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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 P ^{1+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**.

- 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.) - 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)

Journal of Online Mathematics and its Applications