Math 336 - Mathematical Modeling
Spring Semester, 2000
Outline of Chapter
Population of the United States
The United States takes a census of its population every 10 years.
The census has important ramifications for many aspects of our
society, such as budgeting federal payments and representation in
Congress. Accurately predicting this demographic data is important
for planning our communities in the future. There is plenty of
political controversy over the numbers, but at the base of all
calculations for the future is some type of mathematical model.
Current models are quite sophisticated, but first we must appreciate
the basic models behind them. Below we present the census data for
the history of the U. S.
U. S. Census Data
Select the range of Census Data
Growth rate of the range selected
35.1%, 36.4%, and 33.1%,
which averages 34.9% per decade. This
growth rate is almost constant until 1860, so this information should
allow us to estimate the census data up until 1860 using a model with
a constant growth.
Discrete Malthusian Growth
Let the integer n represent
the number of decades after 1790 and define Pn to be the population
for the nth decade
after 1790 (with P0 again representing the
population in 1790). The population for one decade is estimated by
using the population from the previous decade and adding to it the
average percent growth multiplied by the population from the previous
decade. The mathematical model based on this description is given by:
where r is the average growth
rate. Our calculations above suggest that we use r = 0.349 to
estimate the population of the U. S. from 1790 to 1860.
This equation is the Discrete Malthusian
Growth model (named after the work of
(1766-1834)). The Discrete Malthusian Growth
model is a special example of Discrete Dynamical equations or Difference equations, which we will study
in more detail as the semester progresses. Population models using
difference equations are commonly used in Ecological modeling as one
can often determine the population of a species or collection of
species knowing the population of the previous generation of the
species being studied. The Malthusian growth model states that the
population of the next generation is proportional to the population
in the current generation, which is what is written in the equation
Below is a table and graph showing the discrete Malthusian growth
model applied to the U. S. census data up until 1870.
From the table and graph above, you can see that until 1870, the
population of the U. S. follows the Malthusian growth model extremely
well with only a few percent error. However, the error in 1870 jumps
dramatically with a prediction that is 8.2% high. Note that between
1860 and 1870 there was the Civil War, which would cause a dramatic
decline in the population in the U.S., so you would expect this
prediction to be poor. More significantly, the United States was
entering the industrial revolution, which had a more lasting effect
on population growth. The move from agrarian to an industrial society
resulted in a significant decline in the net birth/death rate. The
crowding of industrial societies, where children become more of a
burden than an asset as they are on farms, results in less desire of
the population to have large families.
Clearly, this model is limited to a range of dates where the
growth rate remains relatively constant. If you attempt to continue
using this model until 1920 or 1970, then the model produces the census
values of 192,513,429 and 860,044,211, respectively. Thus, this model
becomes increasingly bad if we assume the constant growth rate of
around 15% and further drops to only 13% in 1970. (The lowest growth
rate can be seen to have occurred during the Great Depression
(between 1930 and 1940) with only a 7.2% growth rate.) Thus, this
simple model can only predict population for a limited time into the
future, but certainly provides good estimates for some community
planning that is required.
Solution of Discrete Malthusian Growth Model
There are not many discrete models that have an explicit solution.
However, it is easy to solve the discrete Malthusian growth model.
From the model above, we see that
Thus, the general solution of this model is given by
This shows why Malthusian growth is also known as exponential growth. By making r into a function which depends on either
time or population or both, then we can see how this model can be
improved. For more information on human population growth you might
want to visit the UN
website on population.
Let us suppose that a population of yeast, satisfying Malthusian
growth, grows 10% in an hour. If the population begins with 100,000
yeast, then find the population at the end of 4 hours. How long does
it take for this population to double?
The population of yeast satisfies the equation
The population after one hour is P1 =
110,000. After two hours, P2 =
= 121,000. Thus, after 4
hours, P4 =
For the population to double, it must reach 2P0 =
200,000. Thus, we must solve
By taking the logarithms of both sides we have
You may have noticed that the discrete Malthusian growth model is
closely related to compound interest problems. If interest is
compounded annually, then the amount of principle in any year
n satisfies the discrete
Malthusian growth model. The general formula for determining the
amount of principle when interest the rate is r (annual), which is compounded k times a year for n years, given an initial amount of
where Pn is the amount of principle after
In population studies, one can use this concept to examine growth
rates for a population growing according to the Malthusian growth
model for differing periods of time. For example, our model above on
the U. S. census had a growth rate of approximately 35% per decade in
the early years. The question arises as to what the appropriate
annual rate of growth would be.
If we let r be the annual
growth rate, then we need to solve the equation
This is easily solved by taking the 10th root of each
Thus, the appropriate annual growth for the population of the U.
