Goals: describe schedules of the major events during the life of species, develop life tables.  Learn to calculate survivorship, various measures of mortality, and measures of generation time and population growth rates from life table data.  Learn to draw and categorize survivorship curves.

The Lecture:

Demography is the study of the schedule of major events in life: births and deaths.  A schedule showing survival to various age classes, mortality at those age classes, and fecundity (offspring produced) by individuals of different ages is called a life table.  Ecologists use life tables to identify periods during the life when there are higher or lower chances or survival, and times when most offspring are produced; this information is crucial to the understanding of the factors that affect abundance.  Life tables can also be used to estimate the measures of per capita population growth we have seen in the previous lectures: R and r.

A cohort life table is made by following all individuals born during one time period until all have died, and keeping track of how many have died and how many offspring they produce at different ages.  The cohort refers to all individuals that were initially produced.  This information can be estimated if we assume that the population is stationary (not increasing or decreasing in size) by what is called a static life table: a schedule of all individuals alive at each age class at one time.  If the population is staying the same size, then these numbers should stay the same over time for all age classes, so they would estimate what would be observed if a cohort were followed.  Since populations are often not stationary, cohort life tables are more accurate.  They are also more time consuming to obtain.

The following hypothetical cohort life table shows many of the typical values that can be calculated from a life table; they are defined below the life table.
 
 

x	ax	lx	dx	qx	kx	mx	lxmx	xlxmx
0	100	1	0.5	0.5	0.30	0	0	0
1	50	0.5	0.2	0.4	0.22	1	0.5	0.5
2	30	0.3	0.1	0.3	0.18	3	0.9	1.8
3	20	0.2	0.2	1.0	--	2	0.4	1.2
4	0	0

x refers to the age class, measured in some time units appropriate to the species.  I will call them years, for convenience, but really they could be hours (for bacteria, for example); for some insect species weeks or months might be appropriate, and for long lived organisms we might want to use decades.

All the other measures in the life table are referenced to the age; they are calculated for each age and the subscript x shown on each of them would refer to the age.  So, for example, in the table above a₀=100, a₁=50, a₂=30...; the subscript numbers refer to the age class.

a_x is the number of individuals surviving to age class x.  In a sexually reproducing species with separate sexes, only females are counted, since population growth depends on the number of females present to reproduce.  These values are some of the basic data that must be taken to create the life table.

l_x is called the survivorship. This means the proportion of individuals originally born (the original cohort) surviving to age class x; it is calculated as:

         l_x = a_x/a₀

d_x is a measure of mortality; it measures the proportion of original cohort dying during age class x.  It is calculated as:
       d_x = l_x - l_x+1

To obtain information on the proportion of the original dying during several age classes combined, values of d_x can be added together.

q_x is called the age specific mortality rate; it tells the probability that an individual who has survived to age class x will die before reaching the next age class.  It is calculated as:

      q_x = d_x/l_x

A disadvantage to q_x  is that the values can not be added meaningfully for more than one age class.  The next column in the table, k_x, is also a measure of age specific mortality; it has the advantage that it can be added meaningfully for more than one age class.  more complicated measure that reflects both the age specific mortality and is additive so it can reflect specific mortality over several age classes:

k_x is called the "killing power" and, as noted, reflects age specific mortality (like q_x).  It is calculated as:

      k_x=log₁₀a_x - log₁₀a_x+1

m_x refers to the number of offspring produced by an average individual of age class x.  In a sexually reproducing species with separate sexes, only females are considered, so it is the number of female offspring produced per female of age class x.  m_x values, like a _x values, are obtained as data from observations of natural populations.  m_x values are required to make any calculations of population growth or generation time from life table data.

The rest of the columns in the life table given above are used to help calculate measures of population growth. The first measure of population growth we will consider is called R₀.

R₀ is the finite rate of per capita population growth over the length of one generation.  In a sexually reproducing population with separate sexes, it can be thought of as the number of  female offspring are produced per female originally present in the cohort.  It is calculated as:

        R₀ = Sl_xm_x

In the life table above, R₀ = 0 + 0.5 + 0.9 + 0.4 = 1.8

R₀ provides some information about population growth. If R₀ is 1, for each female born each generation 1 new female will be produced.  So next generation there will be the same number of females as there were at the start of this generation, so the population size constant (stationary). If R₀ is less than 1, then for each female born each generation there will be less than one female produced --  the population is decreasing in size. If R₀ is greater than 1, then for each female born each generation there will be more than one female produced -- the population is growing.

