Math 536 - Mathematical Modeling Fall 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. The method of taking the census and how it is analyzed has been a very hot issue pitting the Republicans against the Democrats this last year with the issue landing in the Supreme Court. The Republicans wanted a strict interpretation of the Constitution, knowing that a direct head count always undercounts minorities and the poor, who vote predominantly Democratic. The Democrats claimed that the Constitution framers wanted an accurate count of the populace, so that modern statistical methods should be employed. This would naturally give them an advantage in the voting. (The Supreme Court came down in the middle pleasing neither party and saying that the Constitution requires a head count, which will be used for allocating Congressional seats and districting, while the more accurate statistical count will be used for apportioning the money for Federal funding.) Despite the political controversy over the numbers, accurately predicting these demographic data are important for planning our communities in the future. At the base of all calculations for the future population predictions 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
3,929,214
31,433,321
122,775,046
5,308,483
39,818,449
131,669,275
7,239,881
50,155,783
151,325,798
9,638,453
62,947,714
179,323,175
12,866,020
75,994,575
203,302,031
17,069,453
91,972,266
226,545,805
23,191,876
105,710,620
248,709,873
The growth rate between each decade can be determined by dividing the census at one date by the census a decade earlier and subtracting one. The calculation below shows how to compute the growth rate for the decade of 1790-1800.
Thus, the growth rate for this decade is 35.1%. The javascript below performs this computation to give the growth rate for any decade in the history of the U. S.
Select the range of Census Data
1790 - 1800 1800 - 1810 1810 - 1820 1820 - 1830 1830 - 1840 1840 - 1850 1850 - 1860 1860 - 1870 1870 - 1880 1880 - 1890 1890 - 1900 1900 - 1910 1910 - 1920 1920 - 1930 1930 - 1940 1940 - 1950 1950 - 1960 1960 - 1970 1970 - 1980 1980 - 1990
Growth rate of the range selected
From the javascript above, we find that the growth rates for decades following 1790, 1800, and 1810 are 35.1%, 36.4%, and 33.1%, respectively, which averages 34.9% per decade. This growth rate remains 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.
The simplest mathematical model says that the population in the next decade is equal to the current population plus the current population times the average growth rate, r, of the population. The model begins with some starting population, say in 1790. The future populations are predicted at each decade (discrete time intervals) by starting with this initial given population, then finding the next population from the previous population by multiplying by (1 + r). This gives a sequence of predicted populations each based solely on the population from the preceding decade. For example, using the population of 3,929,214 in 1790, we would multiply by 1.349 to get an estimate of the population in 1800. Mathematically, we have
Notice that this prediction is less than 8,000 off the actual census value or a 0.15% error. We could repeat this process to predict each of the succeeding census populations up until 1860 (a period where the growth rate remains fairly constant). Thus, the population in 1840 would be projected to be 1.349 times the population predicted in 1830. Below we provide a table showing this type of constant growth model applied to the census up until 1870.
Notice that the error remains small until 1870 because of the fairly constant rate of growth. Most of the predicted populations are a little high, especially in 1870, suggesting that throughout the 19^{th} century the growth rate declined. The declining growth rate is a general trend that we observe for the U. S. and is something that will be accounted for in an improved model developed later in this section.
Between 1860 and 1870, the Civil War occurred, which is one cause of the dramatic decline in the rate of growth of the population in the U. S. Thus, the 1870 prediction is one that you would expect to be poor. In fact, the shift from the primarily agricultural society to the industrial revolution is more significant in causing the decline in the rate of growth. We will study more about crowding effects in the next lecture section.
Below is a graph of the population model and the data from the table above.
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 192,365,343 and 859,382,645, respectively. Thus, this model becomes increasingly bad if we assume the constant growth rate of 34.9%. A calculation of the growth rate in 1920 gives around 15%, which 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 populations for a limited time into the future, but certainly provides good estimates for some community planning that is required.
Discrete Malthusian Growth
Let the integer n represent the number of decades after 1790 and define P_{n}_{ }to be the population for the n^{th} decade after 1790 (with P_{0 }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 Thomas Malthus (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 above.
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 34.9%. The javascript above shows that in 1920 the growth rate is 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. The solution to the model that is given by the equation above is an exponential function with a base of 1+ r and power n representing the number of iterations after the initial population is given.
Applet for Malthusian Growth for the U. S. population
There is nothing special about starting the Malthusian growth model at the first census point in 1790. The Malthusian growth model assumes a constant growth rate, so is unlikely to do well predicting the population over long periods of time where growth rate varies. Because of the exponential nature of the Malthusian growth model, this model can rapidly diverge from the actual population. Below is an applet where you can vary the starting time, ending time, and growth rate to see how the Malthusian growth model works for the U. S. census data.
If you choose the interval from 1790 to 1860 with a growth rate of r = 0.349, then the Malthusian growth model matches quite well. However, if you try to fit the entire range of data from 1790 to 1990, then you will find no value of r that allows the Malthusian growth model to fit the data. (Using the average growth rate of 23.5% has the Malthusian growth model match the data at 1790 and 1990, but fails to do well for the intermediate data points.) 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.
Example:
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?
Solution:
The population of yeast satisfies the equation
The population after one hour is P_{1} = 1.1P_{0} = 110,000. After two hours, P_{2} = 1.1P_{1} = (1.1)^{2}P_{0 }= 121,000. Thus, after 4 hours, P_{4} = (1.1)^{4}P_{0 }= 146,410.
For the population to double, it must reach 2P_{0} = 200,000. Thus, we must solve
By taking the logarithms of both sides we have
Compound Interest
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 P_{0} satisfies:
where P_{n} is the amount of principle after n years.
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 10^{th} root of each side. So
Thus, the appropriate annual growth for the population of the U. S. near 1800 was about 3% per year.
Improved Malthusian Growth Model
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 20^{th} 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 rate.
The general discrete dynamical population model is given by
where f is a function depending only on the population P at time t_{n}. 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 equation.
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 growth model
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:
where t is the date of the census.
The resulting modified nonautonomous difference equation is given by
where t_{n}= 1790 + 10*n.
The graph below gives a comparative study of these two models. The Malthusian growth model performs poorly as noted above because the growth rate varies too much over the time interval being considered. The nonautonomous discrete Malthusian growth model matches the data quite well though runs a bit high after the severely depressed population growth during the Great Depression. Clearly this would be the better model to predict the 2000 census. (The Malthusian growth model predicts that the 2000 census will be 326,138,498, which is clearly too high, while the nonautonomous Malthusian growth model predicts a 2000 census of 278,244,477, which quite reasonable.) 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.