Population Growth Models
Part 3.3: Solving the Coalition
Model
We saw in Part 3.1 and Part 3.2 that world population does not seem to be growing exponentially. However, this growth might be modeled by a coalition differential equation of the form
- Population growth that may be modeled by a coalition differential equation is often said to be growing "faster than exponential growth" and the growth itself is called "superexponential." What justification can you give for this terminology?
We now use the separation-of-variables technique to obtain a symbolic representation of the solutions of this differential equation. (See Part 5 of the Introduction to Differential Equations module for a discussion of separable differential equations.) Then we will consider the implications of faster-than-exponential growth.
- Separate the variables in the differential equation
and write it in the form
Show that P must satisfy
where C is an arbitrary constant.
- Now show that P must be of the form
How is T related to your constant of integration C?
- This model only makes sense if t is less than T. Why?
- The von Foerster paper calls refers to the time T as Doomsday. Why is this appropriate? What happens as t approaches T?
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.
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.
