Life tables and survivorship curves
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Lab 5. LIFE TABLES AND SURVIVORSHIP CURVES
I. Introduction & Background
The basic information needed to study changes in population size and rates of increase or
decrease in age-structured populations is contained in a life table. A life table is a convenient
way of tabulating age-specific mortality and age-specific fecundity so that we can estimate
changes in population size over time and draw inferences about the sources of mortality
operating upon a population. In this laboratory we will collect mortality data but we will not
have an opportunity to collect fecundity data; consequently, we will be constructing only a
partial life table. Such life tables allow you to analyze patterns of mortality in the population,
and to develop hypotheses, which might explain the ecological forces, such as predation, disease,
competition, or sensitivity to climatic factors. They do not, however, permit you to estimate
population growth rates, which we covered during the Lemna lab. The most reliable method of
obtaining the data for a life table is to begin with a group of individuals all born during the same
time period (a cohort) and record the times at which individuals die until the demise of the last
individual in the cohort. In this way we collect data for column dx in the life table. Using this
approach the completed life table is called a cohort life table or horizontal life table.
Unfortunately, this approach can require a very long time; for example, some turtles live for >
100 years, and bristlecone pine trees may live for 4000 years!
Life Tables
Life tables have a standard form, consisting of a series of columns presenting aspects of
mortality and survivorship. Table 1 gives an example of a life table for meadow grass (Poa
annua) in England.
Table 1. Partial cohort life table for meadow grass (Poa annua) in England.
x
nx
lx
0
843
1
722
2
527
3
316
4
144
5
54
6
15
7
8
3
0
1
0.856
4
0.625
1
0.374
8
0.170
8
0.064
0
0.017
8
0.003
6
0
dx
0.143
6
0.231
3
0.250
3
0.204
0.106
8
0.046
2
0.014
2
0.003
6
0
qx
0.1435
0.2700
0.4003
0.5443
0.625
0.7222
0.8
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Source: M. Begon & M. Mortimer. 1986. Population ecology: a unified study of animals and plants, 2nd ed. Sinauer
Associates, Sunderland, Mass.
The columns of a life table are headed by symbols that are relatively standardized in population
biology and population ecology:
x
= age class
Life tables and survivorship curves
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nx
lx
dx
= number of individuals alive at the start of age class x
= survivorship (proportion of individuals surviving to the start of age class x)
= age-specific mortality (proportion of individuals dying during the age interval x
to x+1)
= death rate (rate of mortality during the age interval x to x+1)
qx
The width of the age class, x, is decided upon by the investigator, based on knowledge of the
biology of the study organism—it may be 4 years for humans, one year for deer, or one month
for deer mice. Depending on the kind of data collected, one of the next four columns (nx, lx, dx,
or qx) is completed. Once any of these columns is filled in, the other three can be calculated.
There is nothing “new” in the remaining columns; they are just different ways of summarizing
the data. The mathematical relationships within and between these columns are summarized in
Table 2.
Table 2. Formulas for calculating nx, lx, dx, and qx columns. The remaining columns can be
derived from data for any one column using these formulas.
_______________________________________________________________________
If you have nx data, then:
lx = nx/n0 (n0 = total number of dead individuals counted) [eq. 1]
dx = lx-lx+1
[eq. 2]
qx = dx/nx
[eq. 3]
If you have lx, then:
nx = lx•n0 (an observed value for n0 is also required)
dx and qx are calculated using equations 2 and 3
If you have dx, then:
nx+1 = nx - dx (beginning with n0; therefore, an observed
value for n0 is also required)
lx and qx are calculated using equations 1 and 3
[eq. 4]
[eq. 5]
If you have qx, then:
dx = qx•nx
(beginning with n0; therefore, an observed [eq. 6]
value for n0 is also required)
lx and nx are calculated using equations 4 and 5
______________________________________________________________________
Survivorship Curves
The most frequently used part of a life table is the nx column, the number of survivors at the start
of age class x. In some studies, this column is standardized, so that the cohort begins with 1000
individuals; if you see a capitalized Nx, instead of a lowercased nx, then the column has been
standardized. For our exercise, we will not standardize the data.
If we plot nx as a function of x, the result is a survivorship curve. Sometimes nx values are
plotted on a logarithmic scale; this convention makes it easy to compare per capita rates of
survivorship or mortality among species. In turn, this allows us to uncover common properties
shared by various species and ultimately to greater general understanding of population
dynamics. Three general types of survivorship curves are recognized; these are illustrated in Fig.
1.
Life tables and survivorship curves
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survivors
1000
100
B.
Type II
A.
Type I
C.
Type III
10
1
age
Figure 1. General types of survivorship curves. A. Type I (convex). Mortality concentrated
near end of physiological life span (examples include large mammals, including humans in
developed countries, clonal plants and bunchgrasses. B. Type II. Rate of mortality constant with
respect to age (most birds, some turtles, many weedy plants). C. Type III (concave). High
juvenile mortality, low adult mortality (many fishes, invertebrates, parasites, trees, and perennial
plants.
II. Objectives of Lab
We will work in two-person groups to collect data on four cohorts (groups of individuals born at
approximately the same time) of Homo sapiens in Sunset Cemetery. Using these data, we will
construct partial life tables and survivorship curves. We will consider people born in 1800-1825
as one cohort, those born from 1826-1850 as a second, 1851-1875 as a third, and 1876-1900 as
our final cohort. You will use age at death to estimate mortality rates and other vital statistics.
We want to investigate whether any dramatic changes have occurred in the demographic
characteristics of people living in Manhattan during the late 19th and early- to mid-20th century.
The combined data from all groups will be compiled by the instructors and will be made
available to each student.
