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2 min read•june 8, 2020

Jacob Jeffries

**Logistic models** describe phenomena using a logistic differential equation:

The *kL*** **term is often grouped together into a single constant.

The most popular (no pun intended) model using a logistic differential equation is modeling **population dynamics**. This is because it accounts for **exponential growth as well as a population capacity**, as seen in **Fig. 1.1**.

There are some properties we are going to derive that are necessary for the AP exam, in particular, the multiple-choice section.

The first one is the solution to the differential equation. It is separable but in a strange way. We will use the form of the differential equation as presented in Eq. 46:

This means we have to do a tough integral. As it is, we cannot integrate the left-hand side with normal techniques. We will assume that the function can be written in the following form:

This means the equation must be true for any value of *f*, given the value of *f* falls within the domain of the function.

The domain is of the function on the left-hand side in Eq. 47 is the following:

Which means we can pick any values of *f* except *0* and *1* to set up a system of equations to solve for *A* and *B*. Let’s choose *L/3* and *L/2*:

Working with the *L/3 *case:

Working with the *L/2 *case:

Substituting Eq. 54 back into Eq. 48 yields a much easier integration:

The second one is the **two relevant maxima**: the maximum value of *f* and the maximum value of *df/dx*.

The maximum value of *f* (*fmax*) is a simple principle: it is simply *L**. The best way to visualize this is by graphing the derived solution and seeing what happens as *x* gets infinitely large. In this case the graph approaches a **“carrying capacity,''** which is expressed as *L* in Eq. 45.

The second value is a bit more difficult to solve. Let’s use implicit differentiation to find the second derivative of *f* using the form of the equation in Eq. 45:

One can solve for *f* to get *fmax = L/2*.

Of course, this isn’t the full story, as this just means *L/2* is a critical point. You can verify this by finding the value of *f’’’* to check that it is positive at *L/2*.

*This is not a maximum in the strict definition of the word; the function does not have a maximum but rather grows monotonically.

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