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The Logistics of the Logistic Curve Dan Kennedy Baylor School Chattanooga, TN dkennedy@baylorschool.org

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A real-world problem from my high school days: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 hours, how many organisms will be produced from a single cell? Solution: 10 hours = 30 20-minute periods There will be 1 ∙ 2^30 = 1,073,741,824 bacteria after 10 hours.

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A problem that seems just as reasonable: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 days, how many organisms will be produced from a single cell? Solution: 10 days = 720 20-minute periods There will be 1 ∙ 2^720 ≈ 5.5 ∙ 10^216 bacteria after 10 days.

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Makes sense… …until you consider that there are probably fewer than 10^80 atoms in the entire universe. Real world Bizarro world

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Why didn’t they tell us the truth? Most of those classical “exponential growth” problems should have been “logistic growth” problems! ExponentialLogistic

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1991 / BC-6 (The Rumor) This was the first logistic differential equation to appear in an AP FRQ.

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The first part did not involve solving the differential equation. The answer was the value of y that maximized 2y (1 – y) … equivalent to finding the vertex of an upside-down parabola!

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The max occurs at y = 1/2.

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The next part required finding an antiderivative using partial fractions.

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Oliver Heaviside Partial fractions in the AP course are best done by the Heaviside Method:

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Oliver Heaviside Oliver, on the heavy side

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Answer: y = 1/2

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Exponential growth occurs when the growth rate is proportional to the size of the population.

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Logistic growth occurs when the growth rate slows as the population approaches a maximal sustainable population M.

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Before solving the differential equation, it is useful to observe how much information can be gleaned from the differential equation itself. The graph of kP(M – P) is an upside- down parabola with zeros at P = 0 and P = M. The maximum is always at M/2.

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When P M, the growth rate is negative. In either case, the population approaches M as t increases. We can see this in a slope field for the differential equation.

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M 0

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Now let’s look at one of the most infamous logistic problems of all: 2004 / BC-5 Suffice it to say that the BC students in 2004 were not amused.

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2004 / BC-5 (Logistic Growth)

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What the students were supposed to do was to answer (a), (b), and (c) using their rich knowledge of logistic functions. (a) 12 ; 12 (b) 12/2 = 6

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Then they had to solve a separable differential equation in (c). The equation was not logistic. In fact, part (d) was designed to let them savor the difference. Sadly, many of them never made it past (a) and (b).

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The logistic equation was: The equation in part (c) was: Notice that this population would grow until t = 12 and then crash fast as t got bigger and bigger.

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Here’s the slope field for 12

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And here’s the solution to the initial value problem in (c):

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Then, finally, the limit in (d):

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As with the rumor problem, solving a logistic differential equation requires the use of partial fractions. In fact, it can be done in the general case:

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2004 / BC-5 (Logistic Growth)

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In the case of 2004 / BC-5:

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John KenellyClemson University Two neat things I learned about logistic functions from John Kenelly: 1.The Wal-Mart Story 2.The Infection Game

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Sam Walton apparently knew that hot-selling items did not sell like hotcakes. They sold like logistic functions. So he tracked sales from all his stores…daily.

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Daily sales are the differences in your total sales from day to day. Positive daily sales mean your total sales are going up. A big sales day means your sales curve has a steep slope that day. But Sam was not looking at slope. He was looking at differences in daily sales. While his competitors were reacting to slope, Sam was reacting to concavity. He was looking for the point of inflection.

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Sam finds the first negative difference in daily sales. Total sales look great, but Sam knows what is coming. While his competitors are stocking up, Sam starts unloading his inventory. When demand lets up, his shelves are free to stock the next hot item!

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The Infection Game is a great way to show students that logistic growth is a real phenomenon. Give everyone in the class a number. (For small classes, make it a hand infection and give each hand a number.) Enter this command and “infect” a random victim. From now on, each new ENTER command will spread the infection.

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Press ENTER again. You now have two infected victims at time t = 2. Press ENTER two times (since each of the two infected victims can spread the infection.) Now press ENTER 4 times…and so on.

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The infection spreads exponentially at first, because you press ENTER more often as it spreads. But soon it slows down because there are fewer new victims to infect. It actually gets exciting when there are two healthy students left and they wonder who will go first and how many ENTERs they can survive. I gathered some data from an actual class…

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Time1234567891011 Infected124813222628292930

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dkennedy@baylorschool.org

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Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try.

Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try.

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