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The Growth of Populations

Mathematics can be used to make predictions about physical systems. But economists and social scientists can likewise use mathematics to make predictions about economic and social systems. The contrast lies only in the details of the mathematical descriptions, not in the language of mathematics itself. For example, the study of linear motion involves equations which are, by definition, linear. The change in position is given simply by multiplying velocity by time in the case of no acceleration. Where constant, non-zero acceleration exists, the velocity changes linearly with time and it is relatively simply to incorporate that effect into determining the change in displacement with time.

There are other observations of nature which appear just as simple, but are not linear. For example, we can ask how a population of organisms changes with time. The organisms could be yeast cells in a petri dish, chickens in a barnyard, or people on a planet. In the most simplistic view of these organisms, we find that their population increases with time. The reason is that organisms reproduce. Since only organisms of the same type or species can reproduce, the most naive way of quantitatively describing the reproduction is to say that the increase in the future depends on the number of organisms around today. For example, say that we are studying yeast cells in a dish. If we make the following assumptions:

1. The petri dish contains only yeast cells.
2. All yeast cells are the same.
3. Each cell lives forever.
4. The birth rate or probability that any cell will produce an offspring stays constant.
5. Cells do not interfere with one another.
6. There is an infinite supply of vital elements (e.g. food, space, etc.).

where P depends on the time at which you count the population and b is the constant birth rate for each individual cell.

Of course, this equation doesn't seem very realistic since yeast cells don't live forever. We might be able to supply them with an always adequate food supply and put them into bigger and bigger dishes as they expand, but eventually some have to die.

How do we model death quantitatively?

Death decreases the population and obviously we can't have more cells dying than there are cells, so the amount of decrease must also depend on the current population. The appropriate equation for describing just death would be:

Does that seem plausible? Let's try it out!

We begin by considering our own United States. We have to choose some convenient time in which to start our observation. The first people to apply it did so around the turn of the 19th century. The first U.S. Census was 1790, so let's start from there. At that time the population in the U.S. was 3.929 million. That's our starting population; we will call it , so in units of millions of people. To start from 1790 instead of zero, let's define our time as t - 1790 instead of just t. That will make things easier to handle for calculation.

Now we can turn to Maple. Note that in the following, the response from Maple is centered and in italic type. Also note that not all of Maple's responses are shown. You can download the Maple file directly to your computer (if you are on a Unix machine or PC) by first selecting the option Load to Local Disk and clicking here. Once you've stored the file, simply call up Maple and load your stored file. Or you can ask Netscape to run Maple for every file ending in .ms and it will open Maple for you.

#First start off by describing the change in population with time by
#writing in Maple the equation we just derived, and asking Maple to
#solve it.  We call the solution psol just to give it a name.  The
#condition in 1790 is a constraint to our solution, so we let Maple
#know about it when we ask for a solution.
dsolve({diff(P(t),t)=a*P(t), P(0)=P0}, P(t));

#The solution gives us the population for any time t we care to use.  If
#we want t to be a year like, e.g. 1860, then our solution is
psol := exp(a*(t - 1790))*P0;

#with P0 being equal to 3.929 in units of millions of people.  Now we are 
#almost ready to compare to real data.  The real data we'll use are the 
#census figures from 1790 to 1970.  Here they are:
uspop := [1790, 3.939, 1800, 5.308, 1810, 7.24, 1820, 9.638, 1830, 12.866,
1840, 17.069, 1850, 23.192, 1860, 31.443, 1870, 38.558, 1880, 50.156, 
1890, 62.948, 1900, 75.995, 1910, 91.972, 1920, 105.711, 1930, 122.775,
1940, 131.669, 1950, 150.697, 1960, 179.323, 1970, 203.185];
#The final thing needed for a comparison is to determine the value of the
#constant a (remember that this is the birth rate - death rate).  To get it,
#we have to look at the data for a year later than 1790.  Since we are 
#assuming that the birth and death rates do not change, we can pick any 
#year we want.  Let's choose 1830.  In that year, the population was 
#12.866 million people.  That's the value we want our Maple equation to
#give.  Let's let Maple tell us what value of a satisfies.
solve(exp(a*(1830 - 1790))*3.939 = 12.866, {a});
#The solution is a = 0.029655.  Let's use that value and plot the solution 
#given to us by Maple on top of the points from the census data in 
#succeeding decades.  A few data points is enough to tell whether all
#of this makes any sense at all.  Let's look from 1790 to 1870.
plot({psol, uspop}, t=1790..1870, P=0..80, style=POINT);

• Do we expect this success to continue for later years? Give your vote yes or no!
• If you vote no, what do you expect to happen?
• If you vote yes, what do you expect to see as the number of years increases?

