The Logistic Function Examples & Exercises
In the past weeks, we have considered the use of linear, exponential, power and polynomial functions as mathematical models in many different contexts. Another type of function, called the logistic function, occurs often in describing certain kinds of growth. These functions, like exponential functions, grow quickly atfirst, but because of restrictions that place limits on the size of the underlying population, eventually grow more slowly and then level off.
As is clear from the graph above, the characteristic S-shape in the graph of 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. 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; in a different context, they can also describe demand curves: the decline of demandfor a product as a function of increasing price can be modeled by a logistic function, as in the figure below.
y C y(0)
The formula for the logistic function, y = C 1 + Ae − Bx
involves three parameters A, B, C. (Compare with the case of a quadratic function y = ax2 + bx + c which also has three parameters.) We will now investigate the meaning of these parameters. First we willassume that the parameters represent positive constants. As the input x grows in size, the term –Bx that appears in the exponent in the denominator of the formula becomes a larger and larger negative value. As a result, the term e–Bx becomes smaller and smaller (since raising any number bigger than 1, like e, to a negative power gives a small positive answer). Hence the term Ae–Bx also becomessmaller and smaller. Therefore, the entire denominator 1 + Ae–Bx is always a number larger than 1 and decreases to 1 as x gets larger. Finally then, the value of y, which equals C divided by this denominator quantity, will always be a number smaller than C and increasing to C. It follows therefore that the parameter C represents the limiting value of the output past which the output cannot grow (see thefigures above). On the other hand, when the input x is near 0, the exponential term Ae–Bx in the denominator is a value close to A so that the denominator 1 + Ae–Bx is a value near 1 + A. Again, since y is computed by dividing C by this denominator, the value of y will be a quantity much smaller than C. Looking at the graph of the logistic curve in Figure 1, you see that this analysis explainswhy y is small near x = 0 and
approaches C as x increases. To identify the exact meaning of the parameter A, set x = 0 in the formula; we find that C C y(0) = = −BA 0 1 + A 1 + Ae Clearing the denominator gives the equation (1 + A)y(0) = C. One way to interpret this last equation is to say that the limiting value C is 1 + A times larger than the initial output y(0) An equivalent interpretationis that A is the number of times that the initial population must grow to reach C The parameter B is much harder to interpret exactly. We will be content to simply mention that if B is positive, the logistic function will always increase, while if B is negative, the function will always decrease (see Exercise 9). Let us illustrate these ideas with an example. Example 1. Construct a scatterplot ofthe following data: x y x y 0 4 8 69 1 6 9 79 2 10 10 86 3 16 11 91 4 24 12 94 5 34 13 97 6 46 14 98 7 58 15 99
The scatterplot should clearly indicate the appropriateness of using a logistic model to fit this data. You will find that using the parameters A = 24, B = 0.5, C = 100 produce a model with a very good fit.
Notice that no matter how large you take x, y will never exceed...