calculate the Tx values (T100 = L100 and T95 = T100 + L95, etc.). Note that this “Tx” means something different than the T in the first life table.
    8. Calculate each ex value (ex = Tx ÷ lx). This is the life expectancy (how many more years they can expect to live) of people in each age category.

Table 4. Life Tables for People Buried in the Virtual Cemetary.

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    1. Is the life expectance better for people who died before or during/after
    2. Can the net reproductive rate (R0) or the maximal rate of increase (rm) be calculated for these groups of people? Why or why not?
    3. In your lab notebook, make graphs of these data as follows:
      1. Survivorship curves are useful for comparing the number of organisms still living at any given time. For each group of people, make a graph of lx on the Y-axis vs. x on the X-axis.
      2. Often, however, survivorship curves are represented by the logarithm of lx. For each group of people, make a graph of log(lx) on the Y-axis vs. x on the X-axis. Which of the three basic shapes is this curve?
      3. Mortality curves show the rate of deaths. For each group of people, make a graph of qx on the Y-axis vs. x on the X-axis. For many organisms, including humans, mortality curves are often “J”-shaped, indicating low mortality at younger ages and high mortality at older ages. Is that true of this curve?
    4. It is difficult to create graphs “on-the-fly” on the Web. Bar graphs are do-able, but line graphs are virtually impossible. Including the scaling on the X- and Y-axes is also difficult to do. Thus, if you push the following button to view the graphs, these are approximations of how yours should look. Your line graph should be the same shape as the top edge of the colored area in each of these graphs.

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Population Age Distribution and Growth Curves

Background

Organisms with overlapping generations and/or which are continuously-breeding tend towards a stable age distribution. The ratio of age groups remains the same as long as birth and death rates remain the same. Any new influence that changes the death rate will also affect the birth rate and age structure.

Ages are easier to get for humans than wild plants and animals. Current data for wild organisms often are less predictive of the future population than in humans because growth is more dependent on local resources. Humans, however, can move resources to meet their needs. Typically rapidly-growing populations have generally low or decilining death rates at younger ages. A graph of such a population would be a broad-based pyramid indicative of many young in the population. In comparison, stable or declining populations have lower birth rates, fewer young, and more older members.


Problems

Age/Sex Distribution by Life Stage

  1. Four beetles of a species known as Confused Flour Beetles were introduced into a container holding 8 g of flour. For the next several months, the numbers of beetles in each stage of their life cycle (eggs-larvae-pupae-adults) were counted. The following data were gathered.
  2. For each TIME:
    1. Add up all the beetles in all stages of the life cycle to determine how many beetles, total, were present (note completed example).
    2. Then, for each time, calculate what percentage of the beetles were in each stage of the life cycle.

Table 5. Numbers of Confused Flour Beetles in 8 gm of Flour.
days eggs larvae pupae adults total
# % # % # % # %
0 0 0 0 4
15 62 17 0 4
30 30 168 0 3
50 47 17.87 75 28.52 51 19.39 90 34.22 263
64 107 47 12 144
78 114 11 20 144
101 185 30 0 156
114 180 20 7 156
134 257 3 0 159
156 236 2 0 157
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  1. Make a population growth curve. On the x-axis, place “Time in Days” (range 0 to 156), making sure to use proper spacing (156 does not immediately follow 134). On the y-axis, place actual “Number of Individuals.” Plot curves for each of the four stages in the life cycle (one each for eggs, larvae, pupae, and adults) as well as for the whole population — a total of five lines on the graph.
    2. Show Me!
  2. Make a series of population age distribution graphs. Make one each for 30, 64, and 114 days (do other days as time and interest permit). The Y-axis represents the four age groups: Eggs, Larvae, Pupae, and Adults. The X-axis represents the percentage of the total population falling into each of those groups, divided by sex. Since we don't know which are males and which are females, we’ll assume they are 50:50. Thus the x-axis will extend 50 percentage units to each side of a center line. For example, the graph for the 50 day sample would be based on the following numbers: 47 eggs = 17.87% (÷2 = 8.94% each of males and of females), 75 larvae = 28.52% (÷2 = 14.26% each), 51 pupae = 19.39% (÷2 = 9.70% each), 90 adults = 34.22% (÷2 = 17.11% each). Percentage of males in each age category are graphed to the left of the center line, and percentages of females are graphed to the right of the center line. The finished graph would look like Figure 2.

