Changing Human Population Growth Rates

(Adapted from Mooney and Swift, A Course In Mathematical Modeling.)

Background

This project addreses the question of whether the human population growth rate can be reduced without limiting the number of children women have (China's method) or shortening life spans (unacceptable to most of the civilized world).

In the late 1800's it was noticed that England's human growth rate was lower than expected when looking at comparable countries. One explanation was that the Church of England required young couples to wait until they could afford a home before they married. The idea was that children would fare better if their parents had the resources to take care of them. The effect was that couples began having children at a later age.

Specifically this project tests whether waiting to have children has any noticeable effect on the population growth rate. It should also convey that even crude models can be used to answer important questions.

Problems

1. Break the female population into three age classes: [0, 20), [20, 40), and [40, 60). We will assume that the average female has 2.3 children, half of whom are female. First assume that the [0, 20) women give birth to 0.5 girls, the [20, 40) women give birth to 0.5 girls, and the [40, 60) women give birth to 0.15 girls. (Note that 0.5+0.5+0.15 = 2.3/2). To avoid confusing birth and survival issues, assume that 100% of the [0, 20) women survive to the [20, 40) age class, and 100% of the [20, 40) women survive to the [40, 60) age class, and all women over 60 leave the studied population. Starting with 100 females in each age class (300 total), track the female population through 400 years (20 time steps). Make a table of the total female population at each time step. Be sure to give the details of your model and how you constructed it.

2. Repeat the same analysis, but with births occurring later. Assume now that the birth rates are 0, 0.75, and 0.4 for the three age classes respectively. Notice that each women still gives birth to 2.3/2 girls. Keep the survival rates the same.

3. Plot the results of parts 1 and 2 on the same coordinate axes. What conclusions can you draw?

4. Using log plots, find the exponential functions which best fit the two plots. From these, compute the percentage growth rates of the two models. Is the hypothesis that delaying births reduces the growth rate supported by this model?

5. Extra Credit: Certain aspects of this model are crude. Does the basic message change if it is made more realistic? Try more realistic survival rates or initial populations. Try more age classes. Try keeping women in the system past the age of 60, but no longer having children. Can you devise a reasonable model where the opposite conclusion can be drawn?