BIO 412 Principles of Ecology Phil Ganter 301 Harned Hall 963-5782
Alligator at Canaveral National Seashore (taken at a safe distance)

Life Table/Demography Laboratory

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Introduction:

Within a population of a species, not all individuals are the same. We have already spoken of differences due to sex and genetic makeup. Today, we will examine the differences due to age. The age of an organism can strongly affect both the rate of mortality and natality. Just think of humans. It is very obvious that not all humans have the ability to reproduce as some are too young and some too old. It may be less obvious, but no less true, that mortality is also dependent on age, with the very young and old more likely to die. When considering a population of mixed ages, an ecologist needs to be able to predict the composition of the population in the future based on the population of today. To do this, we must have a means of taking the proportions of the population in various age classes into account. We can do this with a life table: a schedule of the probabilities of death and reproduction that covers the potential life span of an individual organism. The rates (= probabilities) of dying and reproduction in a life table are referred to as age-specific mortality and fecundity.

How do we get such a schedule? There are two general ways. One can take a snap-shot of the population at a single time and look at the proportions occurring in each age class. However, this method assumes that the population has reached a equilibrium value, called the Stable Age Distribution. A better way, but more difficult, is to follow a group of individuals born at the same time (called a cohort) until all are dead and record when each female reproduces (and how many offspring are produced). From the life table, we can calculate other ecological parameters such as expected life span (adjusted for age), the eventual stable age distribution for the population if the life table does not change with time, and the growth rate of the population.

The techniques used were developed for animal populations and are most highly developed for human populations. For that reason, the science of life tables is called demography (demos is the Greek root for people). Now the term is used for life tables for any organisms, and you often see the phrase "human demography" which is a bit redundant. Recently, plant ecologists have made good use of demographic techniques, which makes good sense if you consider that it should be easier to follow a cohort of plants due to their sessile (fixed in one place) life style.

The Life Table:

The first column of the table gives the ages into which the lifespan of the organisms will be broken.

• This must be, of course, in the time unit appropriate for the organism and for the study. Since fruit flies live only about a month and a half, if they are lucky, the time unit for them will differ from that which is appropriate for humans.
• Often, the age classes are broken into groups (e. g. human life tables often have ages by 5-year classes: 0-5, 5-10, 10-15, etc.) The letter x symbolizes the age class and appears at the head of the first column. The rest of the columns all have specialized symbols (defined below).
  • x = the interval
  n_x = the number alive at the START OF THE INTERVAL

  l_x = age specific survivorship -- fraction of cohort alive at the START OF THE INTERVAL

  d_x = age specific mortality -- number dying in the interval

  q_x = age specific mortality rate -- death rate for individuals of a specific age

  e_x = age specific life expectancy - number of years left for an organism of a particular age. IT IS NOT ALWAYS MAXIMAL AT BIRTH!!

  b_x = age specific natality -- number born in the interval

  m_x = age specific birth rate - the per female rate of offspring production for females of a particular age.

  R₀ = net replacement rate = the number of females alive in the next generation for each female alive in the present generation.

  T_c = generation time

• Many life tables have no age specific birth rate columns and are used only for calculating age specific mortalities and life expectancy. This can be very useful, as we can learn what is the most risky time of life for an organism.
• Life tables are at the base of the life-insurance business. If the table is to be used to calculate the growth rate, then only females are included in the table, as males do not give birth.
• Another convention (especially for human life tables) is to multiply the number in the l_x column by 1000 (if it is actually 0.562, then it reads 562). This gives the number still alive if you had started with a standard population size of 1000 and is done to make comparisons between tables easier. However, if you wish to calculate the net replacement rate (equation given below), then you must either convert l_x back to a proportion or you will be calculating the number of females in the next generation per 1000 females alive in the present generation! We will avoid the problem by expressing l_x as a decimal less than one.

