Journal of Online Mathematics and its Applications
This module investigates a standard model of population growth in a constrained environment.
About this Module and its Authors
Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS.
Ver. 6 or higher |
Ver. 3.0 or higher |
Ver. 5.1 or higher |
Published December, 2001
Journal of Online Mathematics and its Applications
This module investigates a standard model of population growth in a constrained environment.
About this Module and its Authors
Click on the button corresponding to your preferred computer algebra system (CAS). This will download a file which you may open with your CAS.
Ver. 6 or higher |
Ver. 3.0 or higher |
Ver. 5.1 or higher |
Published December, 2001
Journal of Online Mathematics and its Applications
A biological population with plenty of food, space to grow, and no threat from predators, tends to grow at a rate that is proportional to the population -- that is, in each unit of time, a certain percentage of the individuals produce new individuals. If reproduction takes place more or less continuously, then this growth rate is represented by
where P is the population as a function of time t, and r is the proportionality constant. We know that all solutions of this natural-growth equation have the form
where P_{0} is the population at time t = 0. In short, unconstrained natural growth is exponential growth.
Of course, most populations are constrained by limitations on resources -- even in the short run -- and none is unconstrained forever. The following figure shows two possible courses for growth of a population, the green curve following an exponential (unconstrained) pattern, the blue curve constrained so that the population is always less than some number K. When the population is small relative to K, the two patterns are virtually identical -- that is, the constraint doesn't make much difference. But, for the second population, as P becomes a significant fraction of K, the curves begin to diverge, and as P gets close to K, the growth rate drops to 0.
We may account for the growth rate declining to 0 by including in the model a factor of 1 - P/K -- which is close to 1 (i.e., has no effect) when P is much smaller than K, and which is close to 0 when P is close to K. The resulting model,
is called the logistic growth model or the Verhulst model. The word "logistic" has no particular meaning in this context, except that it is commonly accepted. The second name honors P. F. Verhulst, a Belgian mathematician who studied this idea in the 19th century. Using data from the first five U.S. censuses, he made a prediction in 1840 of the U.S. population in 1940 -- and was off by less than 1%.
This table shows the data available to Verhulst:
Date (Years AD) |
Population (millions) |
1790 | 3.929 |
1800 | 5.308 |
1810 | 7.240 |
1820 | 9.638 |
1830 | 12.866 |
1840 | 17.069 |
The following figure shows a plot of these data (blue points) together with a possible logistic curve fit (red) -- that is, the graph of a solution of the logistic growth model.
The next figure shows the same logistic curve together with the actual U.S. census data through 1940. This emphasizes the remarkable predictive ability of the model during an extended period of time in which the modest assumptions of the model were at least approximately true.
On the other hand, when we add census data from the most recent half-century (next figure), we see that the model loses its predictive ability. What (if anything) do you see in the data that might reflect significant events in U.S. history, such as the Civil War, the Great Depression, two World Wars?
For more on limited and unlimited growth models, visit the University of British Columbia. [Ed. Note: This link is not longer operable.]
Journal of Online Mathematics and its Applications
as well as a graph of the slope function, f(P) = r P (1 - P/K). Click on the left-hand figure to generate solutions of the logistic equation for various starting populations P(0). [Note: The vertical coordinate of the point at which you click is considered to be P(0). The horizontal (time) coordinate is ignored.]
An equilibrium solution P = c is called stable if any solution P(t) that starts near P = c stays near it. The equilibrium P = c is called asymptotically stable if any solution P(t) that starts near P = c actually converges to it -- that is,
If an equilibrium is not stable, it is called unstable. This means there is at least one solution that starts near the equilibrium and runs away from it.
Journal of Online Mathematics and its Applications
appear to have an inflection point? Express your conjecture in terms of starting values P(0). [For your convenience, the interactive figure from Part 3 is repeated here. Recall that the vertical coordinate of the point at which you click is P(0) and the horizontal coordinate is ignored.]
Journal of Online Mathematics and its Applications
Then integrate both sides of the resulting equation. (This is easy for the "t" side -- you may want to use your helper application for the "P" side.)
Journal of Online Mathematics and its Applications
n the figure below, we repeat from Part 2 a plot of the actual U.S. census data through 1940, together with a fitted logistic curve. (Recall that the data after 1940 did not appear to be logistic.)
In this part we will determine directly from the differential equation
and, if so,
Our test case will be the U.S. Census data, first up to 1940, then up to 1990. In Part 6 we will study the same questions, but we will use the known form of the logistic solution from Part 4.
To determine whether a given set of data can be modeled by the logistic differential equation,
we have to estimate values of the derivative dP/dt from the data. We will do that by symmetric differences, as shown in the following figure. The slope dP/dt at a given census year t is approximated by the slope of the line joining the points 10 years earlier and 10 years later.
For example, the growth rate dP/dt in 1900 was approximately
Of course, for the period from 1790 through 1940, we can calculate these slope estimates only from 1800 through 1930, because we need a data point before and after each point at which we are estimating the slope.
We may rewrite the logistic equation in the form
In this form the equation says that the proportional growth rate (i.e., the ratio of dP/dt to P) is a linear function of P. Thus, we have a test of logistic behavior:
The same graphical test tells us how to estimate the parameters:
Journal of Online Mathematics and its Applications
In the preceding part, we determined the reasonableness of a logistic fit (up to 1940) and estimated the parameters r and K using only the differential equation, not the symbolic solution found in Part 5. Now we see what we can do by using the solution, which we recall has the form
where P_{0} is the population at whatever time we declare to be time 0. For example, if t = 0 in 1790, then P_{0} = 3.929. For purposes of this exercise, we will make that choice of starting point and measure all times from 1790.
In Part 5, as a step on the way to the symbolic solution, we saw that the solution would have to satisfy
that is, ln (P / (K - P)) should be a linear function of t with slope r. Thus, given a value of K, we can plot ln (P / (K - P)) against t and see if we get a straight line. (Note that there is no need to approximate rates of change for this type of test.) Since you already have an estimate of K from Part 6, it will not be a surprise if the plot looks pretty straight on the first try.
and t in step 2, make a new estimate of r.
Journal of Online Mathematics and its Applications
Journal of Online Mathematics and its Applications