CSC: Modeling the USA Population

Chapter 5: 1950-1990--The Logistic Curve

Abstract : In this chapter, we study the logistic function itself and use it to provide more direct fits to the USA population data for the period 1950-1990. This will yield new estimates of the long-range limiting size of the population.

Prerequisite Knowledge : derivatives; integrals; modeling.

In the last chapter we found that a logistic curve provided a good fit of USA population data for the period 1950-1990. We obtained the fit by looking at relative growth rates and plotting versus . In this chapter we will study the logistic function itself with the aim of obtaining a more direct, and better, fit of the population data. But don't lose sight of the goal; it has never changed. We want to be able to predict with confidence the USA population in future years, and the limiting size of the population if any.

Recall from previous chapters that the logistic model is defined to be the differential equation

, K >0 and 0< P < L .

We have shown that its solution is

, where .

We now are going to apply our knowledge of calculus to the analysis of this function and use the information to fit a logistic curve to the population data.

Analysis of the Logistic Function P(t)

Fitting P(t) to the 1950-1990 Data: A Plan

Fitting P(t) to the 1950-1990 Data: The Results

A Direct Logistic Fit of the 1950-1990 Data