Introduction to Life Tables
||Life tables are used to describe age-specific mortality
and survival rates for a population. When this information is combined
with fecundity data, life-tables can be used to estimate rates of
population change (e.g., r, lambda, and Ro).
I. Types of Life Tables
Cohort or age-specific or dynamic
life tables: data are collected by following a cohort throughout its life.
This is rarely possible with natural populations of animals. Note: a cohort
is a group of individuals all born during the same time interval.
Static or time-specific life tables:
age-distribution data are collected from a cross-section of the population
at one particular time or during a short segment of time, such as
through mortality data. Resulting age-specific data are treated as if a
cohort was followed through time (i.e., the number of animals alive in age
class x must be less than alive in age class x-1). Because
of variation caused by small samples, data-smoothing techniques may be
required (see Caughley 1977).
Composite - data are gathered over a number of years
and generations using cohort or time-specific techniques. This method
allows the natural variability in rates of survival to be monitored and
assessed (Begon and Mortimer 1986).
Semelparity - Individuals that have only a
single, distinct period of reproductive output in their lives, prior to
which they have largely ceased to grow, during which they invest little or
nothing in survival to future reproductive events and after which they
therefore die (Begon et al. 1996:147). For annual species, this results in
nonoverlapping generations. Examples other than annual plants and some
Iteroparity - Individuals that normally
experience several or many such reproductive events. During each period of
reproductive activity the individual continues to invest in future survival
and possibly growth, and beyond each it therefore has a reasonable chance of
surviving to reproduce again (Begon et al. 1996:147). Results in overlapping
Birth pulse - reproductive activity is
restricted to a specific breeding season. Begon et al. (1996) refer to
this as "overlapping semelparity."
Birth flow - reproductive events merge
into a single extended period.
III. Life-Table Notation (columns in a life
||age, measured in years or some other conventional unit.
With longer-lived animals and plants this is often 1 year, but for
voles it might be 1 week and for some insects 1 day. Often expressed
as an interval, e.g., 0-1 years old.
|nx = nx-1
||the number of individuals surviving at the start
of age interval x. Note: n0 = sum(dx) if
dx expressed as numbers dying and the survival schedule is
complete for all members of the cohort.
|dx = nx
- nx+1 =
||the number of individuals of a cohort dying during
the age interval x to x+1. Note: sometimes calculated as
|qx = dx
/ nx =
||finite rate of mortality during the age
interval x to x + 1.
Note: this parameter is least affected by bias in the sample and gives
the most direct projection of the mortality pattern in a population.
It is often used to make comparisons within and between species.
|Sx = (nx-dx)/nx
or 1 - qx =
||finite rate of survival during the age
interval x to x + 1. This parameter is used in harvest
calculations and in population modeling. Note: finite
rates cannot be added to get total survival rate for a longer
period of time; however, finite survival rates are multiplicative
(i.e., survival from age 0 to 3 = S0 x S1 x S2).
|lx = nx
/ n0 =
||the proportion (scaled from 0 to 1) of individuals
surviving at the start of age interval x. You will also
see lx expressed per 1,000, i.e., = (nx
/ n0)1000 . This parameter is used to plot
survivorship curves (see comments below).
|kx = log10 nx
- log10 nx+1 =
||killing power or a standardized measure of the
intensity (rate) of mortality. Unlike qx-values, kx-values
can be added to determine the mortality rate for a number of age
classes. Note: because kx-values are calculated using
log10, you must take the antilog of the common log to convert back to
finite survival (i.e., S = 10-k).
IV. Extended or Expanded Life
Tables = Survival + Fecundity Schedules
||fecundity rate (i.e., average number of female offspring produced per
female in the population over some period of time, generally a
||The mean number of female offspring produced by females in an age
class. This information is used to calculate net reproductive rate (R0)
and the instantaneous rate of change ( r ).
|R0 = [sum (lx
mx)] / l0 =
||the net reproductive rate per generation. Or in other
words, the mean number of female offspring produced by a female during
her lifetime (i.e., the replacement rate). Note: R0
< 1 indicates the members of the population are not
replacing themselves (i.e., the population is declining). R0
> 1 denotes an increasing population. R0
= 1 indicates a stationary or "stable" population.
|G = Sum(lx
mx x) / R0 =
||the mean length of a generation. Other definitions include (1) the
time elapsing between birth of a female and mean time (age) of birth
of her offspring and (2) the average age that adult females give
(lambda) = R01/G = er =
||the finite rate of population change or the net reproductive
rate over some time interval, which is a year in many cases. Lambda
>1 indicates an increasing population, lambda=1
indicates a stationary population, and lambda<1
indicates a decreasing population. The use of this parameter is geared
toward organisms that reproduce during a short breeding season (i.e.,
discrete growth or birth-pulse fertility). Note: your
textbook uses "R" to denote lambda.
|r ~ = ln(R0)/G =
||intrinsic or instantaneous rate of
increase (i.e., the change in population size per individual per
unit of time). An r>0 indicates an
increasing population, r=0 indicates a stationary
pop'n, and an r<0 indicates a declining
population. Note: the equations listed only give you
an approximation of r. You need to solve Euler's equation
for a precise estimate of r (see Begon et al. 1996:165).