III. Data Collection (1st Week) It is imperative that you behave in a respectful manner while in
the cemetery. Note the rules posted on the green sign as you enter the cemetery and follow
them. If other people are in the cemetery for a funeral or a visit, please do not bother them or act
particularly raucous. Use common sense and be considerate.
Sampling Procedure. You and a partner will be assigned to one section of the cemetery (see
map); we do not want to record any gravestone more than once. Move through your assigned
section systematically, checking as many headstones as possible (minimum of 150 data points;
maximum of 250, unless you’re having fun, in which case feel free to fully survey your section).
You will collect data on the age of death for men and women from the four cohorts described
above. Males and females should be recorded separately on the attached data sheet. If only a
person’s initials are inscribed on the gravestone, the individual likely was male. Infants (< 1
year old) without names can be divided, half into one sex and half in the other.
Complete data collection sometime before lab on 25 February 2003, for you must hand in your
group’s data that day (or earlier, if you want to drop it off and lessen the burden on your TAs,
who will be compiling the data for the class to use in your reports). Before you hand in the data,
be sure to count up the marks in each cell of the data sheet, and write the total for that cell in
numbers, circled, so that the value can be easily transcribed.
IV. Analysis (2nd Week; 4 March 2003)
During lab on the second week (actually 2 weeks after data collection lab), we will analyze the
compiled data on age of death collected by all groups. We will use Microsoft Excel to construct
Life tables and survivorship curves
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life tables and graph curves for survivorship (lx), mortality rate (qx), and proportion of original
cohort dying during each stage (dx). Once your instructors have compiled the class data, we will
determine for which cohorts we have sufficient data to construct life tables and graph curves. At
that time, we will tell you for which cohorts we expect to receive graphs and curves. Also at
that time, we will provide you with a few questions to answer relative to interpreting your
graphs.
The computer lab that we use for this lab is relatively small (9 terminals), so we likely will have
to work in groups around the computers for this exercise. The lab will be crowded if the entire
class tries to use this computer lab during class time. Therefore, if you are familiar with Excel,
do your analysis on your own time, in one of the campus computer labs or on your own
computer. Only come to lab this week if you do not feel comfortable working on this alone.
You may work with someone else from the class who does know how to perform the analysis
described herein. As soon as it is available, we will make the data for this lab available on the
course web page so that you may access it from any connection to the Internet.
Accessing the Life Table Data. If you come to class, the Excel file with the compiled data will
be available on the desktop of all terminals in the “Ecology Lab” folder. If you access the data
from the class web page, click on the link to download the data, and then save the file to a disk
or your hard drive. Once you have the data, open Excel and then open the file you just saved. In
the ecology computer lab, the file will be called “SunsetLifeTables.xls”.
Constructing the Life Tables. The data you accessed should have a format like the data in
columns 1-4 in the attached example. There will be several worksheets in the spreadsheet (note
the cohort name on the worksheet’s tab at the bottom of the page, if Excel is maximized to fill
your window. Your job will be to construct the life table from the age at death data (DD, column
2 for males and column 3 for females in this cohort). In this table, note that x simply is the midpoint of the time interval in column 1. It is easier to think of these intervals in this way. From
the raw data for each sex, a good first step would be to copy the male or female data to another
area of the worksheet. In the example, I decided to start with males, so I copied columns 2 and 4
to columns 5 and 6 to separate these data from the female data. Now you may begin calculating
the remaining columns as directed below, or by using the information in Table 2 above.
1) The nx data (column 7) is the number present at the start of each stage. Thus you can
calculate it as nx – DDx-1. Then copy and paste this formula down the rest of the column to fill in
the empty cells.
2) The survivorship data (lx; column 8) can be calculated by nx/n0. Remember that n0 simply is
the total number of dead individuals we counted (this is the sum at the bottom of the DD
column).
3) The dx data (proportion of original cohort dying during each stage) can be calculated as lx –
lx+1 (column 9). Note that this also can be calculated as DD/n0.
4) The qx data (mortality rate or probability of dying during that stage) can be calculated by dx/lx
(column 10 in the example).
Creating the Survivorship Curves. Use the chart wizard to create your curves. You do not
need to plot the lx data on a logarithmic curve, as often is done; you would need multiple y-axes
since dx and qx are on a normal, linear scale. Just use a linear scale for your y-axis and plot all
three curves on the same graph. Try to follow the format of the example graph provide below.
Life tables and survivorship curves
You will need to create separate graphs for males and females in each of the cohorts (a
maximum of 8 graphs to hand in).
Acknowledgments. Much of the background information for this lab was gleaned from
exercises completed in the lab portion of Ecology at Trinity University (Dr. David O. Ribble)
and Biology of Populations at the College of Wooster (Dr. Raymond S. Matlack).
Map of Sunset Cemetery. Be sure to survey the section to which your group was assigned!
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Life tables and survivorship curves
Sample spreadsheet from SunsetLifeTables.xls. For your report, you may print each
spreadsheet showing your calculations directly from Excel. Note that the right half of this
spreadsheet shows the calculations only for the males in this cohort. Your report must also
include calculations for females in the cohorts we direct you to analyze.
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Life tables and survivorship curves
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Sample graph of curves from SunsetLifeTables.xls. Note that this graph shows only the males
in the 1800-1825 cohort. We suggest that you create a separate graph for each sex in each cohort
to minimize clutter in your graphs.
Life tables and survivorship curves
Students Names: ______________________________________
Section of Cemetery: _______________
Cohort
Age at death
<1
1-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-99
100-104
1800-1825
Male
Female
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Data Sheet 1
1826-1850
Male
Female
Life tables and survivorship curves
Students Names: ______________________________________
Section of Cemetery: _______________
Cohort
Age at death
<1
1-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-99
100-104
1851-75
Male
Female
Data Sheet 2
1876-1900
Male
Female
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