S. near 1800 was about 3% per year.
The section above presents a discrete Malthusian Growth model
based on the U. S. population from census data. We noted that if an
average growth rate over the first few decades gives a growth rate
that is much too high for population prediction in the
20th century. In this section we compare the discrete
Malthusian growth model using the average growth rate over all the
data and an improved modified model that uses a time dependent growth
The general discrete dynamical population model is given by
where f is a function depending only on the population
P at time tn. This difference equation is
said to be autonomous as it does
not have a temporal or time dependence. A more general difference
equation is given by
which is a nonautonomous difference
The growth rate for the Malthusian Growth
model is computed by dividing the population at one census date by
the population at the previous census date. When the average is take
over all the census dates in the table above, we find that the
average growth rate
This results in the discrete Malthusian
Clearly, this growth rate is too low for the
early years, and too high for later years. Thus, a modified time
dependent growth rate can be found by fitting a line through the
data. The best fit to the growth rate data as seen in the graph below
satisfies the following:
is the date of the census.
The resulting modified nonautonomous
difference equation is given by
where tn= 1790 +
10*n. The graph below gives a comparative study of these
two models. Though the nonautonomous model is clearly much better
than the autonomous model, there are problems having a population
model that depends on a temporal variable (t). In the next
section, we'll examine improved autonomous models.
In general, population biologists would prefer to have a model
that is not time dependent.
Above we studied the Discrete Malthusian growth model, which
showed exponential growth. This model is appropriate for early phases
of population growth for most animal populations. However, as a
population grows, it encounters crowding pressure due to many factors
such as toxic build up or space and resource limitation.
In 1913, Carlson  studied a growing culture
of yeast. Below is a table of the population for these yeast at one
hour intervals. We would like to develop a mathematical model to
describe the growth of this culture.
The general discrete dynamical system is given by the
This is an iterative map where the population at the (n + 1)st generation depends on
the population at the nth generation. The function
f(p) is called the updating function as it produces the next
population in the iterative scheme. A graph of the updating function
has the (n + 1)st generation on the
vertical axis, while the nth generation is on the
It is clear that the population of yeast in the table above does
not satisfy a Malthusian growth, which has a linear updating function
and grows exponentially without bound. The next obvious addition to
the updating function is the addition of a quadratic term, which
should be negative to reflect a decrease in the growth of a
population due to crowding effects. This is the Logistic Growth model and can be written:
This equation has the Malthusian growth model seen in the previous
section with the additional term -rPn2/M.
The parameter M is called the
carrying capacity of the
The behavior of the Logistic growth model is substantially more
complicated than that of the Malthusian growth model. There is no
exact solution to this discrete dynamical system. The ecologist
Robert May (1974) studied this equation for populations and
discovered that it could produce very complicated dynamics. In its
simplest form the Logistic growth model can be written:
where the parameter m varies between 0
and 4. For a good description of this model complete with Java applet
simulations see the website of
The website of
Fraser shows a process of geometrically viewing the updating
function, which is called cobwebbing.
The updating function, f(pn), is
graphed with the identity map, pn+1 =
pn on a single
graph with the vertical axis being pn+1
and the horizontal axis being pn. The idea is that you start at any
p0, then go vertically to p1 =
you go horizontally to the identity map to locate p1
on the horizontal axis. From there you find p2
by going vertically to p2 =
process is repeated to generate the cobweb of points by this discrete
dynamical system. The sequence of points on the horizontal axis form
the solution set generated by the discrete dynamical system. The
graphical representation allows you make some projections of the
behavior of the system. Below is a diagram showing several steps in
the cobwebbing scheme for the quadratic map
The graph below shows a plot of pn+1 vs. pn
from the table above. (This is accomplished by plotting the
population from one time against the population from the previous
time. For example, the first two points are (9.6, 18.3) and (18.3,
29.0).) The graph of the data is fit applying Excel's polynomial fit
with trendline, using a quadratic passing through the origin, and is
shown below. This equation becomes the updating function.
Our discrete logistic growth model
for the yeast experiment above is given by
Below is a simulation showing both the data and the model. As we
can see, the model does a fairly reasonable job of simulating the
data from this fairly simplistic model.
Qualitatively, we see the same initial roughly exponential growth,
then both models seem to level off at approximately the same value.
This is the carrying capacity of the
population. The equation above shows r =
0.56 and r/M =
0.000861, so M = 650.4. This is clearly a little low based
on the original table.
U. S. Census with Logistic Growth
We saw that the Malthusian growth model did not work well for the
U. S. census data. Can we apply the logistic growth model to the U.
S. census data and get a better fit to the data and avoiding the
problems of the time varying nonautonomous model developed at the end
of the last section?
