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LIFE TABLES,
SURVIVORSHIP CURVES,
12 AND POPULATION GROWTH
Objectives
- Discover how patterns of survivorship relate to the classic three types of survivorship curves.
- Learn how patterns of survivorship relate to life expectancy.
- Explore how patterns of survivorship and fecundity affect rate of population growth.
Suggested Preliminary Exercise: Geometric and Exponential Population Models
INTRODUCTION A life table is a record of survival and reproductive rates in a population, broken out by age, size, or developmental stage (e.g., egg, hatchling, juvenile, adult). Ecol- ogists and demographers (scientists who study human population dynamics) have found life tables useful in understanding patterns and causes of mortality, predicting the future growth or decline of populations, and managing popula- tions of endangered species. Predicting the growth and decline of human populations is one very important application of life tables. As you might expect, whether the population of a coun- try or region increases or decreases depends in part on how many children each person has and the age at which people die. But it may surprise you to learn that population growth or decline also depends on the age at which they have their children. A major part of this exercise will explore the effects of changing pat- terns of survival and reproduction on population dynamics. Another use of life tables is in species conservation efforts, such as in the case of the loggerhead sea turtle of the southeastern United States (Crouse et al., 1987). We explore this case in greater depth in Exercise 14, “Stage-Structured Matrix Mod- els,” but generally speaking, the loggerhead population is declining and mortal- ity among loggerhead eggs and hatchlings is very high. These facts led conserva- tion biologists to advocate the protection of nesting beaches. When these measures proved ineffective in halting the population decline, compiling and analyzing a life table for loggerheads indicated that reducing mortality of older turtles would have a greater probability of reversing the population decline. Therefore, man- agement efforts shifted to persuading fishermen to install turtle exclusion devices on their nets to prevent older turtles from drowning.
Life tables come in two varieties: cohort and static. A cohort life table follows the sur- vival and reproduction of all members of a cohort from birth to death. A cohort is the set of all individuals born, hatched, or recruited into a population during a defined time interval. Cohorts are frequently defined on an annual basis (e.g., all individuals born in 1978), but other time intervals can be used as well. A static life table records the number of living individuals of each age in a popula- tion and their reproductive output. The two varieties have distinct advantages and disadvantages, some of which we discuss below. Life tables (whether cohort or static) that classify individuals by age are called age- based life tables. Such life tables treat age the same way we normally do: that is, indi- viduals that have lived less than one full year are assigned age zero; those that have lived one year or more but less than two years are assigned age one; and so on. Life tables represent age by the letter x , and use x as a subscript to refer to survivorship, fecundity, and so on, for each age. Size-based and stage-based life tables classify individuals by size or developmen- tal stage, rather than by age. Size-based and stage-based tables are often more useful or more practical for studying organisms that are difficult to classify by age, or whose ecological roles depend more on size or stage than on age. Such analyses are more com- plex, however, and we will leave them for a later exercise.
Cohort Life Tables
To build a cohort life table for, let’s say, humans born in the United States during the year 1900, we would record how many individuals were born during the year 1900, and how many survived to the beginning of 1901, 1902, etc., until there were no more survivors. This record is called the survivorship schedule. Unfortunately, different text- books use different notations for the number of survivors in each age; some write this as Sx , some ax , and some nx. We will use Sx here. We must also record the fecundity schedule —the number of offspring born to indi- viduals of each age. The total number of offpsring is usually divided by the number of individuals in the age, giving the average number of offspring per individual, or per capita fecundity. Again, different texts use different notations for the fecundity sched- ule, including bx (the symbol we will use) or mx.^1 Many life tables count only females and their female offspring; for animals with two sexes and equal numbers of males and females of each age, the resulting numbers are the same as if males and females were both counted. For most plants, hermaphro- ditic animals, and many other organisms, distinctions between the sexes are nonexist- ent or more complex, and life table calculations may have to be adjusted.
Static Life Tables
A static life table is similar to a cohort life table but introduces a few complications. For many organisms, especially mobile animals with long life spans, it can be difficult or impossible to follow all the members of a cohort throughout their lives. In such cases, population biologists often count how many individuals of each age are alive at a given time. That is, they count how many members of the population are currently in the 0–1- year-old class, the 1–2-year-old class, etc. These counts can be used as if they were counts of survivors in a cohort, and all the calculations described below for a cohort life table can be performed using them. In doing this, however, the researcher must bear in mind that she or he is assuming that age-specific survivorship and fertility rates have remained constant since the oldest members of the population were born. This is usually not the case and can lead to some
164 Exercise 12
(^1) Some demographers use the term fecundity to be the physiological maximum number of eggs produced per female per year, and the term fertility to be the number of offspring pro- duced per female per year. In this book, we will assume that the two are equivalent unless noted otherwise.
