The construction of a life table begins by gathering information on survivorship by age class. This sounds simple, but is easier described than actually done. For example, one method is to study a cohort of individuals all born at the same time, and follow the survivorship of these individuals until the last member of the cohort dies. At the beginning of such a study, it would be necessary to locate and mark all newborn individuals. Subsequently, one would need to verify when each individual died. Individuals that simply disappeared could not be assumed to have died; they might have emigrated. Studies of cohorts are obviously best done on small populations and on populations that move about in a predictable way. The advantage of studying a cohort is that one knows the exact age of each individual. The disadvantage is that such a study lacks generality, in that cohorts born in different years may have different survivorship schedules. In addition to the difficulties one might encounter in actually marking and gathering information on all members of a cohort, there are also practical problems. A cohort study on most species of turtles, for example, would require the entire professional life span of the investigator (picture a student waiting 50 years to finish her PhD dissertation). A life table developed in this manner is known as a fixed-cohort, dynamic, or horizontal life table.

A second approach is to locate and examine all of the dead individuals in a population during some defined period of time. We would need a method for estimating the age of the animals or plants at death. This approach to the construction of a life table assumes that the rates of survival in the population are fairly constant. If this is not the case, age-specific mortality rates will be confused with year-to-year variation in mortality of the overall population. Data gathered in this manner produce a static, vertical, or time-specific life table.

A third approach is to collect life-history data for several cohorts over as long a period as possible. In most populations there is a large difference between juvenile and adult

Age category |
Number who |
Number alive at |
Sx based on |
Survivorship, lx. |

died in the |
the beginning |
a cohort of |
(Proportion of | |

age category |
of the age class |
1000 |
original cohort alive at the beginning of the age category) | |

0-1 |
0 |
207 |
1000 |
1.000 |

1-4 |
1 |
207 |
1000 |
1.000 |

5-9 |
0 |
206 |
995 |
0.995 |

10-14 |
2 |
206 |
995 |
0.995 |

15-19 |
2 |
204 |
986 |
0.986 |

20-24 |
4 |
202 |
976 |
0.976 |

25-29 |
0 |
198 |
957 |
0.957 |

30-34 |
1 |
198 |
957 |
0.957 |

35-39 |
0 |
197 |
952 |
0.952 |

40-44 |
5 |
197 |
952 |
0.952 |

45-49 |
2 |
192 |
928 |
0.928 |

50-54 |
12 |
190 |
918 |
0.918 |

55-59 |
9 |
178 |
860 |
0.860 |

60-64 |
18 |
169 |
816 |
0.816 |

65-69 |
24 |
151 |
729 |
0.729 |

70-74 |
33 |
127 |
614 |
0.614 |

75-79 |
33 |
94 |
454 |
0.454 |

80-84 |
35 |
61 |
295 |
0.295 |

85-89 |
20 |
26 |
126 |
0.126 |

90-94 |
3 |
6 |
29 |
0.029 |

95-99 |
2 |
3 |
14 |
0.014 |

100-104 |
1 |
1 |
5 |
0.005 |

105-109 |
0 |
0 |
0 |
0.000 |

Total population 207

survivorship. Therefore, even though survivorship data on adults are often relatively easy to gather, such data do not apply to the juvenile age classes. Depending on the age when reproduction begins, it is possible to find the growth rate of the population without specific data on juvenile survivorship, as described later in the chapter.

No matter how the data are gathered, the objective is to produce an estimate of age-specific survivorship and fertility. In human demography, age-specific survivorship is based on a theoretical cohort of 1000 individuals. If we let Sx equal the number of individuals surviving to age x, we set S0 equal to 1000. Then, Sj = the number of individual surviving to age 1, S2 = the number surviving to age 2, etc. Table 4.1 is based on data gathered by an ecology laboratory from the Fairfax City (Virginia) cemetery. In this case 207 male gravestones were examined and the ages at death calculated. (Data were gathered only from

Type II, Constant probability -■—Type III

Type II, Constant probability -■—Type III

Age classes

Figure 4.1 Survivorship "type" curves: (a) on an arithmetic scale; (b) on a log scale.

