The logistic difference equation in Section 1.6 of the textbook, Equation 5, is widely used to model population dynamics. Although the model can be studied in this form by exploring how N changes through time, it is usually re-written in a simpler form by defining the new variable
The variable xt can be interpreted as the size of the population at time t, measured as a scaled fraction of the carrying capacity K. Writing the model in terms of this new variable allows us to see the form of the difference equation more clearly. It also reduces the number of constants in the model from two to one (compare Equations 5 and 7 of Section 1.6 in the textbook). This simplified difference equation is sometimes called the logistic map, but be mindful of the different ways in which the constant r is defined by different authors.