Age-Specific Survivorship ( gx ). Standardized survivorship, lx , gives us the proba- bility of an individual surviving from birth to the beginning of age x. But what if we want to know the probability that an individual who has already survived to age x will survive to age x + 1? We calculate this age-specific survivorship as gx = lx +1/ lx , or equiv- alently,
Equation 2
Life Expectancy ( ex ). You may have heard another demographic statistic, life expectancy, mentioned in discussions of human populations. Life expectancy is how much longer an individual of a given age can be expected to live beyond its present age. Life expectancy is calculated in three steps. First, we compute the proportion of survivors at the mid-point of each time interval ( Lx —note the capital L here) by aver- aging lx and lx +1; that is,
Equation 3
Second, we sum all the Lx values from the age of interest ( n ) up to the oldest age, k :
Equation 4
Finally, we calculate life expectancy as
Equation 5
(note the lowercase lx ). Life expectancy is age-specific—it is the expected number of time-intervals remain- ing to members of a given age. The statistic most often quoted (usually without quali- fication) is the life expectancy at birth ( e 0 ). As you will see, the implications of e 0 depend greatly on the survivorship schedule.
Population Growth or Decline
We frequently want to know whether a population can be expected to grow, shrink, or remain stable, given its current age-specific rates of survival and fecundity. We can determine this by computing the net reproductive rate ( R 0 ). To predict long-term changes in population size, we must use this net reproductive rate to estimate the intrin- sic rate of increase ( r ).
Net Reproductive Rate ( R 0 ) We calculate net reproductive rate ( R 0 ) by multiplying the standardized survivorship of each age ( lx ) by its fecundity ( bx ), and summing these products:
Equation 6
The net reproductive rate is the lifetime reproductive potential of the average female, adjusted for survival. Assuming survival and fertility schedules remain constant over time, if R 0 > 1, then the population will grow exponentially. If R 0 < 1, the population will shrink exponentially, and if R 0 = 1, the population size will not change over time. You may be tempted to conclude the R 0 = r , the intrinsic rate of increase of the expo- nential model. However, this is not quite correct, because r measures population change in absolute units of time (e.g., years) whereas R 0 measures population change in terms of generation time. To convert R 0 into r , we must first calculate generation time ( G ), and then adjust R 0.
R l bx x x
k 0 0
=
∑
e
T
x (^) l
x x
T (^) x Lx x n
k
=
∑
L
l l x = x^ +^ x +^1 2
g
S
x (^) S
x x
= +^1
166 Exercise 12
Generation Time. Generation time is calculated as
Equation 7
For organisms that live only one year, the numerator and denominator will be equal, and generation time will equal one year. For all longer-lived organisms, generation time will be greater than one year, but exactly how much greater will depend on the sur- vival and fertility schedules. A long-lived species that reproduces at an early age may have a shorter generation time than a shorter-lived one that delays reproduction.
Intrinsic Rate of Increase. We can use our knowledge of exponential population growth and our value of R 0 to estimate the intrinsic rate of increase ( r ) (Gotelli 2001). Recall from Exercise 7, “Geometric and Exponential Population Models,” that the size of an exponentially growing population at some arbitrary time t is Nt = N 0 ert , where e is the base of the natural logarithms and r is the intrinsic rate of increase. If we consider the growth of such a population from time zero through one generation time, G , it is
NG = N 0 erG
Dividing both sides by N 0 gives us
We can think of NG /N 0 as roughly equivalent to R 0 ; both are estimates of the rate of population growth over the period of one generation. Substituting R 0 into the equation gives us
R 0 ≈ erG
Taking the natural logarithm of both sides gives us
ln ( R 0 ) ≈ rG
and dividing through by G gives us an estimate of r :
Equation 8
Euler’s Correction to r. The value of r as estimated above is usually a good approx- imation (within 10%), and it will suffice for most purposes. Some applications, how- ever, may require a more precise value. To improve this estimate, you must solve the Euler equation:
Equation 9
The only way to solve this equation is by trial and error. We already know the values of
lxbx , and e (it is the base of the natural logarithms, e ≈ 2.7183), so we can plug in various
guesses for r until Equation 9 comes up 1.0. That will tell us the corrected value of r. Fortunately, a spreadsheet is an ideal medium for such trial and error solution-hunting. Finally, we can use our estimate of r (uncorrected or corrected) to predict the size of the population in the future. In this exercise, you will adjust survivorship and fecundity sched- ules and observe the effects on population growth or decline. This kind of analysis is done for human populations to predict the effects of changes in medical care and birth control programs. If we assume that all age groups are roughly equivalent in size, a similar analy- sis can be done for endangered species to determine what intervention may be most effec- tive in promoting population growth. The same analysis can be applied to pest species to determine what intervention may be most effective in reducing population size.