Age classes

Figure 4.1 Survivorship "type" curves: (a) on an arithmetic scale; (b) on a log scale.

graves in which the birth date was between 1800 and 1890. Why was this done?) Because humans are so long-lived, the data were placed into five-year age intervals. The first step is to use the number who died in each age interval (column 2) to produce column 3, the number of survivors by age class. Next, we normalize the population to a theoretical cohort of 1000 by dividing each number by 207 and multiplying by 1000. This produces the Sx column. However, in most ecological studies we do not, in fact, use the Sx data. Instead, each number in the Sx column is divided by 1000 to produce the survivorship function, lx. Each value in the lx column stands for the proportion of the population that survives to a given age, x. It is measured from birth until the last or oldest member of the population dies. lx is known as age-specific survivorship and can be thought of as the probability, at birth, of living to a specific age class. By definition, l0 = 1.00.

The survivorship table is used to construct the survivorship curves found in all ecology textbooks. In a survivorship curve, age (x), the independent variable, is graphed against survivorship. The y-axis may be on a straight arithmetic scale; however, many authors prefer a log (base 10) scale for survivorship. Pearl (1927) introduced the idea that biological populations routinely fit one of three "types" of survivorship curves (Fig. 4.1a). The type I curve, known as the "death at senescence" curve, is characterized by excellent survivorship at all ages from birth until "old age," at which time the death rate rapidly accelerates a a 1.000

0.800

0.600

S 0.400

0.200

0.000

a 1.000

0.800

0.600

S 0.400

0.200

0.000

6 8 10 Age classes

Figure 4.2 DaLL sheep (Ovis dalli) in Denali National Park, Alaska. (a) Survivorship (/J; (b) log of survivorship (Sj. After Deevy (1947).

6 8 10 Age classes

Figure 4.2 DaLL sheep (Ovis dalli) in Denali National Park, Alaska. (a) Survivorship (/J; (b) log of survivorship (Sj. After Deevy (1947).

b and survivorship plummets. The type II curve is linear and assumes that either a constant number or a constant proportion of the population dies in each age interval. Examine Fig. 4.1. When survivorship is expressed on an arithmetic scale, a constant number of deaths per age interval produces a linear curve. When log to the base ten of survivorship is used (Fig. 4.1b), the constant probability of death per age interval produces a straight line. Finally, the type III curve applies to the vast majority of biological populations. In this curve there is very high mortality among the juvenile age classes while adult survivorship is relatively high. This is illustrated most dramatically in Fig. 4.1a, using the arithmetic scale for survivorship.

How realistic are these three survivorship "types?" Probably few populations exactly match any particular one. Furthermore, as found by Petranka and Sih (1986) for the salamander species Ambystoma texanum, survivorship curves may vary from year to year and place to place for the same species. (Recall our discussion in Chapter 1 of population viability analysis, in which we emphasized that demographic traits are subject to both temporal

■ US Males 1910 Carlisle, UK 1782

■ Northampton, UK 1780

20 40 60 80

Figure 4.3 Human survivorship curves from two eighteenth-century English and two modern United States populations: (a) on an arithmetic scale; (b) on a log scale. After Lotka (1925) and Peters and Larkin (1989).

20 40 60 80

Age classes

Figure 4.3 Human survivorship curves from two eighteenth-century English and two modern United States populations: (a) on an arithmetic scale; (b) on a log scale. After Lotka (1925) and Peters and Larkin (1989).

and spatial variability.) However, we can make some general comments. The least realistic of the three types is type I. A type I curve applies to laboratory populations of animals such as Drosophila. If provided with ample food, the population has a high rate of survivorship until the end of its maximum life span, when individuals die more or less simultaneously (Hutchinson 1978).

Natural populations of mammals such as Dall mountain sheep (Ovis dalli) (Deevy 1947), and many African ungulates (Caughley 1966), have a type I survivorship curve, although notice that 20% of the Dall sheep die in the first year of life (Fig. 4.2).

