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 eggseedling larvaimmature plant pupamature (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 wemake 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 diagramsome adults (0.5) survive to year t+1 after reproducingIn 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 anyextended 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 germination4) 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 life- spans. 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…