0
=
∑ e^ l b
rx x x x
k
r
R
G
ln( 0 )
N
N
G (^) erG 0
G
l b x
l b
x x x
k
x x x
= (^) k
=
∑
∑
0
0
Life Tables, Survivorship Curves, and Population Growth 167
This corresponds to Equation 2:
Note that all cell addresses are relative.
These are all literals, so just select the appropriate cells and type them in.
Select cells A4–A15. Copy. Select cell A19. Paste.
In cell B19 enter the formula =E4. Copy this formula into cells C19 and D19. Copy cells B19–D19 into cells B20–D30. Doing it this way, rather than copying and pasting the values, will automatically update this part of the spreadsheet if you change any of the Sx values in cells B4–D15.
In cell E19 enter the formula =(B19+B20)/2. Copy this formula into cells F19 and G19. Copy cells E19–G19 into cells E30–G30. This corresponds to Equation 3:
In cell H19 enter the formula =SUM(E19:E$30)/B19. Copy the formula from cell H into cells I19 and J19. Copy cells H19–J19 into cells H20–J29.
The portion SUM(E19:E$30) corresponds to Equation 4:
The entire formula corresponds to Equation 5:
e
T
x (^) l
x x
Tx Lx x n
k
=
∑
L
l l x = x^ +^ x +^1 2
g
S
x (^) S
x x
= +^1
- Enter titles and column headings in cells A17–J as shown in Figure 3.
- Copy the values of age from cells A4–A15 into cells A19–A30.
- Echo the values of lx from cells E4–G15 in cells B19–D30.
- Enter formulae to calcu- late the number of sur- vivors at the midpoint of each age, Lx.
- Enter formulae to cal- culate life expectancy, ex , for each age.
Life Tables, Survivorship Curves, and Population Growth 169
17
18 19 20 21 22 23 24 25 26 27 28 29 30
A B C D E F G H I J Age-specific life expectancy Age ( x ) (^) Type I l^ x^ : Type II^ l^ x^ : Type III^ l^ x^ :^ l^ x^ : ^ l^ x^ :^ l^ x^ :^ e^ x^ :^ e^ x^ :^ e^ x^ : 0 1.0000 1.0000 1. 1 0.9900 0.5000 0. 2 0.9700 0.2500 0. 3 0.9400 0.1250 0. 4 0.9000 0.0625 0. 5 0.8500 0.0313 0. 6 0.7500 0.0156 0. 7 0.5000 0.0078 0. 8 0.2000 0.0039 0. 9 0.0400 0.0020 0. 10 0.0010 0.0010 0. 11 0.0000 0.0000 0.
Type I Type II Type III Type I Type II Type III
Figure 3
Do not copy the formula into row 30, because lx there is zero, and so Equation 5 would be undefined.
Select cells A3–A15. Select cells E3–G15 and create an XY graph. Edit your graph for readability. It should resemble Figure 4.
Double-click on the y -axis and choose the Number tab in the resulting dialog box. Set the number of decimal places to 3. Choose the Scale tab. Check the box for Logarithmic Scale. Set the Major unit to 10, and set Value (X) axis Crosses at to 0.0001. Your graph should resemble Figure 5.
- Your spreadsheet is complete. Save your work.
- Graph standardized survivorship, lx , against age.
- Change the y -axis to a logarithmic scale.
170 Exercise 12
Survivorship Curves
0 2 4 6 8 10 Age ( x )
Standardized survivorship (
lx: Type I lx: Type II lx: Type III
l^ ) x
Figure 4
Survivorship Curves
0 2 4 6 8 10 Age ( x )
Standardized survivroship (
l^ ) x
lx: Type I lx: Type II lx: Type III
Figure 5 Survivorship curves are always plotted with a logarithmic y - axis. Can you see why?
In cell C4 enter the formula =B4/$B$4. Copy this formula into cells C5–C8. Do not copy into cell C9. Again, this corresponds to Equation 1. Note the use of a relative cell address in the numerator, and an absolute cell address in the denominator.
Enter the value 0.00 into cells D4, D5, D7, and D8, Enter the value 4.00 into cell D6.
