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Lecture 13: Life Table Analysis
Reading: None.
Life Table Analysis
Definitions:
x = stage or age interval
ax = total number
of individuals alive at start of age x
lx = proportion of
original cohort surviving to the start of age x
mx = proportion of
individuals dying between age x and x+1
(age specific mortality rate)
sx = age specific
survival rate
bx = fecundity at
age x (individual fecundity)
mean number of offspring (eggs) per
surviving adult at age x
lxbx =
offspring (eggs) per original cohort member at a given age x
Fx = total number
of offspring (eggs) produced by a cohort at age x
Basic Reproductive Rate (Ro)
is the reproduction per original cohort member.
Expectation for further life (ex)
for individuals at age x is the weighted mean of survival through
each age interval after age x. The units of ex
are the age intervals used in the life table.
where i is an age interval of value
x or older, (i-x) is time after age x, and 
is the proportion of a cohort surviving in a given age interval.
Types of Life Tables
Cohort Life Table (Horizontal Table):
Follows a group (cohort) from birth to death. This is often very difficult
and may only be possible retrospectively, by cohort estimation.
Methods:
1. Direct observations on survivorship
(Ricklefs, 1996, p. 334, Table 15.6)
2. Age at death observed (or determined
indirectly) (cohort estimation) (Ricklefs, 1996, p. 336, Table 15.7)
Assumptions for cohort estimation: Stable population, and constant
birth and death rates in each age class
Life table of the grass Poa annua
(direct observations).
| Age (x)* |
Number Alive
|
Survivorship (l x) |
Mortality Rate (m x) |
Survival Rate (s x) |
Expectation of life (ex) |
Fucundity (bx) |
| 0 |
843 |
1.000 |
0.143 |
0.857 |
2.114 |
0 |
| 1 |
722 |
0.857 |
0.271 |
0.729 |
1.467 |
300 |
| 2 |
527 |
0.625 |
0.400 |
0.600 |
1.011 |
620 |
| 3 |
316 |
0.375 |
0.544 |
0.456 |
0.685 |
430 |
| 4 |
144 |
0.171 |
0.626 |
0.374 |
0.503 |
210 |
| 5 |
54 |
0.064 |
0.722 |
0.278 |
0.344 |
60 |
| 6 |
15 |
0.018 |
0.800 |
0.200 |
0.222 |
30 |
| 7 |
3 |
0.004 |
1.000 |
0.000 |
0.000 |
10 |
| 8 |
0 |
0.000 |
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*Number of 3-month periods: in other words, 3 = 9 months
Source: M. Begon and M. Mortimer, Population Ecology (2nd.
ed.) Sinauer, Sunderland, Mass. (1986). After data of R. Law.
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On the next page, the methods
for calculating the expectation for further life (ex)
are described in detail for these data on Poa annua.
Life Table data for Poa annua showing
the calculations for estimating expectation for further life.
| x |
lx |
 |
(i-x) |
|
|
|
|
|
|
|
| |
|
|
i-7 |
i-6 |
i-5 |
i-4 |
i-3 |
i-2 |
i-1 |
i-0 |
| 0 |
1.000 |
0.143 |
|
|
|
|
|
|
|
0 |
| 1 |
0.857 |
0.232 |
|
|
|
|
|
|
0 |
1 |
| 2 |
0.625 |
0.250 |
|
|
|
|
|
0 |
1 |
2 |
| 3 |
0.375 |
0.204 |
|
|
|
|
0 |
1 |
2 |
3 |
| 4 |
0.171 |
0.107 |
|
|
|
0 |
1 |
2 |
3 |
4 |
| 5 |
0.064 |
0.046 |
|
|
0 |
1 |
2 |
3 |
4 |
5 |
| 6 |
0.018 |
0.014 |
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
| 7 |
0.004 |
0.004 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
| 8 |
0.000 |
|
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|
|
|
|
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| x |
 |
 |
| 0 |
2.114 |
(0+0.232+0.5+0.612+0.428+0.23+0.084+0.028)/1.000 |
| 1 |
1.467 |
(0+0.25+0.408+0.321+0.184+0.07+0.024)/0.857 |
| 2 |
1.011 |
(0+0.204+0.214+0.138+0.056+0.02)/0.625 |
| 3 |
0.685 |
(0+0.107+0.092+0.042+0.016)/0.375 |
| 4 |
0.503 |
(0+0.046+0.028+0.012)/0.171 |
| 5 |
0.344 |
(0+0.014+0.008)/0.064 |
| 6 |
0.222 |
(0+0.004)/0.018 |
| 7 |
0 |
(0)/0.004 |
| 8 |
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Note that the values of ex are
in units of age intervals, the units of x. In this case, those
units are three-month intervals so the expectation of further life for
an individuals alive at the start of age class 2 (6 months old) is 3.033
months (1.011*3 months).
Life table for Dall mountain sheep based
on age at death (cohort estimation).
| Age Interval (years) |
Number dying during age interval |
Number Surviving at beginning of age interval |
Number survival as a fraction of newborns (lx) |
| 0-1 |
121 |
608 |
1.000 |
| 1-2 |
7 |
487 |
0.801 |
| 2-3 |
8 |
480 |
0.789 |
| 3-4 |
7 |
472 |
0.776 |
| 4-5 |
18 |
465 |
0.764 |
| 5-6 |
28 |
447 |
0.734 |
| 6-7 |
29 |
419 |
0.688 |
| 7-8 |
42 |
390 |
0.640 |
| 8-9 |
80 |
348 |
0.571 |
| 9-10 |
114 |
268 |
0.439 |
| 10-11 |
95 |
154 |
0.252 |
| 11-12 |
55 |
59 |
0.096 |
| 12-13 |
2 |
4 |
0.006 |
| 13-14 |
2 |
2 |
0.003 |
| 14-15 |
0 |
0 |
0.000 |
| Source: Based on data of O. Murie, The Wolves of
Mt. McKinley. U.S. Department of the Interior, National Park Service,
Fauna Series No. 5, Washington D. C. (1944); quoted by E.S. Deevey,
Jr., Quarterly Review of Biology 22:283-314 (1947). |
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