We return to the U. S. census data in the previous section. As
with the yeast model shown above, we need to find the updating
function by plotting Pn+1 versus Pn. So it remains a question of how
to best fit an updating function of the quadratic form given by the
Logistic growth model through this data. It turns out that if you
simply perform the least squares best fit of a quadratic function
(without the constant term) through this data (as trendline does in
Excel), it gives too much weight to the later years of the model and
grossly underestimates the rate of growth r. Based on the assumptions of the
Logistic growth model that increasing population should decrease the
growth rate, I chose r = 0.351
(matching the initial growth rate between 1790 and 1800), then
performed a least squares best fit to the data. This gave the
Logistic growth model:
The graph of this updating function with the data is seen below
(the line Pn+1 =
Pn is usually
shown for reference and can be used in a process called cobwebbing
that we will not discuss at this time):
The simulation of this discrete Logistic growth model compared to
the two models of the previous section and with the data is seen
It is clear that the discrete Logistic growth model follows the
data better than the Malthusian growth model (even the Malthusian
model with r = 0.349), but not as well as the non-autonomous
growth model of the previous section. The Logistic growth model shown
above predicts the population relatively well up until 1900. However,
it even has the wrong concavity after 1930, suggesting a premature
leveling off of the population of the U. S. Thus, a time varying
growth rate seems the best of these three models. This shows that
human populations have a more complicated dynamics than these models
can predict with both time-varying and crowding factors.
(Sociological and technical factors are especially difficult to
incorporate into mathematical models.) Your ecology courses should
help explain more details underlying the assumptions in these models,
so explain a little better when the models are applicable and why
they fail in other predictions.
Later we will study the qualitative behavior of discrete dynamical
equations in more detail, but for now let's begin with the first
elementary step of our analysis. Consider the general discrete
The first step in any analysis is finding equilibria, which is simply an algebraic
equation. An equilibrium point of a discrete dynamical system is a
point where there is no change in the variable from one iteration to
the next. Mathematically, this occurs whenever there is a solution to
Graphically, this is when f(P) crosses
the line y = P, which is one
reason why this line was shown above.
Consider the original discrete Logistic equation listed above with
r > 0. The equilibria are found by solving:
Thus, the equilibria for the Logistic growth model are either the
trivial solution 0 (no population) or
the carrying capacity M.
If we examine the Logistic growth model proposed above for the U.
S. census data, then we find the equilibria satisfying:
Thus, Pe =
0 or 2.849*108. Note that this later population will
be reached in less than 20 years, while actual projections have the
U. S. population rising to over 400 million by the middle of the next
century. It is unlikely that human population can continue its
current course, but what will be the actual scenario? Mathematical
modeling can provide reasonable estimates for short term growth and
allows one to predict several different possibilities for longer term
depending on the assumptions that are entered into the model.
Other Behavior of
the Logistic Growth Model
Below we present an applet that shows simulations of the
logistic growth model for various choices of r.
Robert May (1974) demonstrated that the discrete Logistic growth
model could display very complicated dynamics. Watch what happens in
the applet above as we choose different values of r. For example, try the values r = 0.5, 1.8, 2.3, and 2.65. (Note that the
solution of the discrete Logistic equation only gives solutions at
the integer values of n, so the connecting lines are only
drawn to help visualize the behavior of the system.) The first value
of r shows the curve smoothly ascending to carrying capacity
of 1000. The second value of r has the population ascend and
actually overshoot 1000, then oscillates about 1000, getting closer
to the carrying capacity as n increases. In both of these
cases, the equilibrium population of 1000 is said to be stable. When r = 2.3, the
solution oscillates about 1000, taking on the values of approximately
690 and 1180. This solution is said to have a period of 2. The last case shows the
population oscillating almost randomly about 1000. This last
situation could have either a very high period of oscillation or
actually be chaotic. Later in
this course we will study more of the mathematics behind this
sequence of changes, known as bifurcations. We will find that
derivatives play an important role in the study of the behavior of
 T. Carlson Über Geschwindigkeit und
Grösse der Hefevermehrung in Würze. Biochem. Z.
57: 313-334, 1913.
 Statistical Abstracts of the
United States (1993) 113 th ed., U.
S. Department of Commerce, Bureau of the Census, Washington, DC.
1. Take r = 0.15 and P0 = 75,994,575 (the
population in 1900). Use the discrete dynamical system with n
= 1,2,3,..8,9 to estimate the population through the 20th
century. Compare your results to the actual census data.
2. Consider an annual interest rate r = 6% and an initial
investment of P0 = 10,000. Use the discrete dynamical
system with interest compounded annually, semiannually, quarterly,
and monthly to determine the value of the investment after two years.
3. a. A population of herbivores satisfies the growth equation
yn+1 = 1.05 yn. If the
initial population is y0 = 2000, then determine the
populations y1, y2, and
y3. Also, give an expression for the population
b. A competing group of herbivores satisfies the growth equation
zn+1 = 1.07 zn . If
the initial population is z0 = 500, then determine
how long it takes for this population to double.
c. Find when the two populations are equal.
