Because of limits on food, living space, disease, current technology, war, and other factors, most populations have limited growth as opposed to unlimited exponential growth which is modeled by the classic exponential growth equation **P = P _{0}b^{t/k}**. A limited growth population starts growing almost exponentially, but reaches a critical point in time where its growth rate slows, and the population starts to asymptotically approach an upper limit as time increases. There are several models that are used to describe limited growth of a population.

In this post, I will discuss the logistic function which was used by the Belgian mathematician Pierre Francois Verhulst (1804-1849) to study limited population growth. The logistic function also has applications in artificial neural networks, biology, chemistry, demography, ecology, economics, biomathematics, geoscience, mathematical psychology, sociology, political science, probability, and statistics.

The two text boxes below describes the key parameters and relationships between the parameters of a logistic function. Graph (A) shows a typical logistic function curve and how equation parameters can be calculated from known characteristics of the population. If p_{max}, p_{0}, and t_{c }are known, then a, b, and k can be calculated. Likewise, if a, b, and k are known, then p_{max}, p_{0}, and t_{c }can be calculated. Keep in mind that all limited growth models can only give us a good approximation of a population value at some point in time.

In most cases, the key parameters of a logistic equation are unknown, but an observed set of data-pairs is known. The **least-squares logistic equation** of a data set is the best of all possible logistic equations that describes the relationship between the data-pair variables. Best possible equation means that the sum of the squared errors (difference between observed value and predicted value) is minimized. Modern graphing calculators have the capability of findings a least-squares equation for a variety of models such as linear, quadratic, cubic, quartic, sinusoidal, log, exponential, and logistic. When given a logistic type data set, I will use a graphing calculator to find the least-squares logistic equation of the data set, and then calculate various characteristics and properties of the resulting logistic model. I will now take a look at two problems that illustrate how the logistic function can be used to describe limited population growth.

**Problem 1 solution:** Use math software to do a scatter plot of the data, find the least-squares logistic equation **p = 12.0121 / (1 + 10.6694e ^{-0.023856x})** of the data set, and then do the appropriate calculations. Refer to graph (B) below. From graph (B) we see that the world population growth rate started to slow in 1999, and the upper limit of the world population is about 12 billion. Keep in mind that this least-squares equation is our current best description of world population growth. Future unknowable events will alter this model.

**Problem 2**: The logistic function N(t) = 3,600 / (1 + 29.4e^{-0.2t} ) models the spread of a disease in a town. N(t) = the total number of people infected at time t, and t = the number of days after the first reported infections.

(a) How many people were initially infected?

(b) How many people were infected after 10 days and after 30 days?

(c) When did the rate of infection start to slow?

(d) What is the upper limit of the number of infected people?

**Problem 2 solution:** Use math software to graph the equation, and then do the appropriate calculations. Refer to graph (C) below.

(a) About 118 people were initially infected.

(b) After 10 and 30 days, 723 and 3,355 were infected.

(c) About day 17, the infection rate started to slow.

(d) The upper limit of the number of people infected = 3,600.

Comments:

- It’s fun and interesting to experiment with different logistic function parameters. Experimentation always gives a better learning experience.
- With my graphing calculator, it took about 8 seconds to compute the parameters of a logistic equation. This is an indication of the complexity of the algorithms for computing the parameters of a least-squares equation. I tell my students that they should be ever thankful that they have access to such wonderful computation tools.
- Computer math software allows students to focus on math concepts, and not get lost in gory computational details. This is why graphing calculators have revolutionized the way we teach statistics. Just getting the ‘answer’ is no longer sufficient. Students must be able to interpret and explain the meaning of the answer in the context of the problem.