The Logistic Map

(This page is a bit more mathematical)

Another interesting example of feedback loops is the example of population growth. Do not let the mathematics of this section worry you, just leave this section and carry on if you wish.

Assuming that the population at any time will depend on the population the year before and the rate to which it increases or decreases every year. We can write this mathematically as

X_(n+1)=B*X_(n)

If this year is the year n, then the population this year is X_(n). The population next year will then be X_(n+1). B is the rate of increase or decrease of the population over the year. The population next year will be the population this year multiplied by the rate of increase or decrease. If B=1, then the population will stay the same, because  X_(n+1)=1*X_(n). If B=2, the population next year will be twice that which it was in the past year because X_(n+1)=2*X_(n). If B is less than one, say ½ the population will decrease the next year.

If we run this formula for ten years then if B=1, the population at year 10 will remain the same as in the first year, but if B=2, however, the population doubles each year and in ten years will be 512 times what it was in years one. If B=½ then the population 1/512 of the original size.

This type of growth is known as exponential growth and can grow extremely rapidly. Populations do not grow like this in reality, of course, as they would just grow and grow with no end fuelled by the positive feedback loop. What happens is that a population will grow until there one or more negative feedback loops kick in pulling the population towards a stable point that the population can sustain itself within the resources of its environment. A lack of food and resources, disease, illness and accidents all reduce the population. As more people get born, others die. Positive feedback loops combine with negative feedback loops in complex ways, interacting with each other, so the outcome becomes less obvious.

 X_(n+1)= B*X_(n)*(1-X_(n))

There is an understood mathematical convention in regard to the part of the formula (1-X_(n)). Here X_(n) (the population this year) is set as fraction of the upper limit that the population can reach. So if the maximum possible value for the population is one million, then if the population at a particular time happens to be 750,000, then X_(n) = 0.75 (1,000,000/750,000). (1-X_(n)) is therefore 1-0.75 or 0.25.

When the rate of population growth, B, is between 1 and 2, the population growth is more straight forward. When the population is just beginning to grow X_(n)  the real proportion divided by the maximum will be very small. (1-X_(n)) is therefore close to 1. While X_(n)  is small, B*X_(n)*(1-X_(n)) will be close to B*X_(n). As the population grows the fraction X_(n) will increase. As the population gets nearer and nearer to the upper limit, X_(n)will get closer and closer to 1. As X_(n) grows nearer and nearer to 1, (1-X_(n)) gets nearer and nearer to zero. The more the B*X_(n) part of the equation increases, the more the (1-X_(n)) decreases. The overall result is that the graph flattens out to the maximum limit value for the population.

Some interesting things happen when we take this formula and look at higher values of B.

When B=3.45 the population does not flatten out to one value, but rather it falls into a cycle, fluctuating between two different values. 

As we further increase the value of B the population, we reach a point where the population changes to fluctuate between four values. Increasing B further to around 3.54, we reach a point where the system cycles between eight values and later still 16. As we increase the value of B even more to around 3.57, the system lapses in chaos, where there the fluctuations vary so much that they are unpredictable. This process is known as period doubling. Mitchell Feigenbaum discovered that the rate of period doubling, where the system goes from two attractors to four, to eight and so on approaches the same ratio of 4.669 in chaotic systems as a universally constant number no matter what type of chaotic attractor was described.