Introduction to Demography
We now move from the logistic model, that does not
consider structure within a population, to ways to include
aspects of sex, age, and/or size that will make it possible to
better describe the dynamics of populations.
In quantitative dynamics the usual tool is the life table,
however for simple life histories a lot can be learned from
what are termed diagrammatic models. That’s where we’ll
start.
What kinds of simple life histories are amenable to study
using diagrammatic models?
Simple ones with the following sorts of stages:
A simple plant life history
animal life history
(a holometabolous insect)
seed
egg
seedling
larva
immature plant
pupa
mature (reproductive) plant
adult
In addition, we have to know whether generations overlap
(parents survive through a significant part of their
offsprings’ lifespan, and may reproduce again) or do not
overlap.
Case I – Non-overlapping generations
First, the dynamics of a simple, annual, higher plant
population; one in which adults and offspring do not coexist.
We'll begin with the specific, i.e. an annual plant, then
consider the more general model.
What happens to adult plants each year? They have some
probability of surviving from time t to time t+1. Now if we
make our count just prior to the annual reproductive effort,
Then essentially all Nt alive at the start of our cycle
reproduce, with an average fecundity of f seeds/individual.
Of the total Nt x f seeds produced, only a fraction g
germinate. For simplicity we'll assume the others die, but
more complicated models could include a seed bank.
Finally, of the Nt x f x g germinating seeds, only a portion e
successfully become established. Establishment is a time of
relatively high mortality in most plant populations.
The number (in theory) in the population is made up of
survivors and newborns, i.e.
Nt+1 = Nt + Nt x fge
Since this is a model of non-overlapping generations, for
annual plants and other similar species, the Nt adults do not
carry over; the Nt term = 0.
We could construct the same sort of diagram for an insect
with a simple sequence of life stages, for example a
grasshopper.
When reproduction occurs repeatedly at different ages, the
usual approach is the life table. However, it is possible to
use a diagrammatic approach. Here’s an example for the
English great tit, Parus major:
Note that in this diagram
some adults (0.5) survive
to year t+1 after reproducing
In year t.
What Happens If a Semelparous Species Reproduces at
Varying Ages?
This would mean we can’t simply diagram a life cycle from
birth to reproduction; things are happening at different times
to different individuals.
One useful approach has been size or stage-based models.
Among the best examples are studies of teasal (Dipsacus
sylvestris) and mullein (Verbascum thapsus). Both are
biennials. Theoretically, such plants should germinate and
grow one year, then continue growth, flower, set seed and
die in a second growing season.
In these two species, a rosette of leaves grows (without any
extended stem) in the year of establishment, and continues
growing the following year. If this rosette reaches sufficient
size, a flowering stalk is sent up in that second year, followed
by reproduction and death of the adult.
If, however, environmental conditions (population density,
shading by taller plants of other species) slows rosette
growth, the key size of the rosette for flowering is not
reached, and the supposed biennial may survive additional
years, until that critical size for flowering is achieved.
This presents a complication for simple, age-based modeling.
However, a size-stage model can easily represent this life
cycle.
The stage classifications for the teasel life history are:
1) seeds,
2) seeds which remain dormant in the first spring, rather than
germinating,
3) seeds which remain dormant through 2 cycles of
germination
4) small rosettes < 2.5 cm in diameter,
5) medium rosettes between 2.5 and 18.9 cm in diameter,
6) large rosettes greater than 19 cm in diameter, and
7) flowering plants.
Here is the diagram that represents this life history for one of
eight fields studied by Werner and Caswell (1977):
If this were a perfect model (no errors, lost plants, etc.
resulting from the field situation) then the sum of all
Transitions (arrows leaving a box) should add up to 1.0, i.e.
Something identifiable happens to each individual in the
population, but note that the transitions indicated do not
include mortality.
We assume that the difference between the sums of indicated
transitions and 1.0 is the fraction dying while in that stage.
This population of teasel is growing at a growth rate/unit
time, , of 1.26, or equivalently (=er) an 'r' of .233. Those
results come not from the diagram, but from the use of a
stage-based matrix and its analysis. That comes later…
Case 2: Populations With Overlapping Generations
When generations are overlapping (the reproductive pattern
is called iteroparity, the more usual approach is to use a
life table.
A few ‘rules’ about life tables:
1)Traditionally, life tables give values for females only.
Males are either assumed to have identical survivorship
(they don’t bear young) or are tabled separately.
2)At this stage we consider age-dependent birth and death
rates to be invariant with population density (and any
measure of environmental variation, as well).
There are two ‘forms’ of life tables. They look the same
after creation, but data are collected in different ways.
A.The horizontal (or static) life table. Here we sample a
population made up of individuals of varying ages. For
those in each age group, measure the survivorship of
individuals through that age and the number of young
born, on average, to a female of that age.
