# World Population Growth - Solving the Coalition Model

Author(s):
David A. Smith and Lawrence C. Moore

We now take up the solutions of the coalition model and consider the implications of faster-than-exponential growth. We may separate the variables in the differential equation

dP/dt = k P1+r

to write it in the form

P-(1+r) dP = k dt.

Then we may integrate both sides to get an equation relating P and t.

1. Carry out the integrations, and then solve for P as a function of t. You may do the calculations on paper or in your worksheet, as you choose. (Don't forget the constant of integration. The solution makes no sense without it.)

2. Show that your solution can be written in the form

P = 1/[r k (T - t)]1/r

for some constant T. Specifically, how is T related to your constant of integration C?

This calculation shows that there is a finite time T at which the population P becomes infinite -- or would if the growth pattern continues to follow the coalition model. The von Foerster paper calls this time Doomsday.

It's clear that Doomsday hasn't happened yet. To assess the significance of the population problem, it's important to know whether the historical data predict a Doomsday in the distant future or in the near future. We take up that question in the next Part.

David A. Smith and Lawrence C. Moore, "World Population Growth - Solving the Coalition Model," Loci (December 2004)

## JOMA

Journal of Online Mathematics and its Applications