Applications of Differential Equations

Population Dynamics

(continued from last page...)

You should now have:

malthus/.t ->2050

verhulst/.t->2050

Remembering that these results are in scientific notation, how did the models do? The U.S. Census Bureau predicts that the actual 2050 population will be 398 million. So one model overestimates the population at 687 million, while the other underestimates it at 341 million. The Malthusian model is wildly off, overestimating by 72%.The Verhulst model is closer, missing the mark on the low side by about 17%.

Could we have done better? The accuracy of both models is subject to our estimation of their parameters, k in the case of Malthus, and a and b in the case of Verhulst. We found these by forcing our solution curve through certain points in our data set, but did we choose these points wisely? In fact was it actually a good idea to force our solution curve through specific points at all? After all, we're really trying to find a curve that models the entire data set, not just a few points in it. It seems like a better idea would have been to estimate our parameters using the entire data set! But how do we do that?

Mathematica has the ability to fit a curve defined by a function with arbitrary parameters through a set of data. Thanks to our two models, we actually do know the form of the functions we're after, it's just the values of their parameters that we're having a hard time nailing down. If Mathematica can find a best-fitting choice for these parameters which takes all the points into consideration then our models should become better than we just found using only a few points. The command we need is FindFit. Ask Mathematica for the syntax of this command, then come back here.

Let's go look at what you should have gotten...

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ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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