To examine what happens, let's go back to Maple and plot more points.

plot({psol, uspop}, t=1790..1970, style=POINT);

There are other factors such as immigration which affect the population. In addition, the birth and death rates are not constant with time. Wars, disease, cultural habits (e.g. smoking, pollution of the environment, etc.) affect the death rate while improved medicine, food, working conditions, etc. increase the birth rate.

Even for our simple yeast cells, we would have to say that there is some limit to the size of the petri dish we could use to hold them. Eventually, crowding and competition for resources would limit the exponential growth. In every environment, there is a maximum number of any given species that can be supported. This one fact must clearly be added into our equation to make it more accurate. With the inclusion of only this one additional constraint, let's answer the following questions.

• How would we account for limits to the size of the population in predicting changes to the population? In mathematical terms would we expect the change in population as the size approaches the maximum to be positive, negative, or zero?
• Sketch the shape of the curve that allows for limiting the number of individuals in the population.

where a and c are positive constants. This equation is called the logistic equation or Verhulst-Pearl equation. It was first developed by Pierre-Francois Verhault and later used by Raymond Pearl and Lowell Reed to model the population of the United States.

The logistic equation describes a situation in which the change is positive for a while if a > cP but eventually the population gets large enough so that cP > a and the change becomes negative. This is the simplest way to flip the sign of the slope.

The justification of this type of equation for our case of a limiting environment is relatively straightforward. If the constant c were zero, then we get our original equation in which the rate of growth is aP. This corresponds to an environment with unlimited space and resources. To correct for a non-infinite environment, we note that the maximum number of organisms in the environment is M. At any given time, the number of organisms that can still be born is M - P(t). The fraction of maximum attainable population that is still possible is therefore,

               
In this way, the growth is the maximum value for small P (i.e. the
fraction is 1) and goes to zero as the fraction approaches 0.  The
equation describing the finite environment model would then be
              
If we identify c = a/M, then we see that this is our logistic equation.
Now let's see how it fares against the census data.  Let's return to Maple.

#We consider here the solution to the Verhulst-Pearl or logistic equation
#
# First redefine the constants used in the equation as variable
a := 'a';
P0 := 'P0';
dsolve({diff(P(t),t)=P*(a - c*P), P(0) = P0}, P(t));
  


#There are two unknowns in this equation, the values of a and of c.  If
#we wish to match our logistic expression to data, we have to find their
#values.  Since there are two unknowns, we will need at least one more
#data point to find their values.  Let's choose the data point at
#1800.  Since the population is known for 1800 (we can call it P1), let's 
#solve the logistic expression for the unknown constant c
solve(a*P0/(c*P0 + exp(-a*t)*(a - c*P0)) = P1, {c});
  


#Now our constraint on the value of c is that it must be positive.
#We can plug in our expression for time as t - 1790, and evaluate which
#values of a make it possible for c to be greater than zero.
cconst := (a*P0 - P1*exp(-a*10)*a)/(-P1*P0 + P1*exp(-a*10)*P0);
P0 := 3.929;
P1 := 5.308;
plot(cconst, a=.025..0.035);



#
#The plot shows that we need a about 0.030, which is close to our old value
#for a with the exponential equation.  In this case, we find that
evalf(subs(a=.030, cconst));
   

#We can actually fit several points of the data to find the optimal values
#of a = .031 and c = .00016.  Now plot the result.
a := 0.031;
c := .00016;
P0 := 3.929;
pmax := a*P0/(c*P0 + exp(-a*(t-1790))*(a - c*P0));
plot({pmax, uspop}, t=1790..2000, style=POINT);

The result is pretty spectacular. We can fit the data over a much larger span of time. Of course, there is still a large discursion of the census data toward large values compared to theory at about 1940. Pearl and Reed did the theoretical curve shown in the mid-1920's, found a good fit, and predicted that the maximum size of the U.S. population would be less than about 200 million. To find out the current estimate for the U.S. population today, go to the Census Bureau's population clock.

You can ponder the logistic equation and its implications further by doing exercises 2-19, 2-20, and 2-21.