Join our community

Learn Morecaret down

caret down

Join our community

Fiveable Community students are already meeting new friends, starting study groups, and sharing tons of opportunities for other high schoolers. Soon the Fiveable Community will be on a totally new platform where you can share, save, and organize your learning links and lead study groups among other students! ๐ŸŽ‰

group of students
group of students

ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

7.9 Logistic Models with Differential Equations

#logisticmodels

#differentialequations

โฑ๏ธย ย 2 min read

written by

Jacob Jeffries

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.

continue learning

Slide 1 of 9
Fiveable

Join Our Community

Fiveable Community students are already meeting new friends, starting study groups, and sharing tons of opportunities for other high schoolers. Soon the Fiveable Community will be on a totally new platform where you can share, save, and organize your learning links and lead study groups among other students!๐ŸŽ‰

Fiveable Logo

2550 north lake drive
suite 2
milwaukee, wi 53211

92% of Fiveable students earned a 3 or higher on their 2020 AP Exams.

*apยฎ and advanced placementยฎ are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.

ยฉ fiveable 2020 | all rights reserved.