One type of mathematical model of population growth for a species is a differential equation whose solutions exhibit some of the same behavior as the actual population. If we have a good model of this type, we might even be able to make predictions about the future real-world population. One such model is the logistic model we have studied in class. If our population is living in a fixed habitat with limited, but self-renewing, resources (for instance, plants as food for the individuals of an animal species to eat), then one reasonable assumption is that there is a limited sustainable maximum population M that the habitat can support. In biology, M would be called the carrying capacity. The logistic model was proposed by the Belgian scientist P.F. Verhulst in the 1830's, and has been intensively studied by biologists and demographers since. Verhulst's idea was to consider the following relationship:
(*) The rate of change of the natural logarithm of the population is proportional to the difference between M (the maximum sustainable population) and the population.
Verhulst's model can be rephrased as the differential equation
As we saw previously, the general solution of this equation
can be written in the form:
In a "real-world" population study in biology or demographics, the idea would be to
We are going to do this, in two different ways using the following table of estimated data for the world human population starting from the beginning of the 20th century:
| (year) | 1900 | 1910 | 1920 | 1930 | 1940 | 1950 | 1960 | 1970 | 1980 | 1990 | 2000 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| P (pop. in millions) | 1550 | 1750 | 1860 | 2070 | 2300 | 2555 | 3039 | 3707 | 4456 | 5283 | 6080 |
If these points were exactly on some logistic equation solution graph, then for each of these t values, the points (P, P'/P) would lie on some line z = k(M - P) in the (P,z)-plane. To begin,
A) Using the table of values above, estimate the rate of
change P' (in million people per year) at each of the given times,
using a forward difference
B) Make a list of your approximate data points (P,P'/P) in the following format. Your first data point (corresponding to the year 1900) should have P' = 20 as above and P = 1550 from the table. Then 20/1550 = .0129 (approx.) is your first P'/P value, and the first point in the list is as here:
(Since there's no obvious good way to estimate P'(2000) from the table, only include data points for 1900 through 1990 -- ten points in all.) You can plot your list of (P,P'/P) points to see what they look like, using a plot command like this
Try this, and note the output. You should see what looks roughly like two different groups of points. The one to the left corresponds to the years 1900, 1910, 1920, 1930, 1940. The group to the right corresponds to the years 1950, 1960, 1970, 1980, 1990. From (*) above again, if our population values were all on the graph of a logistic function, then the points we have plotted here would all lie on a single straight line z = k(M - P) in the (P,z)-plane. That isn't true here, so we'll need to do some data analysis. In fact, what we will do will be to look at the trend from the first and second halves of the 20th century separately(!) You may have seen a method called linear regression or least-squares curve fitting for doing this in another course. We will do this now, in two different ways, using Maple's least-squares routine.
C) First we will find a logistic model that fits the data well for the first half of the 20th century. For technical reasons we will omit the 1910 data point. This is not entirely ``kosher'' but will be necessary to get reasonable results by our method.
To put the data into the form Maple wants, we will need
to separate out the P- and z=P'/P-coordinates of the data
from dl, as follows:
If all your [, ]'s and (, )'s match, this should generate the equation of a line in the form
and generate the equation of a second line in the form
From the slope and intercept values of this new line, determine the best-fit k and M for the logistic model, for this data. Determine a formula for the solution of the logistic equation that should model the population in 1950 - 2000. Plot your solution curve as a function of time for the period 1900 - 2050. (Take t = 0 as the year 1900.)
E) Discuss your results. What world population do your models predict for 2010,2020,2030,2040,2050? In what year is the maximum population growth rates experienced? What are the carrying capacities? Also, think about the following questions:
Solutions will be due on Monday May 1. Because it is more extensive than some of the other labs we have done this term, this assignment will count as two of the parts of the lab/discussion grade for the course.