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. Thats where well 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 historyanimal life history (a holometabolous insect)seedeggseedlinglarvaimmature plantpupamature (reproductive) plantadult
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. Heres 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 cant 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 GenerationsWhen 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 dont 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, andb) 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 havent 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 dont have significant affects on the populations 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, lets 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. Heres interesting life expectancy data:ex
There are 3 categories into which survivorship patterns are usually divided: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
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 senesce