Applications of Fractals

Population Growth

We hear about the rapid growth of population in developing countries the all the time. With the problems that it is constantly causing, it is rather obvious how important it is to analyze the population growth. Last century, Thomas Malthus came with a theory in which he said that with every generation, the population increases a certain amount of times depending on the growth rate. Mathematically, if we make r the percent growth rate, and P the population, our formula will become

new P = (1+r) · P

For example, if r = 1/2 the population will increase 50%, or become 1.5 times larger. However, something about this theory seems not right... According to this theory, the population will increase infinitely. However, the population is really limited by natural resources, such as space and food. Let’s pretend the maximum possible population the environment can hold is 1, so P is a number from 0 to 1. As the population gets closer to 1, the growth rate is going to decrease and get close to 0. We can achieve this by multiplying the growth rate by (1–P). This way, as P is getting closer to 1, the growth rate will be multiplied by a number that is getting close to 0. We now determined that the growth rate should really be r(1–P). If we use it in the above formula, we get

new P = [1 + r(1–P)] · P

If we now do some algebra

new P = [1 + r – rP] · P
new P = P + rP – rP2
new P = (1 + r) · P – rP2

Will now use this formula. Knowing this formula, it is easy to determine what the population becomes after a long period of time. For example, when r is between 0 and 2, the population becomes 1 and stays there, no matter what it was at the beginning. When it is 2.25, it will always end up jumping between 1.17 and 0.72. When r is 2.5, it ends up jumping between 1.22, 0.54, 1.16, and 0.70. When it is 2.5, it ends up jumping between 8 values, and when r gets higher, it jumps between 16 values. As we increase r, the number of these values doubles. We call this bifurcation. Sounds familiar? Of course! We have already seen this in the lesson on chaos and fractals with a different formula. So.. maybe this will give us a fractal pattern as well! Let’s make a graph in which we plot the values of the population for all values of r from 1.9 to 3:

This is identical to the Feigenbaum Fractal! Indeed, you can unexpectedly find fractals even in the growth of population.