9 Life tables and survivorship types

9.1 Background

Life tables are powerful analytical tools used in various fields to understand and quantify mortality patterns and life expectancy within populations. Originating in population biology and demography, life tables have found extensive applications in epidemiology, human mortality studies, and even product life-cycle management. By systematically tracking and analysing the survival and mortality rates of individuals within specific cohorts, life tables reveal essential information about population dynamics, age-specific risks, and the factors that influence longevity. When they also record reproduction, they tell us how fast a population grows.

The basic algebra used in life tables is explained in Neal, in the chapters Life Tables and Growth of Age-Structured and Stage-Structured Populations, and in the lecture “Life tables and age-structured population growth”. This practical uses the same notation as Neal and the lecture (see the table below). Gotelli’s A Primer of Ecology (optional) covers the same material with different symbols: $g_x$ for $p_x$, $b_x$ for $m_x$ and $G$ for $T$.

In addition to providing valuable insights into mortality patterns and life expectancy, life tables also allow us to consider mortality patterns in the context of distinct survivorship types that result from the way mortality risk changes over the life course. Three primary survivorship types are commonly identified, based on survivorship curves ($l_x$) plotted on a log scale:

Type I (convex curve): mortality is low early in life and increases with age. Most individuals survive to old age, and most deaths occur in older age groups. Type I survivorship is seen in humans and other long-lived species with strong parental care and low early-life mortality.
Type II (straight line): mortality is constant with age, so individuals have about the same chance of dying at any age. Some small mammals and many birds come close to this pattern.
Type III (concave curve): mortality is high early in life and decreases with age. Survival is low in the early stages of life, but those that survive the vulnerable period are likely to live a long time. Type III survivorship is typical of species that produce many offspring with little parental care, such as many fish and trees.

Real species are often mixtures: for example, Dall sheep and many birds have high juvenile mortality (Type III-like), followed by mortality that rises with age in adults (Type I-like).

By identifying and understanding these survivorship types, researchers gain critical insights into the demographic structure of populations and the selective pressures that influence survival and longevity.

Learning outcomes:

Increased competence in using Excel formulae for mathematical modelling.
Competence in using mathematical models in Excel to strengthen your own understanding of biological processes.
Building a cohort life table, and interpreting survivorship ($l_x$), survival probability ($p_x$), probability of death ($q_x$) and life expectancy ($e_x$).
Calculating the net reproductive rate ($R_0$), generation time ($T$) and the capacity for increase ($r_c = \ln(R_0)/T$), an approximation to the intrinsic rate of increase $r_m$.
Understanding the three types of survivorship curve and how they relate to the way mortality changes with age.
Understanding why changes in reproduction early in life matter more than changes late in life (the declining “force of selection”).

9.2 Notation

Symbol	Meaning	Calculation
$x$	Age at the start of the interval ($x = 0$ means age 0 to 1)
$S_x$	Number alive at the start of age $x$	counted
$l_x$	Survivorship: probability of surviving from birth to age $x$	$S_x/S_0$
$p_x$	Survival probability: probability of surviving from age $x$ to $x+1$	$S_{x+1}/S_x$, or $l_{x+1}/l_x$
$q_x$	Probability of dying between age $x$ and $x+1$	$1 - p_x$
$e_x$	Life expectancy: further years expected at age $x$	$\left(\sum_{y \ge x} l_y\right)/l_x - 0.5$
$\mu_x$	Force of mortality: the instantaneous death rate of individuals still alive at age $x$ (a rate, not a probability)	if $\mu$ is constant over the year, $q_x = 1 - e^{-\mu_x}$
$m_x$	Female offspring per female aged $x$
$R_0$	Net reproductive rate: female offspring per female per lifetime (per generation)	$\sum l_x m_x$
$T$	Generation time: mean age of mothers when their daughters are born	$\sum x l_x m_x / R_0$
$r_c$	Capacity for increase: approximate population growth rate per year (an approximation to $r_m$)	$\ln(R_0)/T$
$r_m$	Intrinsic rate of increase per year (exact, from the Euler-Lotka equation); $\lambda = e^{r_m}$	solve $\sum e^{-r_m x} l_x m_x = 1$

9.3 Worked example

The Worked example sheet in the spreadsheet is the life table from the lecture: a cohort of 500 newborn females, with $S_x$ = 500, 400, 200, 50 and 0 at ages 0 to 4, and $m_x$ = 0, 2, 3, 1 and 0. Click on the cells to see the formulas. For example:

