Prosperity, War, Immigration, and "Cultural" Factors in Modeling United States Population Growth Since 1790

The growth of the United States population since 1790 can be modeled in terms of a nonlinear differential equation with time-dependent coefficients. The growth process represented by the model is subject to the effects of immigration, war, prosperity, and a so-called "cultural" factor. Specifically, the equation assumes that the rate of growth for a particular year is the result of positive factors' working against limiting factors. The positive factors comprise (1) a term proportional to the population size, p, for that particular year, and (2) a term equal to the immigrant population, p m ', for the same year. The limiting factors are all proportional to the square of the present population size, p 2 and comprise a war term, w, that aids the growth-limiting process; a prosperity term, ε g ', that inhibits the limiting process; and a culture term C, that supports the limiting process, too. The term, w, is taken to be proportional to the wartime size of the country's armed forces, assuming that the war effects on population growth of a land faced with either the immediate prospect or the reality of armed conflict include both the casualties and the disruption of family life and planning of those directly involved. The term ε g ' is intended to be a measure of the general prosperity level. It is taken to be proportional to the ups and downs of the GNP, above or below a smooth trend line, the latter being found by curve-fitting a growth curve to the actual GNP data. Finally, the culture term, C, is assumed to vary with time, but not to be tied explicitly to known environmental, social, or other causes. It is probably connected to age distribution within the population, average age of partners of first marriages, number of children per woman of childbearing age, advances in medicine, the set of prevailing values, the trend of family planning, and so on. The time history of C emerging from the computer-based optimization of the model shows a downward trend. The use of the word "cultural" leaves something to be desired, since such phenomena as war, prosperity, and immigration excluded from C are nevertheless products of culture. With the analytical form of the differential equation describing population growth set up in this way, all proportionality constants (parameters) involved were determined by means of a digital computer and on the basis of an optimization process which utilized the actual data of population size p * over the period 1790-1965. The residua