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Life Tables

How do ecologists investigate the number of births andthe number of deaths?

They use the age structure of the population.

Births and deaths are summarized from data in a “life table”

The life table, however, can take a number of forms.

The simplest is called a diagrammatic life table...

This is the diagrammaticlife table for an annual plant.

F is the fecundity (number of seeds per plantg is the probability of a seed germinatinge is the chance of a seedling becoming establishedp is the chance of an adult surviving (if an annual = 0)

What is the population size attime t=1? Nt+1 = Ntp + NtFge

As a numerical example…suppose you plant a rare prairie grass at Ojibway to re-establish it here. You put in 100 plants.

N(t) = 100F = 100g = 0.02e = 0.05p = 0.3 (We’re now looking at a perennial plant.)

N(t+1) = 100(0.3) + 100(100)(0.02)(0.05) = 30 + 10 = 40

N(t+2) = 40(0.3) + 40(100)(0.02)(0.05) = 16

N(t+3) = 6.4; N(t+4) = 2.5; N(t+5) = 1; N(t+6) < 1 (extinct)

To successfully re-establish this species, g, e, and/or p must increase (assuming fecundity is a species characteristic).

Here’s another diagrammatic life table, with observedtransition probabilities included. It’s for the great tit inWytham Wood in England over a one year period.

Diagrammatic life tables are useful to illustrate age structureand dynamics when the age structure is simple, when thereare few age classes (or stages) in the population.

When there are many age classes in the population, adifferent form of life table is used.

It is called a cohort life table.

A cohort is a group of individuals of the same age. Wefollow them from the time they are all newborns until thelast one dies. By convention (and because they bear the babies) cohort life tables follow the schedules of females only!

The appropriate age classes into which to divide apopulation differs for different species…

For many insects, the appropriate age classes might be days or weeks.

For rodents, the age classes might be weeks or months

For large mammals the age class intervals are likely to be years.

The basic variables in the cohort life table are:

a) age structure (or age classes)

b) survivorship - how many from the cohort (number or fraction) are still alive at each age? This may be expressed as a number or as the proportion of the original cohort surviving.

c) age specific natality - how many young are born to females of each age during that time period

By convention age class 0 are the newborns. (Newborns could be seeds, eggs, or live births)

Using numbers alive (and from it number dying) at each time, here’s what the life table looks like:

Age class #alive #dying 0 100 20 1 80 20 2 60 20 3 40 0 4 40 20 5 20 20 6 0 ---

In more usual use, the number surviving is converted tothe proportion of the size of the newborn cohort still alive.This number is called the survivorship, and it’s given thesymbol lx. We’ll add survivorship to the table…

proportion surviving Age class #alive lx

0 100 1.0 1 80 .8 2 60 .6 3 40 .4 4 40 .4 5 20 .2 6 0 0

Survivorship is frequently viewed as a graph. Pearl (1930)identified 3 general patterns in graphs of survivorship(as categories from a continuum)…

Type I - organisms live out a very large fraction oftheir genetically programmed maximumlifespan. Humans and other large mammalshave this survivorship pattern. Organisms in zoos

frequently show or approach this type of survivorship.

Type II - organisms suffer a constant proportionalmortality over time, e.g. most of the sample life tablethat you saw. Perching birds and bats are good examples of this survivorship.

Type III - suffer very high mortality in initial periods of life, but have high survivorship thereafter. A maple tree or a salmon are good examples here. For example, a salmon may produce ~ a million eggs, but less than 10 succeed to become fry.

There also may be differences between the sexes...

Why is the survivorship of male grey seals lower than that for females?

Graphs of survivorship can be used to compare populations living in different habitats...

(Cactus seedlings in the desert)

You can also compare different, closely related species…Here are 3 lizard species. Among them all 3 survivorshippatterns are found...

The next important variable that can be calculated fromour basic life table is age-specific mortality or qx. It is the fraction of the number alive at the start of an interval that die during that interval.

Age class #alive # dying qx

0 100 20 0.2 1 80 20 0.25 2 60 20 0.33 3 40 0 0 4 40 20 0.50 5 20 20 1.0 6 0 ---

The typical pattern in mammals (and many other species) is to have a somewhat higher qx in the earliest phases of life,

then qx drops to a low value through the reproductive years

And in the post-reproductive period qx increases until the cohort is gone.

A real qx curve can be far more complex, with explanationsfor at least parts of the complexity. Here is data for the reddeer...

Why should mortalitysuddenly rise at around 7years old?

As you’ll see when I show you birth data, peak reproduction occurs overages 7 - ~10. Reproductionhas costs, evident here asincreased mortality.