In cell E4 enter the formula =C4D4*. Copy this formula into cells E5–E8.
In cell E9 enter the formula =SUM(E4:E8). This corresponds to Equation 6:
In cell B10 enter the formula =E. We do this because you will soon change the values of Sx and bx , and this layout will make it easier to compare the effects of different survival and fertility schedules on pop- ulation growth or decline.
In cell F4 enter the formula =E4A*. This is an intermediate step in calculating generation time, G. Copy the formula from cell F4 into cells F5–F8.
In cell F9 enter the formula =SUM(F4:F8). This is another intermediate step in calculating generation time, G.
In cell B11 enter the formula =F9/E. This corresponds to Equation 7:
R l bx x x
k 0 0
=
∑
- Enter a formula to cal- culate standardized sur- vival, lx.
- Enter the values shown for age-specific fertility, bx.
- Enter a formula to cal- culate the product of stan- dardized survival times age-specific fertility, lxbx.
- Enter a formula to cal- culate net reproductive rate, R 0.
- Echo the value of R 0 in cell B10.
- Enter a formula to cal- culate the product lxbxx.
- Enter a formula to cal- culate the sum of the products lxbxx.
- Enter a formula to cal- culate generation time, G.
172 Exercise 12
1 2 3 4 5 6 7 8 9
10 11 12 13 14
A B C D E F G Cohort Life Table: Fertility, Survival, and Population Growth
Age ( x ) S (^) x l (^) x b (^) x ( l (^) x )( b (^) x ) ( x )( l (^) x )( b (^) x ) ( e ^- rx )( l (^) x )( b (^) x ) 0 1000 1.0000 0.00000 0.0000 0.0000 0. 1 900 0.9000 0.00000 0.0000 0.0000 0. 2 250 0.2500 4.00000 1.0000 2.0000 1. 3 10 0.0100 0.00000 0.0000 0.0000 0. 4 0 0.0000 0.00000 0.0000 0.0000 0. Total 1.0000 2.0000 1. R (^) 0 1. G 2. r est. 0. r adj. 0. Should be 1 1.
Figure 8
In cell B12 enter the formula =LN(B10)/B. This corresponds to Equation 8:
Follow the procedures in Steps 12 and 13 of Section A. Your graph should resemble Figure 9.
Start by entering the estimated value of r from cell B12. You will see how to use this guess below.
In cell G4 enter the formula =EXP(-$B$13A4)E**. This is an intermediate step in applying Euler’s correction to the estimate of r calcu- lated in Step 11 of Section B. Note that the formula uses your guess for the value of r.
In cell G9 enter the formula =SUM(G4:G8). This corresponds to the right side of Equation 9:
If your guess for r is correct, this formula will yield a value of 1.0.
In cell B14 enter the formula =G. Again, this is simply a convenient layout for comparing the effects of changing Sx and bx.
e rx l bx x x
k − =
∑ 0
r
R
G
ln( 0 )
G
l b x
l b
x x x
k
x x x
= (^) k
=
∑
∑
0
0
- Enter a formula to esti- mate the intrinsic rate of increase, r.
- Your spreadsheet is complete. Save your work.
- Create a survivorship curve from your Sx values.
C. Euler’s correction (Optional)
- (*Optional) Enter a guess for the correct value of r into cell B13.
- Enter a formula to cal- culate e − rxlxbx.
- Enter a formula to com- pute Euler’s equation.
- In cell B14, echo the re- sult of the formula in cell G9.
Life Tables, Survivorship Curves, and Population Growth 173
Survivorship Curve
0 1 2 3 Age ( x )
Standardized survivorship (
) lx
Figure 9
Appendix: SAMPLE SURVIVORSHIP SCHEDULES FROM
NATURAL POPULATIONS OF ANIMALS
In all cases, assume Sx for the next age after the oldest in the table is 0.
Life Tables, Survivorship Curves, and Population Growth 175
Table A. Survivorship schedule for Dall Mountain Sheep ( Ovis dalli dalli ).
Age (years) S x Age (years) Sx
Data from Deevey (1947). Numbers have been standardized to S 0 = 1000.
Table B. Survivorship schedule for the Song Thrush.
Age (years) S x Age (years) Sx
Data from Deevey (1947). Numbers have been standardized to S 0 = 1000.
Table C. Survivorship and fertility schedules for the barnacle Balanus glandula.
Age (years) S x Age (years) Sx
Data are from Connell (1970). Values of S 4 and S 6 have been interpolated and rounded to the next integer.
