Intuition

Why does the mathematical form of the logistic map make sense? Let’s begin from the simple context of modeling a population, where we might ask how some population will progress over time. Immediately it’s useful to have the current population $x_t$, as well as a measure of how quickly the population is changing. This rate is described by $r$, which gives us the average number of offspring a single organism will produce by the next generation. Since we are assuming all organisms are actively reproducing each generation, the population will grow at a rate proportional to its current size. For example, if $r=2$ and $x_t=8$, we will have $x_{t+1}=16$, $x_{t+2}=32$, and so on, growing exponentially over time.

At this point, we are modeling growth with the equation

$$x_{t+1} = rx_t$$

You’ll notice this model can simply grow without bounds, which doesn’t quite match reality. Most real-world populations have an implicit bound on their size; as a population grows, resources are depleted faster than they can be produced, resulting in unsustainable size. In this case, the population will shrink as organisms die off. Here we might expect an equilibrium size to emerge over time as the population settles.

How do we model this? To start, instead of using $x_t$ to explicitly represent the number of organisms in the population, we will instead treat it as a ratio of the current population size to its maximum possible size (often determined by the environment, commonly referred to as its carrying capacity). Thus, $x_t \in [0,1]$ at any given time, where a value of $1$ implies the population is at the environment’s carrying capacity. We then need a way of encoding the detrimental effects of quickly growing, large populations. An intuitive approach might be to decrease the growth rate as the population grows; although there are more organisms, fewer of them are able to reproduce due to the increasing likelihood of starvation. The logistic map employs a simple rule to address this: the growth rate should decrease by larger amounts when there is a larger population size, and by smaller amounts with smaller population sizes. The value $1-x_t$ gives us a multiplicative factor that changes in this way, decreasing in magnitude as the population $x_t$ increases and vice versa. As a result, we can wrap up the growth rate for the current generation as the new rate $r^\prime = r(1-x_t)$. Plugging this in to our simple exponential growth model from earlier, we have

$$x_{t+1} = r^\prime x_t = rx_t(1-x_t)$$

This, of course, is the form of the logistic map defined above.

As a growth model

The logistic map as a growth model is flawed in many ways.