CSC: Modeling the USA Population

Chapter 4: The Verhulst Model

Abstract : In this chapter, we study the relative growth rate of the USA population, not as a function of time as we did in Chapter 2, but as a function of the size of the population. We investigate the period 1860-1990 when the unrestricted growth of the earlier years came to an end. We again will see that the periods 1860-1940 and 1950-1990 require different modeling functions, although both will be logistic curves. Moreover, we will be able to estimate the long-range limiting size of the population.

Prerequisite Knowledge : derivatives; integrals.

We have seen that the exponential function provides a good model of the USA population for the years 1790-1850. This was a period of rapid expansion when the country was growing. However, unrestricted growth cannot continue forever. In fact, in population studies, the relative growth rate often becomes a function of the size of the population. To see if this kind of functional relationship is the case for the USA population, we will plot versus for the years 1860-1990. As we have done previously, we will use the average of the right- and left-sided estimates for .

> dataCensus:=readdata(USAPopCensus17901990,[integer,float]):
ndc:=nops(dataCensus):

dataAll:=[seq([dataCensus[i][1],round(dataCensus[i][2]),100*(dataCensus[i+1][2]-dataCensus[i-1][2])/(20*dataCensus[i][2])],i=2..ndc-1)]:

dataPvsRel:=[seq([dataAll[i][2],dataAll[i][3]],i=1..nops(dataAll))]:

headers:=["year","population","relative growth rate (%)"]:
printtable(dataAll,"Relative Growth Rate (two-sided) Estimates",headers);

plot(dataPvsRel, style=point,title="Relative Growth Rate (%)",labels=["P (millions)","rel gr rate"]);

                       Relative Growth Rate (two-sided) Estimates

                    year             population     relative growth rate (%)

                    --------------------------------------------------------

                    1800                 5297               3.110251

                    1810                 7224               2.990725

                    1820                 9618               2.951237

                    1830                12901               2.907527

                    1840                17120               3.025701

                    1850                23261               3.093805

                    1860                31513               2.640815

                    1870                39905               2.349204

                    1880                50262               2.303032

                    1890                63056               2.048338

                    1900                76094               1.928601

                    1910                92407               1.643111

                    1920               106461               1.440434

                    1930               123077               1.042477

                    1940               132122               1.082598

                    1950               151684               1.600334

                    1960               180671               1.472151

                    1970               204879               1.148263

                    1980               227722                .989035

The situation here looks familiar from our previous work. We see that the period 1860-1940 is clearly different from the period 1950-1980. But in each case there appears to be a linear relationship between the relative growth rate and the size of the population. As we did in Chapter 2, for each of these time-periods we will fit a line to the data, and then solve the resulting differential equation to obtain a formula for .

Fitting a Line to the 1860-1940 Data

Fitting a Line to the 1950-1980 Data