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Life Tables How do ecologists investigate the number of births and the 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 diagrammatic life table for an annual plant. F is the fecundity (number of seeds per plant g is the probability of a seed germinating e is the chance of a seedling becoming established p is the chance of an adult surviving (if an annual = 0) What is the population size at time 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) = 100 F = 100 g = 0.02 e = 0.05 p = 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 observed transition probabilities included. It’s for the great tit in Wytham Wood in England over a one year period.

Diagrammatic life tables are useful to illustrate age structure and dynamics when the age structure is simple, when there are few age classes (or stages) in the population. When there are many age classes in the population, a different form of life table is used. It is called a cohort life table. A cohort is a group of individuals of the same age. We follow them from the time they are all newborns until the last 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 a population 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 to the proportion of the size of the newborn cohort still alive. This number is called the survivorship, and it’s given the symbol 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 of their genetically programmed maximum lifespan. Humans and other large mammals have this survivorship pattern. Organisms in zoos frequently show or approach this type of survivorship. Type II - organisms suffer a constant proportional mortality over time, e.g. most of the sample life table that 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 survivorship patterns are found...

The next important variable that can be calculated from our 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 explanations for at least parts of the complexity. Here is data for the red deer... Why should mortality suddenly rise at around 7 years old? As you’ll see when I show you birth data, peak reproduction occurs over ages 7 - ~10. Reproduction has costs, evident here as increased mortality.

Before moving on, a quick review: Much of analysis of population dynamics is based on the use of life tables… There are two types covered here… diagrammatic life tables cohort life tables The life table parameters we’ve looked at so far are… survivorship lx - the probability of living from birth to age x age specific mortality qx - the probability of death while in age class x

The patterns of survivorship and reproduction in human populations generate yet another way of looking at age structure. This tool is called a demographer’s curve. It isn’t really a curve. Instead, it’s a bar plot or histogram of the proportion of the total population that is of each age. The shape of this “curve” can indicate a lot about whether a 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 parameter to 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 the fecundity 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 peak reproductive 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 female of age x. For example, if there were 10 females of age 2, and they produced, among them, 20 daughters, then the fecundity of this 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 mother in this population will produce 2.2 daughters if she lives to the maximum age. However, the gross reproductive rate ignores the mortality schedule evident in the life table. We know that 100% of the 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 multiplying each mx times the corresponding lx. The summed result is called the Net Reproductive Rate, and called 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 an average female in this population leaves behind 1 daughter over her lifetime. (It is an assumption that there is 1 male offspring to replace the father, as well.) Since the female parent is exactly replaced by her female offspring, this population will remain constant in size from one generation to the next. Very small changes in survivorship or fecundity could shift this population to one that would grow or one that would 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 starting population of 100 grow over the first few generations? Generation N first 100 second 116 third 134 fourth 155 fifth 180 sixth 209 seventh 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 growing population? During the period of exponential growth, the environment and needed resources were not limiting to growth. These are the same conditions favoring explosive growth of exotic, invading species like the zebra mussel.