The Logistic Map is a model of population growth and decay, where a population size, given by x, is updated generation by generation. A population of 1 means maximum size, and 0 is extinction.

The rate of population growth is determined by a constant r that ranges in value from 0 to 4. If there was no death, each generation the new population would be r times the current population, i.e. r x, and the population would grow exponentially. But the model includes death (because of competition for limited resources) and this reduces the population by r x2 each generation. Thus, the logistic map models the interaction between birth and death processes in a population.

The equation that describes the population change each generation is therefore xnew = r x (1 - x).



Quadratic Chaos
Infinite complexity from a simple equation

Exploring the Logistic Map

This app allows you to explore the behaviour of the logistic map xnew = r x (1 - x), including the ability to compare the evolution form different initial conditions.

Use the slider to choose a value for the growth parameter r, and an initial size for each population. Click the Next buttons to move to a new generation. The value for the new population (a normalised value from 0 to 1) is shown and also plotted.

r:     

Population 1:   Start   Current 0.5  

Population 2:   Start   Current 0.5  

     

Study the behaviour of the populations for various values of the growth parameter r, and for different initial conditions.

Some things to consider

Two aspects of this population process are particularly important:

  1. How does the specific value of r effect the long term behaviour of x?
  2. How does the series of x values depend on the initial population value?
To explore these, try various values for r, and test the sensitivity to the initial value by looking at the convergence or divergence of the two populations over time.

Look at the behaviour for the following ranges of r and make a table to record what you find: (you can type directly into the table below)

rResults
0 – 1
1 – 2
2 – 3
3 – 3.45
3.5
3.56
3.74
3.83
4

When the population ends on a constant, fixed value, can you calculate what that value must be? (Hint: think about quadratic equations.)

Bifurcations and Chaos

A bifurcation occurs when a solution to an equation suddenly splits into two, such as you will have seen in the logistic map when the value of r crosses 3 (if you didn't notice, run the above app with r = 2.95 and then r = 3.05). Repeated bifurcations lead to cycles with 4 values, then 8, 16 etc, before things get really crazy around r = 3.57.

Chaos is the mathematical term that describes when a system is infinitely sensitive to its starting conditions, i.e. even the tiniest difference in starting point leads to a completely different outcome. The mathematics of chaos is behind the somewhat over-dramatised Butterfly Effect that says the tiny change due to the beating of a butterfly's wings in Brazil can set off a tornado in Texas.

For values of r close to 4 it seems the logistic map could be chaotic, but how to tell for sure? Well, there is a famous result that says if a map has a 3-cycle, that is a solution where it keeps repeating the same three values over and over again, then it is chaotic.

In the next section you can visualise the long term population dynamics for a range of r values using a bifurcation diagram. This is a diagram that shows the long time behaviour of the map for each r — if it is a fixed value, then a single point is plotted, if it is a 2-cycle, the both points are shown, and so on. The first few progressive bifurcations are clearly seen, as is the chaotic region, including its contained islands of stability.

See if you can find a value of r that leads to a 3-cycle. Test your value with the app above.

About the activity

Prior to the development of chaos theory, it was believed that the ability to perfectly describe a physical system with a mathematical model implied the ability to fully predict its future behaviour. Chaos theory undermines this view since even with a perfect description, the predictability of a chaotic system downgrades very rapidly. In fact, maintaining predictability within a perfectly described chaotic system requires the initial conditions to be known to infinite precision — clearly impossible in real life.

The key aspect of these systems is nonlinearity. Mathematially, this means the models involve terms like x2 and other powers, and physically it means that there is feedback in the system; as the system grows it begins to negatively affect its own growth.

The importance of chaos and related concepts (such as fractals) in real world systems has been one of the most significant developments in our mathematical and scientific understanding of nature over the last half century. The idea that the incredible complexity we see in nature may often arise from very simple equations (the logistic map is just one of many simple models that exhibit this behaviour) and that equations with unpredictable outcomes do not necessarily involve random inputs, represents a major shift in our understanding of the mathematical modelling of nature.