Before moving on, a quick review:

Much of analysis of population dynamics is based on theuse of life tables…

There are two types covered here…diagrammatic life tablescohort life tables

The life table parameters we’ve looked at so far are…survivorship lx - the probability of living from birth

to age xage specific mortality qx - the probability of death

while in age class x

The patterns of survivorship and reproduction in humanpopulations generate yet another way of looking at agestructure. This tool is called a demographer’s curve.

It isn’t really a curve. Instead, it’s a bar plot or histogramof the proportion of the total population that is of each age.

The shape of this “curve” can indicate a lot about whethera population is growing or declining.

What do the demographer’s curves show us?

In the curve for Sweden, note that there are fewer pre-reproductives than there are currently reproducing. Assuming that family size doesn’t change, what does that predict for the next generation?

The curve for Mexico looks kind of like a pyramid. There are larger proportions in younger age classes, fewer in reproductive ages, and a much smaller proportion in post reproductive years. What does that suggest for future generations?

The curve for the U.S. is pretty much flat-sided except for the bulge in mid-reproductive age classes. What’s that? (It’s the ‘echo’ of the post-war baby boom.)

The second key component of the life table is a parameterto measure the birth rate…

It is usually called fecundity.

It is the number of individuals (in whatever form - hatched,eggs laid, seeds, live young, as appropriate for the species,born to females of each age class.

N.B. remember that the life table normally only counts females; for births in most species, you can assume that there are an equal number of male births, even if they aren’t counted.

There are some basic patterns in fecundity with age…

First, an atypical pattern - the red deer. Compare peaks in thefecundity curve to the earlier mortality curve...

More frequently seen are 2 basic patterns:A relatively rapid rise to peak reproductive activity, followed by a more-or-less rapid decline to 0 reproduction. The age of first reproduction is termed . The age at which reproduction ceases is called .

for milkweed bugs:

Other species show a generally more gradual rise to peakreproductive activity, them maintain this level for many years, finally declining late in life to 0. This curve is for white-tailed deer...

Now we can add fecundity rates to the life table…

By convention, fecundity is not the total number of offspring.It is the number of daughters born to the average femaleof age x.

For example, if there were 10 females of age 2, and theyproduced, among them, 20 daughters, then the fecundity ofthis age class is 2.0.

Now, add mx to the life table...

proportion surviving fecundity Age class #alive lx mx

0 100 1.0 0.0 1 80 .8 0.2 2 60 .6 0.3 3 40 .4 1.0 4 40 .4 0.6 5 20 .2 0.1 6 0 0 ----

mx= 2.2

The mx is called the Gross Reproductive Rate.

The gross reproductive rate indicates that a motherin this population will produce 2.2 daughters if she lives tothe maximum age.

However, the gross reproductive rate ignores the mortalityschedule evident in the life table. We know that 100% ofthe cohort does not survive to the maximum age.

So, to determine the real contribution of an average female,we need to incorporate mortality. You do so by multiplyingeach mx times the corresponding lx.

The summed result is called the Net Reproductive Rate, andcalled R0 in short form.

survivorship fecundity Age class lx mx lxmx

0 1.0 0.0 0 1 .8 0.2 0.16 2 .6 0.3 0.18 3 .4 1.0 0.4 4 .4 0.6 0.24 5 .2 0.1 0.02 6 0 ---- -----

R0 = lxmx = 1.0

R0 = lxmx

The sum for this life table is 1.0. That means that anaverage female in this population leaves behind 1 daughterover her lifetime. (It is an assumption that there is 1 maleoffspring to replace the father, as well.)

Since the female parent is exactly replaced by her femaleoffspring, this population will remain constant in size fromone generation to the next.

Very small changes in survivorship or fecundity couldshift this population to one that would grow or one thatwould decline over time...

First, increase the fecundity of age class 4 from 0.6 to 1.0…

survivorship fecundity Age class lx mx lxmx

0 1.0 0.0 0 1 .8 0.2 0.16 2 .6 0.3 0.18 3 .4 1.0 0.4 4 .4 1.0 (was .6) 0.4 (was .24) 5 .2 0.1 0.02 6 0 ---- -----

R0 = lxmx = 1.16

If the net reproductive rate is 1.16, then how does a startingpopulation of 100 grow over the first few generations?

Generation N

first 100second 116third 134fourth 155fifth 180sixth 209seventh 242

(population sizes are rounded to whole individuals)

Remember, the potential for explosive growth, if growth remained exponential, was recognized by Thomas Malthus. His work was a strong influence on Darwin in developing the idea of evolution.

Malthus’ example of exponential growth: the human population in North America after colonization.

What does exponential growth say about the growingpopulation?

During the period of exponential growth, theenvironment and needed resources were notlimiting to growth.

These are the same conditions favoring explosivegrowth of exotic, invading species like the zebramussel.