## Applications of Differential Equations

### Population Dynamics

(continued from last page...)

You probably remember that when we solved the Verhulst model's differential equation we included an initial condition. The resulting solution contained two parameters and was an incredible mess! We're not going to use that messy version of the solution this time, and we're not even going to need the initial condition, as we'll use something else instead, (more on that later.) Mathematica will find both parameters for us. The Verhulst model's solution can actually be written in a much tidier form than we used earlier. We'll define it using the command below. Enter it into Mathematica, evaluate it, then return here.

usverhulstfit = l/(1 + E^(-k (t - 1990)))

This is basically the version of the model you will find in Wikipedia if you were to look it up there. (The numerator here is a lowercase L, though it may look like a one in your browser.) Admittedly this is still not pretty, but it is definitely better. You may be wondering where the 1990 came from in the equation. This should be the t-value of the curve's inflection point, and it is what we're using instead of an initial condition. With our census data this location is not glaringly obvious, but it appears to be happening around 1990 if you squint your eyes just right and ignore that there seem to be about three inflection points!

Next we'll see if FindFit can find the unknown parameters without us giving it any help (like we had to with Malthus.) Enter the command:

FindFit[uscensus, usverhulstfit, {l, k}, t]

Let's move on and 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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