10 Life tables and survivorship types

10.1 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 analyzing 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. As a result, life tables play a vital role in shaping policies, making informed decisions, and gaining a deeper understanding of the complex interplay between age, health, and mortality.

The basic algebra used in life tables is explained in Neal Chapter 6 and in Gotelli Chapter 3.

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 distribution of mortality risk through a life course. Survivorship types are crucial in understanding the overall survival patterns and demographic characteristics of different populations.

Three primary survivorship types are commonly identified, based on survivorship curves on a log scale:

Type I (Convex Curve): This survivorship type is characterized by a low mortality rate during early life, with an increase in mortality rates at older ages. It indicates that individuals in the population have a high probability of surviving to advanced ages, and the majority of deaths occur in older age groups. Type I survivorship is observed in humans and other long-lived species with strong parental care and relatively low early-life mortality.
Type II (Straight Line): In this survivorship type, the mortality rate remains relatively constant across all age groups. It suggests that individuals have an approximately equal chance of dying at any age throughout their life. Type II survivorship is commonly observed in species where mortality rates are consistent, regardless of age. Some small mammals and birds exhibit this survivorship pattern.
Type III (Concave Curve): Type III survivorship is characterized by high early-life mortality rates, followed by a decline in mortality at older ages. This pattern indicates that survival rates are low in the early stages of life, but those who survive the initial vulnerable period have a greater likelihood of reaching older ages. Type III survivorship is often seen in species that produce numerous offspring but provide limited parental care, where only a fraction of the offspring survive to adulthood.

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. Survivorship patterns play a significant role in shaping the dynamics of populations and help scientists make informed decisions regarding conservation efforts, healthcare planning, and population management strategies.

Learning outcomes:

Increased competence in using Excel formulae for mathematical modeling.
Competence in using mathematical models in Excel to strengthen own understanding of biological processes.
Understanding how life tables are calculated.
Understanding the three types of survivorship curve and how they relate to mortality and survival probabilities (and their trajectories with age).
Understanding the decline in the “force of selection”.

10.2 Your task

Download and open the Excel file Life tables exercises.xlsx.

The file has three worksheets (“Life table”, “Survivorship Curves” and “Gotelli Table 3.1 example” (don’t worry if you don’t have the text book).

10.2.1 Life table

Let’s start with “Life table”.

The aim now is to use Excel as a modeling 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) babies (on average) are produced by each female.

Start by calculating survivorship ($l_x$). Survivorship is the probability of survival to a particular age. Therefore, at time 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 generalize 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 age-specific survival probability. 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., what is the probability that an individual currently aged 2 will survive to become age 3. In this case, the $254/298 = 0.852$. The calculation is $g_x = l_{x+1}/l_x$, or $S_{x+1}/S_x$.

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

Now complete the remaining columns $l_x m_x$ and $l_x m_x x$, and use them to calculate (a) $R_0$; (b) Generation time; and an approximation of $r$.

Refer to the sheet “Gotelli Table 3.1 example” if you get stuck (click on the cells to see the formulae used there).

10.2.1.1 Declining force of selection

The force of selection describes the relative importance of events that happen during the life course on population growth (e.g. $R0$ or $r$).

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 0-1, or 1-2 etc.?”

Specifically, we can look at the effect of this experiment on $R0$ or $r$, which are measures of fitness?

Try graphing your results as age vs. change in $R0$.

What you should see is that changes early in life matter more than changes later in life.

10.2.2 Survivorship curves

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

The aim here is to start 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 changes 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, take the natural log (using 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 survival ($g_x$) and of death ($q_x$). This is the proportion of individuals that enter the age interval that do survive and do not survive to the end of the interval, respectively. You can calculate it as =B4/B3 and =1-(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 $x$ (2) able to sketch a cartoon of mortality trajectory if shown one of these survivorship curve.

10.3 Questions

What are the three main types of survivorship curves, and which organisms typically exhibit each type?

What does a net reproductive rate (R0) 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 ($g_x$)?

How does altering reproduction or survival/mortality at different ages affect $R_0$ and population growth rate?