## Applications of Differential Equations

### Population Dynamics

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### The Verhulst Model

Sometimes called the logistic model, recall that this model takes into account the internal competition within the population for resources. The initial value problem is:

dp/dt = a p - b p2
p(to) = po

Here po represents the initial population, and a and b are parameters accounting for the birth/death rate, and the "competition rate."

#### Solving the Initial Value Problem

For the U.S. population you know from your data that when to is 1790 the value of po is 3929700, so you can use DSolve to solve the initial value problem. Your answer will contain a and b, as well as t. Assign the solution to the variable verhulst.

You can look back at other labs if you're stuck, but you should be capable of setting this one up for yourself by now.

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

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ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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