(done June 18, 1999) This work was done at the request of Prof. Glen Paige for his new book, "Nonkilling Political Science" A simple model of population growth is the exponential model where the population at time t, P(t), is given by Ce^{rt} for some appropriate constants C and r. This model is broadly accurate, especially when an organism invades a new ecological niche and the environment seems to have infinite capacity for growth. This model accurately describes the initial growth of a bacterial culture in a fresh dish of growth medium, or the human population as it discovered agriculture. More refined models are needed as environmental and predatory constraints impinge, but this model works well with most human populations because many human populations rapidly shift from one environmental disequilibrium to another (and thus present a moving target that the environmental constraints fail to effectively constrain). It is of course never that simple: Black Death in Europe caused populations to drop dramatically for a few centuries. Regardless of how one models P(t), the population at time t, P(t) may be used to estimate the total number of humans that have lived in a given time period. From a time A to a later time B, the integral of P(t) on [A,B] gives the total number of personyears for that time interval. If one has an estimate of the average lifespan, say 25 years as in Keyfitz's work, the number of people who lived from time A to time B is approximately (1/25) of the integral of P(t) on the [A,B]. Keyfitz assumed that exponential growth occured in various historical time intervals [A, B], but with possibly different constants C and r for each time interval. He chose C and r to match given values of P(t) at A and B. This requires solving two equations in 2 unknowns:
Keyfitz then integrates Ce^{rt} on the interval [A,B]. An antiderivative is (C/r)e^{rt} and the definite integral is the antiderivative at B minus the antiderivative at A. The total personyears from A to B is (C/r)[e^{rB}e^{rA}]. By algebra, this is equal to (1/r)[P(B)P(A)]. By substituting the value of r described just above, the total personyears from A to B is [P(B)P(A)](BA)/{Ln[P(B)]Ln[P(A)]}. This formula is now applied to a system of time intervals from 1,000,000
BC to the present. The sum of the total personyears for all these
intervals gives the total personyears for all human life. By dividing
by the average lifespan, one obtains an estimate of how many people have
ever lived. The data at ends of intervals are taken from a recent
textbook on population, except for the first data point of 2 people at
1,000,000. That data point was proposed by Keyfitz.