Survivorship: $l_1 = 400/500 = 0.8$
Survival probability from age 1 to 2: $p_1 = 200/400 = 0.5$, so $q_1 = 1 - 0.5 = 0.5$
Life expectancy at birth: $e_0 = (1 + 0.8 + 0.4 + 0.1)/1 - 0.5 = 1.8$ years
Net reproductive rate: $R_0 = 0 + 1.6 + 1.2 + 0.1 = 2.9$
Generation time: $T = (1 \times 1.6 + 2 \times 1.2 + 3 \times 0.1)/2.9 = 4.3/2.9 = 1.48$ years
Capacity for increase: $r_c = \ln(2.9)/1.48 = 0.72$ per year, so $\lambda \approx e^{r_c} = 2.05$

Note that $r_c = \ln(R_0)/T$ is an approximation to the intrinsic rate of increase, $r_m$. The exact value for this life table ($r_m = 0.776$, $\lambda = 2.17$) comes from the Euler-Lotka equation or from a matrix model (next practical). The spreadsheet labels this quantity “r (approx.)”.

9.4 Your task

Download and open the Excel file Life_Tables_Exercises.xlsx.

The file has three worksheets: “Life table”, “Survivorship Curves” and “Worked example”.

9.4.1 Life table

Let’s start with “Life table”.

The aim now is to use Excel as a modelling tool to produce a life table.

I have provided some initial data collected from a cohort of animals. I know how many individuals start each year (i.e. how many “enter an interval”) and therefore I can see how many survive each year (i.e. how many “enter the next interval”). I also know how many female offspring (on average) are produced by each female of each age ($m_x$).

Start by calculating survivorship ($l_x$). Survivorship is the probability of surviving from birth to a particular age. Therefore, at age 0, $l_0 = 1$, since everyone is alive at this point. The next value ($l_1$) must be calculated based on the number alive at that point. In this case it is $352/500 = 0.704$. You must generalise this calculation into a formula that can be dragged to fill column D in the worksheet. In algebraic form, the equation is $l_x = S_x/S_0$.

Tip You need to understand the use of the $ symbol in Excel, and how to drag the selected area to place the formula in the column.

Next, you can calculate the age-specific survival probability, $p_x$. Note that this is different from $l_x$. Survival probability is simply the probability that an individual will survive its current age class, i.e. the probability that an individual currently aged 2 will survive to become age 3. In this case, it is $254/298 = 0.852$. The calculation is $p_x = l_{x+1}/l_x$, or $S_{x+1}/S_x$.

From there you can calculate the age-specific probability of death, $q_x$. This is simply $1 - p_x$: the probabilities of survival and death must sum to 1.

Then calculate life expectancy, $e_x$, in column I: add up the $l_x$ values from age $x$ to the end of the table, divide by $l_x$, and subtract 0.5 (because, on average, individuals that die during a year die halfway through it). For example, in the worked example $e_1 = (0.8 + 0.4 + 0.1)/0.8 - 0.5 = 1.125$ years. Tip =SUM(D3:$D$13)/D3-0.5 in cell I3 can be dragged down the column.

Now complete the remaining columns, $l_x m_x$ and $l_x m_x x$, and use them to calculate (a) the net reproductive rate, $R_0 = \sum l_x m_x$; (b) the generation time, $T = \sum x l_x m_x / R_0$; and (c) the capacity for increase, $r_c = \ln(R_0)/T$, which approximates the population growth rate per year. Is this population growing, stationary or shrinking?

Refer to the sheet “Worked example” if you get stuck (click on the cells to see the formulae used there).

9.4.1.1 Declining force of selection (optional, if time)

This experiment is optional here: you will do a fuller version in the practical Trade-offs and the declining force of selection. Do it if you have finished the rest of the life table and the survivorship curves.

The force of selection describes the relative importance of events that happen during the life course for fitness (measured, for example, by $R_0$ or $r_m$).

To investigate this you can do a computer “experiment” by altering reproduction at different ages. You can think of these as “what if?” experiments, e.g. “What if there was a mutation that added 1 to reproduction at age 1, or at age 2, etc.?”

Before you start, predict: will adding one daughter at age 2 or at age 8 have the bigger effect on $R_0$? Why?

