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Unit 7

7.9 Logistic Models with Differential Equations

2 min readโ€ขjune 8, 2020

Jacob Jeffries


https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(565).png?alt=media&token=b152ffb0-a14c-4d0e-94a1-222f4e93efa5

Logistic models describe phenomena using a logistic differential equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1004).png?alt=media&token=a06cf19f-87d1-40ba-bf0f-76bed2f2c83f

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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1001).png?alt=media&token=ca7db6cb-9da7-4ad0-895d-0cc84fc45f74

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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1000).png?alt=media&token=a46d7cf1-c194-4c13-8548-e8537c59ea2c

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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(999).png?alt=media&token=d382533f-6a42-4231-888c-b9a16902dd74

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. Letโ€™s choose L/3 and L/2:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(996).png?alt=media&token=71db7a92-87d6-425e-866a-7ff7ea8c6f1e

Working with the L/3 case:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(995).png?alt=media&token=f757c677-0f9d-46a6-8347-3ed8e37ede44

Working with the L/2 case:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(993).png?alt=media&token=5503186a-2aac-4ad8-923d-8a22b7844334

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(991).png?alt=media&token=b9f78390-07e6-4e1b-9525-340e1cec956e

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 โ€œcarrying capacity,'' which is expressed as L in Eq. 45.

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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(989).png?alt=media&token=ea8f8653-aad0-418c-a9ee-953ba74efa4e

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

Of course, this isnโ€™t the full story, as this just means L/2 is a critical point. You can verify this by finding the value of fโ€™โ€™โ€™ to check that it is positive at L/2.


Footnotes

*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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