|ex = Tx
/ Lx =
where Lx = (lx + lx+1)
Tx = (Sum (Lx)) - Lx-1
|mean expectation of further life for individuals alive
at the start of age interval x. The mean expectation of further
life can be used as one way of compressing an entire life table into
one number. It has limited application for wildlife studies, but is
commonly used in the insurance business.
||= stable age distribution (i.e., where the proportion
of the population in each age class remains constant over time). This
is only achieved if the observed survival and fecundity schedules
remain constant over a long period of time. Note:
population projections from simple deterministic growth models (e.g.,
the exponential and logistic growth models) are based on the
assumption that the population has a stable age distribution.
ecologists do not all use the same life-table notation. For example,
Begon et al. (1996) use different symbols to denote the finite rate
of increase and mean generation length. Nevertheless, data and
calculations in the respective columns have the same meaning.
Age at Death Recorded Directly - the number of
individuals dying in successive intervals of time is recorded for a group of
individuals born at the same time. This is the most precise type of data
available because it is based on a single cohort followed through time. The
observed data are the dx column of the life table.
Cohort Size Recorded Directly - The number of
individuals alive in successive intervals of time is recorded for a cohort
of individuals. These data are similar to those obtained with Method 1,
except that those surviving are tallied, not those dying. These data
are also precise and specific to the cohort studied. Observed data are
entered into the nx column of the life table.
Age at Death Recorded for Several Cohorts -
Individuals are marked at birth and their death recorded, as in Method 1,
but several cohorts are pooled from different years or seasons. These data
are usually treated as if the individuals were members of one cohort and the
analysis of Method 1 is applied.
Age Structure Recorded Directly - The number of
individuals aged x in a population is compared with the number of
these that died before reaching age x+1. The number of deaths in that
age interval, divided by the number alive at the start of the age interval,
gives an estimate of qx directly.
Ages at Death Recorded with a Stable Age Distribution and
Known Rate of Increase - Often it is possible to find skulls or other
remains that give the age at death of an individual. These data can be
tallied into a frequency distribution of deaths and thus give dx
Age Distribution Recorded for a Population with a Stable
Age Distribution and Known Rate of Increase - In this case, the age
distribution is measured directly by sampling. The number of individuals
born is calculated from fertility rates.
||Note: Methods 5 and 6 are based on
the critical assumption that the rate of population change is
known (or the population is stationary, i.e., r=0) and the age
distribution is stable. Although methods 5 and 6 appear more
realistic in terms of data collection, there are numerous ways in
which time-specific life tables have been calculated incorrectly,
e.g., from hunter kills (see Caughley 1977).
VI. Sample size, Precision, and Accuracy of Life
Caughley (1977) recommended that the cohort consist of at least 150
individuals when basing survival estimates on the stable-age assumption.
However, age determination is difficult for many species, and if age is not
measured carefully the resulting life table may be very inaccurate.
VII. Value of Life Tables
The main value of a life table lies in what it tells us about the
population's strategy for survival, i.e., life tables help us to understand
the dynamics of populations. For example, time-specific life tables, although
often not meeting the assumptions necessary to estimate survival rates, are
valuable to a manager of exploited populations because they show the existence
of strong year classes or help identify weak age classes.
Although we have used age-structured schedules, for some organisms age is
not the best life history variable on which to develop analyses of population
change. For example, the stage of development (egg, larval, pupal, adult
stages) of some insects may be a more important life history variable. If
predation on fish is size-dependent, then size rather than age would be the
Opportunities to follow cohorts for long periods of time are rare, which
precludes cohort analysis, and the critical assumption of a stable-age
distribution is so difficult to meet that it makes the use of time-specific
life tables (Johnson 1994).
IX. Survivorship and Mortality Curves
Survivorship curves are usually created by
plotting lx on the y-axis and age on the x-axis.
Occasionally you may see nx plotted on the y-axis. The y-axis
is usually logarithmic, i.e., log10 (lx), to
allow comparisons among different studies and species. In other words, log
transformations standardize the survivorship curve.
Mortality curves are created by plotting qx
or kx against age.
Compare your survivorship/mortality curves to those
described in Begon and Mortimer (1996:153-154) and Johnson (1992:433).
Begon, M., J. L. Harper and C. R. Townsend.
1996. Ecology: Individuals, populations and communities. 3rd ed. Blackwell
Scientific Ltd., Cambridge, Mass. 1068pp.
Begon, M., and M. Mortimer. 1986.
Population ecology: a unified study of animals and plants, 2nd edition.
Blackwell Sci. Publ., Boston, Mass.
Caughley, G. 1977. Analysis of vertebrate
populations. John Wiley & Sons, New York. 234pp.
Johnson, D. 1994. Population analysis.
Pages 419- 444 in T. Bookhout, ed. Research and management techniques
for wildlife and habitats. Fifth ed. The Wildl. Soc., Bethesda, Md.
Krebs, C. J. 1989. Ecological methodology.
Harper & Row, Publ., New York. 654pp.