Add 1 to $m_x$ at one age at a time, from age 1 to age 10 (put it back before moving to the next age), and record the new value of $R_0$. Do not include age 0: newborns cannot breed at birth. Graph your results as age vs. change in $R_0$.

What you should see is that changes early in life matter more than changes later in life. (Adding one daughter at age $x$ raises $R_0$ by exactly $l_x$: a female must survive to age $x$ to have that extra daughter.)

Use $R_0$, not $r_c$, for this experiment. The approximation $r_c = \ln(R_0)/T$ can even fall when reproduction is added late in life, because $T$ increases; the exact $r_m$ always increases, but by less and less with age.

9.4.2 Survivorship curves

In the second part of this class we focus on the Survivorship Curves worksheet.

The aim here is to explore how different types of organisms with different ways of life (“life history strategies”) can have qualitatively different kinds of life tables. The most important thing to observe is the difference in survivorship curves ($l_x$). These differences become very obvious when you plot the log-transformed survivorship against age.

In the Excel worksheet, I have placed tables showing the fate of cohorts of three populations of different species. Your job now is to calculate the survivorship curve ($l_x$) for these species, and take the natural log (using the formula =LN(C3) for the first population, =LN(J3) for the second population, etc.).

You should see that the graphs automatically fill up with lines. These show Type I, II and III survivorship.

Next, calculate the age-specific probability of death ($q_x$) and of survival ($p_x$). These are the proportions of individuals that enter the age interval that do not and do survive to the end of the interval, respectively. You can calculate them as =1-(B4/B3) and =B4/B3 for the first row of the first population.

Plot graphs of these.

Make sure that you are (1) able to “diagnose” survivorship type from looking at a graph of $\ln(l_x)$ vs $x$; and (2) able to sketch a cartoon of the mortality trajectory if shown one of these survivorship curves.

9.5 Questions

What are the three main types of survivorship curves, and which organisms typically exhibit each type?
What does a net reproductive rate ($R_0$) of less than 1 signify for a population?
How does the shape of a survivorship curve reflect trade-offs in life history strategies, typically?
How is survivorship ($l_x$) different from survival probability ($p_x$)?
How does altering reproduction or survival/mortality at different ages affect $R_0$ and population growth rate?
Two populations have the same $R_0$, but one has a shorter generation time. Which grows faster per year, and why?
The probability of dying in a year, $q_x$, can never be greater than 1, but the force of mortality, $\mu_x$ (see the Notation table and the lecture), can. Why?

9.6 Takeaways

Life tables summarise age-specific survival and reproduction in a compact, comparable way.
Survivorship type reflects how mortality changes with age, and so different life history strategies.
$R_0$ is growth per generation; dividing $\ln(R_0)$ by the generation time $T$ gives the capacity for increase, $r_c$, an approximation to the growth rate per year, $r_m$.
Early-life changes usually have the greatest impact on $R_0$ and population growth.

Symbol	Meaning	Calculation
\(x\)	Age at the start of the interval (\(x = 0\) means age 0 to 1)
\(S_x\)	Number alive at the start of age \(x\)	counted
\(l_x\)	Survivorship: probability of surviving from birth to age \(x\)	\(S_x/S_0\)
\(p_x\)	Survival probability: probability of surviving from age \(x\) to \(x+1\)	\(S_{x+1}/S_x\), or \(l_{x+1}/l_x\)
\(q_x\)	Probability of dying between age \(x\) and \(x+1\)	\(1 - p_x\)
\(e_x\)	Life expectancy: further years expected at age \(x\)	\(\left(\sum_{y \ge x} l_y\right)/l_x - 0.5\)
\(\mu_x\)	Force of mortality: the instantaneous death rate of individuals still alive at age \(x\) (a rate, not a probability)	if \(\mu\) is constant over the year, \(q_x = 1 - e^{-\mu_x}\)
\(m_x\)	Female offspring per female aged \(x\)
\(R_0\)	Net reproductive rate: female offspring per female per lifetime (per generation)	\(\sum l_x m_x\)
\(T\)	Generation time: mean age of mothers when their daughters are born	\(\sum x l_x m_x / R_0\)
\(r_c\)	Capacity for increase: approximate population growth rate per year (an approximation to \(r_m\))	\(\ln(R_0)/T\)
\(r_m\)	Intrinsic rate of increase per year (exact, from the Euler-Lotka equation); \(\lambda = e^{r_m}\)	solve \(\sum e^{-r_m x} l_x m_x = 1\)