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.