Life Tables and Survivorship Curves

In this exercises below, you will calculate life table information for a hypothetical species. The table lists the age (x), and the number of individuals alive at the beginning of each age interval (n_x). From this, you will calculate the number dying during each interval (d_x), the mortality rate (q_x), and the average life expectancy for individuals alive at the start of each interval (e_x).

In order to calculate life expectancy, you will need to calculate two additional parameters: L_x and T_x. L_x is the number of individuals alive, on average, from the age interval x to x+1. T_x is the total number of years that remain to be lived by all individuals in the population. It is the sum of the L_x column beginning at age x. It may be easiest to calculate the values in the T_x column by adding the values in the L_x column beginning at the bottom of the column. For example, 0 + 1 = 1. The value of T₈ is therefore 1. Next, 0 + 1 + 2 = 3. The value of T₇ is 3. Similarly, 0 + 1 + 2 + 4.25 = 7.25. This is T₆. 

The values of T_x are the total years that will be lived by the population at that age. The average life expectancy can be calculated by dividing by the number in the population.

e_x = T_x/n_x.

The formulas for each of these life table parameters and an example are given in the table below. In this example, formulas are shown in the third row and sample calculations in the fourth row.

Sample calculations for a hypothetical species.

Age	Number Alive	Number Dying	Mortality Rate	Average Number Alive	Total Years of Life Remaining	Expectation of  further life
x	nx	dx	qx	Lx	Tx	ex
		= nx - nx+1	= dx/nx	nx + nx+1   = ___________ 2	x     =    Lx         xmax	= Tx/nx
0	612	612-539=73	73/612=0.119	(612+539)/2=575.5	45.5+160+291+402.5 +495.5+575.5=1970.0	1970/612=3.22
1	539	87	0.161	495.5	1394.5	2.59
2	452	99	0.219	402.5	899.0	1.99
3	353	124	0.351	291.0	496.5	1.41
4	229	138	0.603	160.0	205.5	0.90
5	91	91	1.000	45.5	45.5	0.50
6	0	--	--	0	0	--

Exercise 1

Calculate d_x, q_x, L_x, T_x, and e_x in the table  below. Show your calculations for the items in second row (x = 0). It is not necessary to show your calculations for the other rows. You may wish to refer to the table above for help with this table.

A life table for a hypothetical species.

Exercise 2

The net reproductive rate (R₀), instantaneous growth rate (r), and mean generation time (T) can be calculated if you know the age-specific fertility (fecundity). R₀ is equal to the average number of offspring produced per individual. Frequently in population studies, we count only the number of female offspring produced per female. This should be approximately equal to the total number of offspring produced per individual. This information is given in the column labeled m_x. Before calculations can be done, the number of individuals in each age category (n_x) must be converted to a proportion of the starting population size. These proportions are referred to as l_x. The column labeled l_xm_x is obtained by multiplying m_x times l_x. The sum of this column is equal to the net reproductive rate. R₀ = S(l_xm_x). The mean generation time (T) is calculated by multiplying the l_xm_x column by the age (x) and summing this column. The result must then be divided by R₀. R₀ is a finite rate and can be converted to an instantaneous rate (r) by: r = (lnR₀)/T. Because our estimate of mean generation time (T) is approximate, the calculated value for r is also approximate.

R₀ = l_xm_x

 

T =	xlxmx _____ R0
r =	lnR0 _____ T

6. Calculate Ro, T and r for the population in the table below. Put these values in the area underneath the table where indicated. Show your calculations for the items in second row (x = 0). It is not necessary to show your calculations for the other rows.

Graphs

Draw a survivorship curve on semi-log paper using the n_x data for the population in Exercise 1.

Draw a survivorship curve on semi-log paper using the n_x data for the population in Exercise 2.

Click here to print semi-log graph paper.

**** Important note - The Y axis must be a log scale. If you use a graphing program, be sure that the Y- axis is a log scale. Some graphing programs do not create a log scale on the Y-axis (Create A Graph does not). To use these programs, you must take the logarithm of the nx value before you enter it into the computer. If your graphing program creates a log scale on the Y-axis (Excel does), you can enter the nx numbers directly and set the program to create a log scale. Either way will produce the same curve.

Questions

1) Which type of theoretical survivorship curve (type I, type II, or type III) does the population in Exercise 1 most closely resemble? 

2) What does the survivorship curve tell you about survivorship and mortality rates experienced by the population in Exercise 1?

3) Suppose that the organism in Exercise 1 is a plant. Age 0 refers to seeds and age 1 are seedlings. Why might the life expectancy of the plant increases at the seedling stage (age 1) of its life?

4) Which type of theoretical survivorship curve (type I, type II, or type III) does the population in Exercise 2 most closely resemble?

5) What factors might cause the type of survival in Exercise 2?

6) What does the survivorship curve tell you about survivorship and mortality rates experienced by the population in Exercise 2? 

7. Calculate the following for the population in Exercise 2 and show your work.

R₀ = 

 

T =

 

r =