R₀ measures finite per capita population growth over the length of one generation.  Since generation lengths vary from species to species, it is not useful for comparing growth rates among species.  It can be used, however, to estimate R, the standard finite measure of per capita population growth, and, if a population is growing exponentially, it can also be used to estimate r.  To relate R₀ to either standard measure of per capita population growth, we first need to known the length of a generation.

T stands for the length of a generation, which means the length of time from the time individuals are born to the time most offspring on average are produced for a population.  It is calculated from the life table information as:

    T = Sxl_xm_x/ Sl_xm_x

In the life table above, T=(0+0.5+1.8+1.2)/1.8 = 3.5/1.8 = 1.9

Now let's relate R₀ to R.  Remember that R is the finite measure of per capita population growth, so:

R=N_t+1/N_t

To relate this to R₀ , we need to determine how much a population growing according to the finite rate R would increase over the length of a generation, rather than just one time unit (I will refer to years as the time unit for the rest of this.)  To do this, we will first develop a formula for how much a population growing according to finite rate R would increase over any amount of time (then we can apply this to one generation.)

Suppose we start with an initial population size N₀ .  To determine the population size one year later, we would multiply  N₀  by R:

    Number of individuals after one year = N₁ = RN₀

To determine the population size after another year, we multiply by R again:

    Number of individuals after two years = N₂ = RN₁   and since we know that N₁=RN₀  N₂=R(RN₀ ) = R²N₀

Now let's look at another year:

    Number of individuals after three years = N₃ = RN₂   and since we know that N₂=R²N₀ = R³N₀

By now you should see the pattern.  To get the number of individuals after any time of length t, we multiply the initial population number by R raised to the t power.  Stating this as a formula, we get:

     N_t=R^tN₀

So over the length of a generation, T, N_T=R^TN₀.   But we know that over the length of a generation, the growth rate is R₀ .  So R₀ must equal R^T.

Now that we can relate R to R₀ we can relate R to r and then use our estimate of  R₀ to estimate the value of r.  To relate R and r, we must assume that a population is growing exponentially (remember that this will be true for early stages of logistic growth as well as for actual exponential growth.)

If a population is growing exponentially, remember that the equation for exponential growth is

     N_t=N₀e^rt.

R measures population growth over one year, so t=1.  For population growth over one year, we can write the exponential growth equation as:

     N_t+1=N_te^r(1) ₌N_te^r

Dividing both sides by  N_t gives:

     N_t+1/N_t=e^r

Since R=N_t+1/N_t, we can substitute R into the above equation:

     R=e^r

Taking the natural log of each side gives:

    lnR=r

Now that we have related R and r, we can figure out how to estimate r from life table information: R₀ and T.

Remember from above that  R^T=R₀.  To relate this to r, take the natural log of each side:

     lnR^T=lnR₀.  Remember from the rules of logarithms that log(X^Y)=YlogX.  Applying this gives:

     TlnR=lnR₀.  Remember  that lnR=r, so:
    Tr=lnR₀
     r= lnR₀/T

So if we assume that a population is growing exponentially, then we can estimate r from life table data.
One way to test for exponential growth is to see whether there is a stable age distribution, that is, whether the proportions of individuals in each age stays constant over time.  If we have made cohort life tables starting different years, we can look at the proportions of individuals in each age class and see whether or not the age distribution is stable.

For the life table given above, we can calculate r:

     r = nR₀/T  = ln(1.8)/1.9 =  0.31

A final kind of information we obtain from life tables is a graph called a survivorship curve.  A survivorship curve is a plot of the log of survivorship (lx) versus age class.  By plotting the logarithm, we make our measure of survivorship proportional, so that the slope of the curve reflects the probability of dying during a particular age regardless of the number of individuals in each age class.

Ecologists define three types of survivorship curve, I, II, and III, as shown on the following graph:

A type I curve shows a high probability of survival early in life, and a much lower probability for later ages.  This is typical of humans and some other large mammals.  A type II curve shows a constant probability of survival throughout life.  This is typical of songbirds.  A type III curve shows a low probability of survival for early ages, and a much higher probability of survival later.  Many invertebrates have type III curves.

Real survivorship curves may show one of these three standard patterns, or they may be mixtures.  For example, a common curve starts out looking type III, but then becomes type I.  This occurs because young ages are often vulnerable and have lower survival than older ages,  but once adulthood is reached survival may be quite high until old age.   This is shown here: