Biology · Introductory biology · Concept

Survivorship curves and life tables

A survivorship curve follows a cohort, a group of individuals born at the same time, and plots the fraction still alive at each age, lₓ, usually on a logarithmic scale. Type I curves stay high and fall late, because most deaths come in old age; Type II curves fall as a straight line on the log scale, because a constant fraction dies at every age; and Type III curves plunge early, because most young die.

Building a life table

Start with nₓ, the number alive at the start of each age interval. Survivorship lₓ = nₓ/n₀ is the fraction of the starting cohort still alive; dₓ = nₓ − nₓ₊₁ counts the deaths during the interval; and qₓ = dₓ/nₓ is the fraction of those alive at age x that die before the next age.

l_{x} = \frac{n_{x}}{n_{0}}, q_{x} = \frac{n_{x} - n_{x + 1}}{n_{x}}

\begin{array}{c} l_{x} = \frac{n_{x}}{n_{0}} \\ q_{x} = \frac{n_{x} - n_{x + 1}}{n_{x}} \end{array}

Three idealized curves

Real species fall between these shapes, and a species’ curve shifts with its environment.

Survivorship curve shapes
Curve	Deaths come	Examples
Type I	Late in life	Humans and other large mammals
Type II	Evenly: a constant fraction per interval	Many birds
Type III	Early in life	Oysters, most fish and trees

Why the axis is logarithmic

On a log scale, equal ratios are equal steps. A constant fraction dying each interval multiplies the survivors by the same factor, so a Type II cohort plots as a straight line. On an ordinary axis the same curve bends, which can look like early loss when it is not.

Life expectancy

Average the survivors at the start and end of each interval and multiply by its length to count the individual-years lived in it. Adding those from age x onward and dividing by nₓ gives eₓ, the average remaining life at age x. For a Type III species eₓ can rise after infancy: surviving the dangerous early stage improves the odds.

What a cohort needs

Follow one cohort from birth, without adding later arrivals. An individual lost from view before it dies has an unknown fate, not a known death. A static life table, built from the ages present at one moment, assumes that birth and death rates are steady.

Common mistakes

Plotting survivors on an ordinary axis and reading a constant-rate curve as early loss: on a log axis it is a straight line.
Confusing lₓ, the fraction of the original cohort alive, with qₓ, the fraction of those alive at age x that die in the interval.
Adding newborns from a later year to the starting cohort.
Treating a species’ curve as fixed: survivorship changes with conditions.

Key terms

Cohort: A group of individuals born around the same time and followed through their lives. Individuals that join later aren’t counted in it.
Survivorship: The fraction of the original cohort still alive at each age, lₓ. It differs from the chance of dying during the next interval.
Life table: A table following a cohort by age: how many survive to each age, how many die, and the survival and death rates. Life expectancy comes from it, with an assumption about the last age class.
Interval mortality: The fraction of the individuals alive at the start of an age interval that die during it, qₓ. Compare intervals of different lengths with care.
Survivorship types: Type I: most survive until old age, as in humans. Type II: a steady death rate at every age, as in many birds. Type III: most die young, as in many fish and plants. Real populations can fall in between.
Individual-age exposure: The total time lived by all the individuals during an age interval, in individual-years (Lₓ). It is estimated from the survivors at the start and end of the interval, not just the number alive.
Survival censoring: When an individual’s exact time of death isn’t known, for example because the study ended while it was still alive. A censored survivor must not be counted as a death.

Work through an example

A cohort of 1,000 newly hatched frogs has 400, 200, 120, 80 and 0 alive at ages 1, 2, 3, 4 and 5 years. Build the life table, describe the survivorship curve, and find the life expectancy at hatching and at age 1.

Build a life table and read a survivorship curve →

Plot a survivorship curve on a log scale →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax Biology 2e — Population demography

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