In this first module on Population in Chapter 5,
Module 5.1, I will cover foundational concepts
including derivation of the balancing equation
and the relationship of arthithmetic and geometric change
as well as between geometric and exponential growth.
Population models are concerned
with rate of change in population numbers.
If the number in a population is 100
at the beginning of a week,
and 120 at the end of the week
then its rate of change
would be the arithmetic rate of 20 per week.
To convert this to a demographic rate,
this population is growing at the geometric rate
of 20/100, which means that every individual
is increasing its number by 20%
or 1.2-fold per week.
The arithmetic rate,
denoted r* is given by the model
[Equations on screen, N(t+1) is r* rearranged]
The demographic rate, denoted r,
is given by the model
[Equation on screen, N(t+1) is r rearranged]
Projections of these rates yield different results
Arithmetically — a population of 1,000 individuals
growing at 250 individuals per year
will have grown by 500 at the end of 2 years
and by 750 by the end of 3 years
Geometrically — a population growing at 25% per year
will have grown by 562 in 2 years
and 953 in 3 years.
Note that the arithmetic series
will have a common difference between numbers,
for example, 1, 3, 5, 7 and 9
is an arithmetic series with a difference of 2,
and the geometric series will have a common ratio
For example the series 1, 3, 9, 27 and 81
is a geometric series with a common ratio of 3
Population growth rate involves geometric growth in discrete time
as well as an exponential growth in continuous time.
Balancing equation
The simplest population model
is the crude rate model
that relates the total population this year
to the total population last year.
This is based on the balancing equation
where if we let last year represent the initial population
at time 0, denoted Pop0,
then the balancing equation is given as
[Equation given on screen]
Assuming that the population is closed to migration
and therefore excluding in-and out-migration
we can define the following:
[Equations on screen]
where b and d denote per capita birth and death rates, respectively.
Therefore we have
[Equation on screen]
Note that if b -d = 0
the population is at replacement
if b > d the population is increasing
and if b < d the population decreasing.
The population at time two,
that is t=2 or population at time two, is
[Equation on Screen]
This can be expressed as
[Second equation on screen]
Because the formula for Pop(1) can be substituted.
Thus Pop(2) can be given as
[Third Equation on screen]
The relationship for t number of time units is
[Equation on screen]
By substituting N(t) for Pop(t)
and lamda for (1+b-d) yields
[Equation on screen]
This expression gives the discrete or geometric version
of population growth.
This model can be rephrased in terms of exponentials
by taking logs of both sides and defining r = ln (e^lamda).
Therefore [Equations on screen]
Note that the numerical relationship
between geometric and exponential growth
has two parts.
First, at low r-values,
e^t =1+r.
Second, geometric and exponential growths
are interrelated as:
[Equations on screen]
In other words, the geometric rate
of population increase
equals the exponent e
raised to the rth power.
There are a number of assumptions
of the crude rate model or balancing equation
including (1) the population is homogeneous
and has no age structure
(2) birth and death rates are fixed,
(3) the population is closed to migration.
Whereas some of these assumptions may be limiting,
this model has provided a foundation for the development
of more complicated models
as will be described in subsequent modules.
This ends this first module of Chapter 5 on population.