Calculations from the life table:

First, lets start with a life table that has no births in it, so we can work with life expectancy exclusively. We gather n_x and calculate l_x, q_x, and e_x from it! First, I should mention that, by convention, the first age inerval has the notation 0, so the number in the cohort at the start is n₀.

To calculate e_x, we will need to tack on another pair of columns to make it easier to do. This column is L_x, the average number alive in interval x. This is simple to do:

From this column, we can get Tx, which is the sum of all of the Lx values from x until the age that the last member of the cohort dies (this is symbolized by L_°). We will use this to calculate life expectancy. To understand how this is done, one must consider the units we are using (T_x and L_x will have the same units). Remember that we defined L_x as the average number of individuals alive in an age class. You might wonder how we can reconcile these units with life expectancy, as L_x seems to be in the units "individuals" and life expectancy (e_x) is in time units. However, L_x is the sum of individuals that live a particular period of time (= the duration of an age class), and so the units are actually "individuals-age class." This is not obvious from the way we calculate L_x, but it is necessary for calculating life expectancy, as we will see below.

Now, we can calculate the age specific life expectancy. This is the total

What are the units of life expectancy? T_x is in "individuals-age class" and n_x is in "individuals". Dividing T_x by n_x cancels out "individuals" and we are left with "age class". Thus, you will get an answer in terms of age classes. If you want to use a more universal unit, you have to convert between age class and, say, years. A life expectancy of 10 age classes is 50 years if one age class is ten years long.

If we eliminate the males and add a column containing births (total number of births by mothers in each age class), then we can calculate the net replacement rate (R₀).

M_x, the age-specific birth rate is the number of births from mothers in the x age class divided by the total number of individuals in that age class

R₀, the net replacement rate, is the sum over all age classes of the product of the age-specific survivorships times the age-specific birth rate. Thus, the population growth rate depends on the number of offspring each age class produces reduced by the probability of surviving to that age (remember that lx will be between 0 and 1).

The net replacement rate is a measure of population growth rate.

• If it is 1, then the population in neither growing or declining in size (a stable population).
• If it is less than 1, the population is declining by that proportion each generation (if it is 0.5, then the population will be half as big each generation).
• A value over 1 indicates a growing population (a value of 2 is a population that doubles each generation).

Before we go on, it is best if we take a second look at what we have calculated to see why the calculation results in the net replacement rate.

• If you do not do this for all of the terms derived here (e_x, T_c), you can't really say you understand demography (or, in general, anything described in mathematical terms). You can only say that you can do the calculation. A computer spreadsheet can also do the calculation, but I would not assert that the computer understands the calculation.
• To accomplish this understanding, we will use a device you should be familiar with from chemistry and physics classes: dimensional analysis.
  • What is R₀, really? The easy answer is the sum of l_x times m_x for each age class (x).
  • What are you multiplying when you multiply mx times l_x? Use dimensional analysis.
    • Survivorship (l_x) is calculated as n_x/n₀ and is individuals alive at time x per each individual alive at the start.
    • The m_x term also has dimensions. It is calculated as b_x/n_x, or is the number of births at time x per individual alive at time x.
    • When you multiply these terms, each l_xm _x term has the dimensions below:

.

• So, l_xm_x is in terms of births at time x per individual at time 0 (or per individual in the cohort).
  • When you sum these, you are getting all of the births (at all times) per individual in the cohort. This will be the number born per individual during the lifespan of the cohort (not just any particular individual). Now look at the definition of R₀ commonly given:

"R₀ is the net replacement rate or the number of individuals in the next generation per individual in the current generation."

• You can see how this is true from the above analysis of the dimensions of the terms that contribute to the calculation of R₀. If the sum is less than one, there is less than one in the next generation for each alive in the current generation, and the population is shrinking. Try dimensional analysis to further your understanding of the calculation of T_c below.

We can also calculate the approximate generation time from the same data that gave us the net replacement rate.

