College Algebra & Trigonometry, 2018a

146 CHAPTER 3. EXPONENTS AND LOGARITHMS
Some phenomena in the natural world exhibit behavior similar to the growth
of this function. However, in the natural world, few, if any, things can grow
unconstrained. Most growth of any kind is limited by the resources that fuel
the growth. Populations often grow exponentially for a period of time, however,
populations are dependent on natural resources to continue growing. As a result,
the simple exponential function is only useful for modeling real-world behavior
if the x-values are limited.
It was this problem with the simple exponential function that led French mathematician
Pierre Verhulst to slightly adjust the differential equation that gives rise
to the exponential function to make it more realistic.
The original differential equation said:
y ′ = k ∗ y
This says that the rate of growth of y is always directly proportional to the value
of y. In other words the larger a population gets, the faster it will grow - forever.
Verhulst changed this to say:
y ′ = k ∗ y(1 − y N )
This is the defining relationship for the Logistic function. Notice that when values
of y are small, this is essentially the same as the simple exponential. If y is small,
then the (1− y ) term will be very close to 1 and will produce behavior very much
N
like the simple exponential.
The N in the formula is a theoretical “maximum population.” As the value of y
approaches this maximum value, y will approach 1 and (1 − y ) will get smaller
N N
and smaller. As it gets smaller, the factor of (1 − y ) will slow down the growth
N
of the function to model the pressure that is put on the resources that are driving
the growth.
The solution of the Logistic equation is quite complicated and results in a standard
form of:
y =
N
1+be −kt