Growth Models
Part 3.2:
The Coalition Model
As we noted at the end of Part 3.1, the von Foerster paper argues that the differential equation modeling growth of world population P
as a function of time t might have the form
where h and k
are positive constants. Before attempting to solve this differential equation,
we explore whether it can reasonably represent the historical data.
The model asserts that the
derivative of P should be proportional to
a power of P, that is, the rate of change should be a power
function of P. If that is the case, then the logarithm of the derivative should be a linear function of the logarithm of the population, i.e.,
where 1 + h is the power.
We can test that assertion by plotting ln(dP/dt) versus ln(P) and seeing whether it is approximately a straight line. As we did in Part 2, we will estimate the derivatives by calculating symmetric difference quotients.
- Construct the symmetric
difference quotients (SDQ) approximating dP/dt from the historical
data.
- Construct a plot
of ln(SDQ) versus the logarithm of the population. Decide whether you think it is possible that
dP/dt is a power function of P. Keep in mind that we have
only very crude approximations to values of dP/dt, and many of them
are constructed on intervals that are not symmetric about the corresponding
year.
- Whatever you think about
the linearity of the plot in the preceding step, use your helper application's least
squares procedure to find the best fitting line.
- How can you determine values of h and k from the slope and intercept
of the best-fitting line? Calculate values for h and k.
- Now construct a slope field
for the model differential equation, and add a sample solution passing through one of the data points. (See Part 3 of the Introduction to Differential Equations module for an explanation of slope fields.) Experiment with the selected data point to see if it makes
any difference in the shape of the solution.
- Display the data points and one solution curve together with the slope field. Now what do you think about the Coalition Model as a description of the historical data?