• Generation time is the average time from the birth of a female until that female gives birth.
• For reasons we will discuss in class, this calculation is an approximation for human data (because we have overlapping generations). However, the statistics you calculate will be good approximations if you use the following formula:

  • 

Gathering the Data:

Data from a local cemetary

Your first consideration should be just that: consideration. We are visiting a graveyard, which is consecrated ground and must be afforded due respect. No walking on graves, no disturbing memorials, and a serious demeanor is expected.

We want to work with a cohort that is old enough so that it is most likely that all in that cohort are dead. You will work in pairs pick a random starting point far enough from others that you will not overlap and resample data already collected by other pairs. Walk along the tombstones and record the sex of the individual, his or her birth and death dates, and calculate the age at death from 100 tombstones. If a date is missing, skip that tombstone. Skip any tombstone later than 1930.

Back in the lab, we will organize the data from the entire class by year of birth. When that is done, we will assign you a cohort to work with. We may have to combine sequential years, as we want at least 50 individuals in each cohort

In-lab data

If we can not go to the cemetery, we will use data from an old cemetery located just outside of Philadelphia, PA. The instructor will assign two sets of years to you. The first set are two to three years of data from the earliest graves and the second set are from the newest graves. You will do the calculations below on both sets of data and compare them for changes in demography.

Data Analysis:

Construct a separate life table for each cohort-by-sex combination. Use the headings below:

Make 5 year age classes, so that the age class column should read 0, 0-4, 5-9, 10-14, etc. until the last of the cohort died. X will be the number of the age class, starting with 0 and going up to 15 or so (depending on how old the oldest person in the cohort was when he or she died).

Separate the data by sex (a separate table for each sex) and calculate the death rates, survival rates, and life expectancies for each sex from your cohorts.

When you have completed the above, look to the table below for the m_x data. This data comes from two populations taken during mid-century censuses. One is an island and the other is a large, industrial country. We will use both of these to see how both mortality and natality contribute to population growth. Use both sets of mx data and the survivorship data from the females in your cohorts to calculate R₀ and T_c. If you are using a spreadsheet, this is easy to do.

Data Presentation and Interpretation:

First, present you dimensional analysis of the calculation of T_c.

Second, present the life tables for your cohort, by sex. Title each table and make sure it includes the years in which they were born in some prominent place.

Third, make two graphs:

• one for mortality rates (of all four cohorts) versus age class (as the x axis)
• one for life expectancy(of all four cohorts) versus age class (as the x axis)

  On each graph, all four datasets should be presented (one line for females from the early cohort, one for females from the later cohort, one line for males from the early cohort, one for males from the later cohort ). Be sure that a legend is presented and that all lines look different enough to make reading the graph easy.

Fourth, present the calculations for R₀ and T_c for both island and continent data.

Fifth, answer the questions below.

Thinking about the experiment:

1. Is there any difference between males and female life expectancies or mortality rate schedules? Be specific. Suggest reasons for what you conclude from the graphs (and be sure what you conclude comes from the graphs and not what you expected). If your expectations and the graphs disagree, this is a good thing to discuss and to suggest reasons for the discrepancy.
2. How do the m_x values differ between the island and contental populations?
3. How does the change in m_x schedules from the island to the continent change both net replacement rates and generation times. Keep in mind that both sets of figures were calculated with the same survivorship data. How would the differences in R₀ affect population growth?
4. Lets look at the effect of timing of reproduction on growth and generation time. Recalculate R₀ and T_c for the early cohort with the island and continental data, but alter one thing. Move all of the m_x data up one age class earlier, so that the 10-14 age class is 0.070, the 15-19 class is 0.175, etc (and ending with the 40-44 age class). What we have done is changed the age of first reproduction. What happens to R₀? How can this be when m_x (which is the number of children born in an age class, so all of the m values represent all of the children born per female) has not changed? What happens to population growth when women choose to have children at a younger age? Make you comments in terms of the changes expected in the growth of a population.

Last updated September 9, 2006