Euler’s method can be used to model population growth. For example, the exponential model (Malthus) is the initial value problem

The applet generates solutions for these differential equa-tions using Euler’s Method, and allows you to determine the growth constant k experimentally. You will also be finding exact solutions using the method of separation of variables, and you can use the applet to plot these solutions and compare them with those found numerically.

Experimenting with the Malthus model: The applet begins with an initial value problem that models the growth of the U.S. population from 1790 to 1990, assuming a population of 3.9 million in 1790. Growth is unconstrained, and the growth constant k has an initial value of 0.02. Note that with this valueof k the model predicts a population after 200 years that falls short of the 250 million reported by the census in 1990. Experiment with the value of k to try to achieve the target value of 250 million in 1990.

Using the Malthus model to predict a future population: Once a value of the growth constant k has been determined that predicts a population in 1990 of 250 million, use the model to predict the population in the year 2100. (Replace the final date 200 by a suitable new value.) The new final value of the population will be be displayed at the top of the graph, but the graph will need to be zoomed out to see the entire curve. (Zooming is described below.) Do you believe that the predicted population for 2100 is realistic? Why, or why not?

Note: Euler’s method generates the approximate solution by taking “small” steps of size h and extending the curve in the direction of the slope field. Smaller values of h yield better Euler approximations (at the expense of more computational time). Experiment with different values of h to see the effect on the predicted final value of the population. You can also click on the button CHECK Euler Solution to gain access to two additional buttons—one that draws the slope field, and one that draws a solution curve using a more accurate numerical method than Euler’s method. (Entering values of h that are excessively small could cause the applet to run interminably. Such values are rejected by the applet and are replaced by more suitable values.)

About Zooming: You may want to zoom in on a part of the graph of special interest, or you may need to zoom out to see more of the graph (as is the case when you are extrapolating into the future). Zooming in is accomplished by dragging a rectangle about the part of the graph that is of special interest. Zooming out is accomplished by clicking on the positive x-axis or the positive y-axis. Clicking on the x-axis expands the coordinate system in the x-direction only. Clicking on the y-axis expands it in the y-direction. A single mouse click anywhere else on the graph will restore the default coordinate system.

Experimenting with the Verhulst model: Enter the differential equation for the Verhulst model into the function input field, using the value M = 750 million as the maximum environmental carrying population. (Enter this formula with the syntax: k*y*(750-y) / 750.) Experiment, as before, to find the value of the constant k that predicts a target population of 250 million in the year 1990. What population does the Verhulst model predict for the year 2100? Is it more realistic than the Malthus model’s prediction?

Entering your own solution obtained by the method of separation of variables: Click on the button ENTER Your Solution. An input field becomes available that allows you to enter your own solution. For example you might have found the solution 3.9*e^(k*t) as the exact solution for the Malthus model. Enter this function and remember to hit the ENTER Key. Click the Draw button to see its graph and its final value. How does it compare with the Euler solution? How does it compare with the more exact solution shown under the CHECK Euler Solution button?

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