4. The model for breathing an inert gas is given by the equation
where c is the concentration of the inert gas in the lung,
q is the fraction of air exchanged with each breath, and g is the ambient concentration of the inert gas.
Suppose that a 2.5 liter lung begins with c0 =
7.0*10-5 M of Helium. Assume that each breath exchanges
1.5 liters of air with the environment which has a concentration
g = 1.0*10-5 M of Helium.
a. Find the concentration of Helium in the lung for each of the
first two breaths.
b. Sketch a graph showing the updating function and the identity
function labeling important points, like the
cn+1 intercept and the equilibrium
point. What is the slope of the updating function and the identity
c. Show with cobwebbing the first two breaths computed in Part a.
Is the equilibrium stable?
d. Suppose that after 5 breaths the lung had a concentration of
1.06*10-5 M of Helium. Determine what the concentration
was in the lung after 4 breaths.
5. Below is data on several populations of herbivores in related
areas. The data fit a population model of the form
where r is the growth rate and m
is the emigration rate.
a. Use the data below to determine the updating function for this
population, i.e., find r and m. Take the first line of the data and find the
population of the next two generations.
b. Graph the updating function and determine any equilibria.
Determine the stability of all equilibria. Sketch the cobweb scheme
for the population computed in Part a.
6. The population of China in 1980 was about 985 million, and a
census in 1990 showed that the population had grown to 1,137 million.
Assume that its population is growing according to the Malthusian
a. Use the data above to find the constant r and then write
the solution Pn. Predict the population in the year
b. How long does it take for China's population to double?
7. Consider the discrete logistic growth model given by
a. Suppose that the initial population P0 is 50. Find
the populations of the next three generations, P1,
P2, and P3.
b. Sketch a graph of the updating function and find the
equilibria. Show the cobwebbing diagram for this model.
8. Consider the discrete dynamical system
xn+1 = f(xn) with the
updating function f(x) = 10x/(5 + x).
a. Starting with x0 = 2, compute
xn for n = 1,2,3. Find all equilibria.
b. Sketch a graph of f(x) showing the y-intercept
and any asymptotes for x > 0. Show the appropriate
cobwebbing diagram for the simulation in Part a.
9. A modified version of the discrete logistic growth model is
a. Discuss the biological significance of each of the terms in the
b. Assume that the initial population is P0 = 500, then
determine the population for the next three generations. Repeat this
calculation starting with P0 = 100.
c. Find all equilibria. Sketch the cobwebbing diagram for each of
simulations in Part a.
Computer Lab Exercises
1. Consider the discrete logistic growth
This problem explores some of the
complications that can arise as the parameter r varies.
a. Let M = 5,000. The first step in studying
this model is to find all equilibria (where the population stays the
same). Determine all equilibria , Pe. Stability of an
equilibrium, Pe, requires |f
'(Pe )| < 1. For each of the two equilibria, find a condition on
r that guarantees stability.
b. Let r = 1.89 with P0 = 2,000.
Simulate the discrete dynamical system for 50 generations. Sketch a
graph the solution of dynamical system and write a brief description
of what happens. Is the larger equilibrium stable?
c. Repeat the process in Part b. with r = 2.1
and r = 2.62. Is the larger equilibrium stable for these
d. Use the applet above to try to find a
parameter value that gives you an oscillation with period 3. This
always implies that the dynamical system has gone through
3. Below are tables of census data (population
in millions) for Canada and India. We want to repeat for these
countries the modeling effort that we performed in class for the U.
a. Find the growth rate for each decade with
the data above by dividing the population from one decade by the
population of the previous decade and subtracting 1 from this ratio.
Associate each growth rate with the earlier of the two census dates.
Determine the average (mean) growth rate, r, from the data above.
Associate t with the earlier of the dates in the growth ratio, and
use EXCEL's trendline to find the best straight line k(t) = a + bt
through the growth data. Graph the constant function r, k(t), and the
data as a function of t over the period of the census
b. The Discrete Malthusian growth model is
where r is computed in Part a. and
P0 is the population in 1861 or 1901 (depending on the
country). Solve this model, not forgetting that n is in
c. Now consider the revised growth
where k(tn) is computed in Part
a. and P0 is again the initial population. Solve this
nonautonomous discrete dynamical system. (Note that
tn = 1861 + 10n or tn = 1901 +
d. Use EXCEL to graph the data and the
solutions to the models in Parts b. and c. for the period from 1861
(or 1901) to 2000. Briefly discuss how well these models predict the
population over this period. List some strengths and weaknesses of
each of the models and how you might obtain a better means of
predicting the population.
f. Find the percent error between the actual census data
and the models in 1921, and 1981. Use each of the models to predict the populations
in 2001 and 2031. Decide if the calculated value is most likely high or low.