While modern human populations have a type I survivorship curve, in the not-so-distant past, living to a ripe old age was not assured (Fig. 4.3). By looking for the age where 500 of the original 1000 in a population are still alive, we have an idea of the average life expectancy at birth (Fig. 4.3a). For the modern (1985) US population, this figure is after the age of 80 (Peters and Larkin 1989). By contrast, for US males living early in the twentieth century, this figure was less than 60 (Lotka 1925). For eighteenth-century English populations, average life expectancy was less than 10 years in Northampton and around 40 years in Carlisle (Lotka 1925)! When these same values are plotted on a log scale, however, they all approximate a type I survivorship curve (Fig. 4.3b).

In order for an organism to have a type II survivorship curve, all stages of the life history must be more or less equally vulnerable to predation or other causes of death. Birds,

Age classes

Figure 4.4 Log survivorship for two cohorts of white-crowned sparrows (Zonotrichia leucophrys). Based on Baker et al. (1981).

Age classes

Figure 4.4 Log survivorship for two cohorts of white-crowned sparrows (Zonotrichia leucophrys). Based on Baker et al. (1981).

Age classes

Figure 4.5 Survivorship curve for a gray squirrel (Sciurus carolinensis) population in North Carolina. Based on Barkalow et al. (1970).

Age classes

Figure 4.5 Survivorship curve for a gray squirrel (Sciurus carolinensis) population in North Carolina. Based on Barkalow et al. (1970).

especially the adult stages, are most commonly cited as having a type II survivorship curve. For example, when Gibbons (1987) examined longevity records of vertebrates in captivity, only birds displayed a type II survivorship curve on an arithmetic scale. In a study of white-crowned sparrows (Zonotrichia leucophrys) (Fig. 4.4) Baker et al. (1981) found a type II survivorship curve on a log scale, which indicates a more or less constant probability of death, irrespective of age. The maximum life span was 49 months in this species. Botkin and Miller (1974), however, argued that birds do not, in fact, have an age-independent mortality rate. The survivorship curve for the sooty shearwater (Puffinus griseus), based on an arithmetic scale, appears to be type II. However, Botkin and Miller showed that, on closer examination, while the mortality rate in the early age classes was 0.07 per year, there was an increase in the mortality rate of 0.01 per year. They concluded that mortality was not in fact age-independent in the sooty shearwater, nor indeed in most species of birds.

Age in days

Figure 4.7 Survivorship schedule for Phlox drummondii on an arithmetic scale. Based on Leverlich and Levin (1979).

Age in days

Figure 4.7 Survivorship schedule for Phlox drummondii on an arithmetic scale. Based on Leverlich and Levin (1979).

The type III survivorship curve, which features heavy mortality among young age classes followed by good to excellent adult survivorship, applies to most biological populations from barnacles to sea turtles to plants (Hutchinson 1976). Even medium-sized mammals, such as gray squirrels (Sciurus carolinensis) (Fig. 4.5; Barkalow et al. 1970) and golden lion tamarins (Leontopithecus rosalia) (Fig. 4.6; Jonathan Ballou, personal communication) have type III survivorship curves, as do most amphibians such as Ambystoma tigrinum (e.g. Anderson et al. 1971).

Actually, most species do not follow any one of the type curves precisely, especially when an arithmetic scale is used. For example, in Phlox drummondii (Leverlich and Levin 1979) survivorship drops from 1.00 to 0.67 in the first 63 days after germination. After 124 days survivorship is down to less than 0.30. Mortality is minimal thereafter until the plants are almost a year old (Fig. 4.7). Yet this plant, when its survivorship is plotted on a logarithmic scale, has been used as an example of a type I curve (Smith 1996). Therefore, although survivorship curves are extremely useful in order to visualize the large amount of data in a life table, there is little agreement as to what constitutes a survivorship "type," and nothing is gained by attempting to fit a life table to any of the three "type" curves.

Age class |
Mean number of female offspring per female, mx |

0-1 |
0 |

1-5 |
0 |

5-10 |
0 |

10-15 |
0 |

15-20 |
0.025 |

20-25 |
0.250 |

25-30 |
0.500 |

30-35 |
0.150 |

35-40 |
0.100 |

40-45 |
0.010 |

45-50 |
0 |

50-55 |
0 |

55-60 |
0 |

60-65 |
0 |

65-70 |
0 |

70-75 |
0 |

75-80 |
0 |

80-85 |
0 |

85-90 |
0 |

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