Figure 2. Population Age/Sex Distribution for 50 Days
AdultsMale AdultsFemale Adults
PupaeMale PupaeFemale Pupae
LarvaeMale LarvaeFemale Larvae
EggsMale EggsFemale Eggs
Male ScaleFemale Scale
% Males% Females

Show me the graph for
| 0 Days | 15 Days | 30 Days | 64 Days | 78 Days |
| 101 Days | 114 Days | 134 Days | 156 Days |

Age/Sex Distribution by Age Categories

Similar curves can also be used where the numbers of males and females are known. Consider the data in Table 6:

Table 6. Census Data from Mid-1980s
Community North Avondale West Norwood Active Members,
Church in Norwood
Age #M %M #F %F #M %M #F %F #M %M #F %F
0-5 173 2.6 162 2.4 367 3.7 318 3.2 4 2.2 8 4.4
5-10 136 2.0 176 2.6 349 3.5 379 3.8 6 3.3 8 4.4
10-15 229 3.4 192 2.8 347 3.5 319 3.3 7 3.9 3 1.7
15-20 514 7.6 449 6.6 465 4.7 482 4.9 0 0 7 3.9
20-25 482 7.1 492 7.3 536 5.4 535 5.4 3 1.7 4 2.2
25-30 260 3.8 300 4.4 334 3.4 411 4.1 5 2.8 8 4.4
30-35 231 3.4 229 3.4 321 3.2 265 2.7 7 3.9 7 3.9
35-45* 349 5.2 370 5.5 505 5.1 573 5.8 7 3.9 13 7.2
45-55* 302 4.5 397 5.9 478 4.8 598 6.0 3 1.7 4 2.2
55-60 157 2.3 185 2.7 252 2.5 305 3.1 3 1.7 7 3.9
60-65 160 2.4 150 2.2 223 2.3 246 2.5 7 3.9 7 3.9
65-75* 178 2.6 158 2.3 236 2.4 392 4.0 8 4.4 13 7.2
75+ 102 1.5 229 3.4 217 2.2 452 4.6 8 4.4 23 12.8
Σ =   3273   48.4   3489   51.6   4630   46.7   5275   53.3   68   37.8   112   62.2
Σ = 6762 9905 180

From these data, similar graphs can be constructed, again with percent on the X-axis. Again, the percentage of males is plotted to the left of center and the percentage of females to the right of center. Age (or age groups) goes on the Y-axis. A graph of data from the community of Evanston would look like Figure 3.

Figure 3. Population Age/Sex Distribution for Evanston

  1. Make age distribution graphs for the three populations for which data are given in Table 6.
    1. Which of the populations has the most even age distribution?
    2. In an increasing population, this sort of age distribution graph should form a triangle which is wider at the base, tapering to a point at the top. In a population of stable (or decreasing) size, the graph will be fairly rectangular with about the same number of people in all the age categories. Which does each of these populations resemble?
    3. Can you generalize about which populations are growing, declining or stable?
    4. In general, based on the age distribution curves, would the Norwood church be a place you could go to meet other people your own age? How does the age distribution graph support your answer? Back in the 1990s, that church closed its doors and sold the property. How do you think the demographics of the congregation would have contributed to that decision?

Show me the graph for
| Bond Hill | Hyde Park | North Avondale | Norwood |
| Norwood Church | Oakley | Pleasant Ridge |


Copyright © 1998 by J. Stein Carter. All rights reserved.
This page has been accessed Counter times since 25 Jun 2001.