## Life Tables

A life table is an age-specific summary of mortality rates operating on a cohort of individuals first developed by human demographers and introduced to ecologists by Raymond Pearl in 1921. The mortality schedule is generally calculated based on the known number of survivors in each age class. In order to create a life table, an age interval must be decided upon in which to group the data. For longer-lived species, such as trees or turtles, an age interval of several years may be appropriate, whereas for shorter-lived species, such as birds, some plants, or insects, 1 year or less may be appropriate. These age intervals are known as age classes. Two different types of life tables can be calculated: a static life table (also called a stationary, time-specific, current, or vertical life table) and a cohort life table (also called a dynamic, generation, or horizontal life table). A static life table is calculated based on a cross section of a population at a specific time. In this case, the mortality schedule would be calculated for each age class at a specific time. A cohort life table is calculated for a cohort of organisms followed throughout life. In this case, one age cohort would be followed throughout their life (Table 1).

The columns of a life table include x (age), nx (number alive at age x), lx (proportion of organisms surviving from the start of the life table to age x), dx (number dying during the age interval x to x + 1), and qx (per capita rate of mortality during the age interval x to x + 1). Given x and n„ the following equations can be used to obtain the remaining survivorship information:

Fertility data can also be included in a life table. Additional columns include bx (fertility schedule or the average number of female offspring produced per female aged x during time x), and lxbx and x(lj>x). The fertility schedule is observed in the field or laboratory, while lxbx and x(lxbx) are used to calculate the net reproductive rate (R0), which is the multiplication rate per generation of the population:

The mean length of a generation

 Table 1 Hypothetical cohort life table including both mortality and fertility data x nx lx dx Mortality rate (qj Fertility schedule (bj lxbx x(lxbx) 0 500 1.00 77 0.154 0.000 0.00 0 1 423 0.846 208 0.492 1.000 0.846 0.846 2 215 0.430 107 0.498 1.000 0.430 0.860 3 108 0.216 36 0.333 1.000 0.216 0.648 4 72 0.144 53 0.736 1.000 0.144 0.576 5 19 0.038 1.000 0.038 0.19

Figure 1 Hypothetical survivorship curves (nx). Adapted from Pearl R (1928) The Rate of Living. New York: Knopf.

Figure 1 Hypothetical survivorship curves (nx). Adapted from Pearl R (1928) The Rate of Living. New York: Knopf.

The intrinsic capacity for increase (r), derived by Alfred Lotka in 1925, combines the natality and mortality demographic parameters. Lotka showed that a population subject to constant mortality and natality rates would gradually approach a fixed or stable age distribution. When a population has reached a stable age distribution, it will increase according to the differential equation:

or in the integral form:

Based on the number of survivors, Raymond Pearl described three general types of survivorship curves on a log-transformed scale (Figure 1). A type 1 curve describes populations with low per capita mortality for most of the life span followed by high mortality of older organisms. A type 2 curve describes a linear survivorship curve that implies a constant per capita rate of mortality independent of age. Lastly, a type 3 curve describes a population with high per capita mortality early in life, followed by a period of much lower and relatively constant mortality. Humans in developed nations tend to follow the type 1 survivorship curve, while many birds exhibit a type 2 curve and many fish a type 3 curve. These curves are idealized and thus few, if any, populations have survivorship curves that exactly follow these curves.

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