27 Logistic Functions

Arkansas Tech University MATH 2243: Business Calculus Dr. Marcel B. Finan

One of the consequences of exponential growth is that the output f(t) increases indeﬁnitely in the long run. However, in some situations there is a limit L to how large f(t) can get. For example, the population of bacteria in a laboratory culture, where the food supply is limited. In such situations, the rate of growth slows as the population reaches the carrying capacity. One useful model is the logistic growth model. Thus, logistic functions model resource-limited exponential growth. A logistic function involves three positive parameters L, C, k and has the from L f(t) = . 1 + Ce−kt We next investigate the meaning of these parameters. From our knowledge of the graph of e−x we can easily see that e−kt → 0 as t → ∞. Thus, f(t) → L as t → ∞. It follows that the parameter L represents the limiting value of the output past which the output cannot grow. We call L the carrying capacity. Now, to interpret the meaning of C, we let t = 0 in the formula for f(t) and obtain (1 + C)f(0) = L. This shows that C is the number of times that the initial output must grow to reach L. Finally, the parameter k aﬀects the steepness of the curve, that is, as k increases, the curve approaches the asymptote y = L more rapidly.

Example 27.1 Show that a logistic function is approximately exponential function with continuous growth rate k for small values of t.

Solution. Rewriting a logistic function in the form

Lekt f(t) = ekt + C

1 L kt we see that f(t) ≈ 1+C e for small values of t.

Graphs of Logistic Functions 185 Graphing the logistic function f(t) = 1+48e−0.032t (See Figure 47) we ﬁnd

Figure 47

As is clear from the graph above, a logistic function shows that initial exponential growth is followed by a period in which growth slows and then levels oﬀ, approaching (but never attaining) a maximum upper limit. Notice the characteristic S-shape which is typical of logistic functions.

Point of Diminishing Returns Another important feature of any logistic curve is related to its shape: every logistic curve has a single inﬂection point which separates the curve into two equal regions of opposite concavity. This inﬂection point is called the point of diminishing returns.

Finding the Coordinates of the Point of Diminishing Returns(Optional) To find the point of inflection of a logistic function of the form P = f(t) = L 1+Ce−kt , we notice that P satisfies the equation dP P = kP 1 − . dt L Using the product rule we find

d2P dP 2P = k 1 − . dt2 dt L

2 dP d2P L Since dt > 0 we conclude that dt2 = 0 at P = 2 . L To ﬁnd y, we set y = 2 and solve for t :

L L 2 = 1+Ce−kt 2 1+Ce−kt L = L 2 = 1 + Ce−kt 1 = Ce−kt ekt = C kt = ln C ln C t = k

ln C L Thus, the coordinates of the diminishing point of returns are k , 2 .

Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time.

Example 27.2 The following table shows that results of a study by the United Nations (New York Times, November 17, 1995) which has found that world population growth is slowing. It indicates the year in which world population has reached a given value:

Year 1927 1960 1974 1987 1999 2011 2025 2041 2071 Billion 2 3 4 5 6 7 8 9 10 (a) Construct a scatterplot of the data, using the input variable t is the number of years since 1900 and output variable P = worldpopulation (in billions). (b) Using a logistic regression, ﬁt a logistic function to this data. (c) Find the point of diminishing returns. Interpret its meaning.

Solution. (a) For this part, we recommand the reader to use a TI for the plot. (b) Using a TI with the logisitc regression we ﬁnd L = 11.5,C = 12.8, k = 0.0266. Thus, 11.5 P = . 1 + 12.8e−0.0266t

3 (c) The inﬂection point on the world population curve occurs when t = ln C ln 12.8 k = 0.0266 ≈ 95.8. In other words, according to the model, in 1995 world population attained 5.75 billion, half its limiting value of 11.5 billion. From this year on, population will continue to increase but at a slower and slower rate.