## Applications of Differential Equations

### Population Dynamics

(continued from last page...)

Your command should have yielded this:

?FindFit

Using the first form given above, the first argument we need to give it is our list of census data, the second is the form of the function we're using for our model, (in our case Malthus or Verhulst,), the third argument is our parameters, and the fourth is our variable, (in our case t.)

Let's give it a shot. We'll create and name our Malthusian modeling function with the following command, (go ahead and enter it now, then come back.)

usmalthusfit = p0*E^(k t)

This is the classic form of the Malthusian model that you'll find in most books. You'll notice that it is slightly different, and somewhat simpler than the form we got when we had Mathematica solve the differential equation for us. That earlier solution employed an initial condition, but we've left this as a second parameter, p0, for Mathematica to find from the data. This "new" form assumes that the initial condition happens at t=0, whereas we earlier assumed it happened in 1790, so we don't actually know the new initial condition anyway. Issue the following command, (but expect a ridiculous result!)

FindFit[uscensus, usmalthusfit, {p0, k}, t]

Let's move on to discuss what just happened...

 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

©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA
e-mail:
cbarker@deltacollege.edu