How would you develop a life table for a tree, say Sugar
Maple, Acer saccharum, in a forest?
a) core trees to count tree rings and age each individual.
b) calculate what fraction survive from age group x to age
group x+1.
c) count the number of maple seeds or keys (usually by
subsampling branches) produced by the average female
of each age group.
B. The alternate type of life table, called a vertical or cohort
life table, collects the same information, but does it by
following a cohort (all the babies born at a given time)
from their birth until the last of them has died. The whole
cohort is the same age.
We measure:
a) what fraction of those alive at one birthday (or time)
are still alive at the beginning of the next time interval,
and
b) how many babies the average female had during that
time.
This method, for any long-lived animal or plant, takes a long
time, and may even be impractical, but it is the usual
theoretical approach.
Caveat emptor:
There are problems inherent in either approach:
In collecting data for a horizontal life table, we seem to be
Assuming that environmental conditions in the past haven’t
materially affected the values we get. Is what is happening
to 10 year olds now the same as what happened to 20 year
olds 10 years ago? It disregards environmental history.
In collecting data for a cohort life table we seem to be
assuming changes in environmental conditions while we are
following the population don’t have significant affects on
the population’s demographic variables. It disregards the
importance of what is happening currently in the
environment.
One last important caveat:
There is a difference between age and age class.
At birth an organism is of age 0.
It belongs to the first age class, i.e. age class = 1.
That difference will be important in calculations, particularly
when we develop matrix approaches (or brute force
equivalents) to assess (predict) population growth.
The variables in the life table:
x - the age of the cohort
lx - the survivorship, or the fraction of the original
cohort that has survived from birth to reach age x.
Expressed as an equation survivorship
lx = N(x)/N(0)
mx (or bx) - the number of female children born to an
average female of age x.
The first of the variables we will add to the life table is the lx.
Before actually filling in values, let’s look at patterns in
survivorship. There are various ways to do that. One is by
means of expectancy of remaining life; the demographic
variable is ex. This is the variable actuaries calculate to
determine the cost of your life insurance. Here’s interesting
life expectancy data:
ex
There are 3 categories into which survivorship patterns are
usually divided:
1. Type I survivorship - organisms well-adapted to their
environments (or well-buffered against them), or which
live in very stable environments.
In these circumstances we can
expect most organisms to live
out a very large fraction of their
genetically programmed lifespans. Examples: humans (at
least in well-developed countries,
most other mammals, many
species in protected, zoo
environments.
2. Type II survivorship - mortality is almost totally random,
resulting from interaction with the environment, and,
therefore, affects a constant proportion in each of the age
classes. That produces a diagonal survivorship curve.
Examples: perching birds and, interestingly, bats.
European robin species
3. Type III survivorship - the youngest age class(es) are
relatively unprotected and undeveloped, thus susceptible
to and suffer severe mortality. Following an initial sorting
out, the death rates are much lower until the onset of
senescence. Examples include many insects, weedy plants
like thistles, and Atlantic or Pacific salmon.
data for mackerel
These 3 categories are pigeonholes; many species have
survivorship patterns intermediate between those categories.
A few examples of deviations:
a) milkweed bugs – they begin like other insects with
a relatively severe mortality, after that their
survivorship is a nearly perfect diagonal.
b) Condors – like a number of other large birds, they
do not have a diagonal survivorship curve; their
survivorships are closer to a typical type I curve.
c) After severe initial mortality in many tree species,
there is a juvenile (sapling) stage during which
mortality appears basically diagonal, then a long
period as mature adult trees during which mortality
appears to be type I.
And bats:
Milkweed bugs:
California condors, now recovering from near extinction, are
very large birds that will be the subject of some sample
calculations a little later. For now, the question is why
condors don’t have type II survivorship?
To persist with a reasonable pre-reproductive survivorship
(> 0.5), they have to maintain an annual adult survivorship
of > 0.7. That is more like type I than type II.
Can we understand why many larger birds don’t have a type
II survivorship?
There are two types of development in birds. Some birds are
altricial. They are born without feathers and require initial
parental care. At hatching they are basically weak and
helpless, and typically have relatively high mortality as
hatchlings, nestlings and immediately after fledging, but a
constant mortality rate thereafter.
Precocial birds, (hatched more completely developed, with
feathers, and capable of independent existence from
hatching) like ducks and geese, delay reproduction for a
longer period. Many of these species form long-term pair
bonds at their first mating; learning processes enhancing
successful reproduction must be completed before their first
serious reproductive effort. Higher mortality extends
through the pre-reproductive period, and the lower rate
characteristic of adult life (the diagonal curve) begins at .
This is the condor pattern.
However, the condor is altricial!
Here are the theoretical survivorship patterns you’ve seen
before:
Next time we’ll begin using a life table and begin
calculations…