## Applications of Differential Equations

### Population Dynamics

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Your command should have produced this result:

FindFit[uscensus, usmalthusfit, {p0, k}, t]
{p0->0., k->1.}

This result is just silly! It is the trivial solution, since if p0 is zero, the function will stay zero forever! What we are seeing here is that the FindFit function isn't something we can rely on blindly. We need to think about what it produces, and maybe give it a little help sometimes. This is one of those times.

The FindFit function allows for the parameter values to be given initial estimates, so perhaps if we gave it a ballpark idea of where to look it will do a better job of finding a sensible answer. Where should be suggest it look? Well the value of k is already known to us from our previous work to be around 0.02, so we'll suggest it uses that value for k, but the value of p0 is a little more tricky. Remember that this is no longer our initial condition in 1790, but a new one for the year 0! There is nothing in our data to give us a clue for what to use for this long ago, so we can't use our previous approach. But remember that we have already solved this problem once, even though we were less than impressed with the accuracy of its predictions. To jog your memory, here is the model, variable name malthus, that we came up with earlier:

Now we're not very impressed with this model, but maybe we could at least use it to get a ballpark idea of what the value of the function should be when t=0. Tell Mathematica to evaluate the following command:

malthus/.t->0

Let's go see what you should have gotten...

 If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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e-mail:
cbarker@deltacollege.edu