In order to make a more realistic model, Verhulst considered that in nature, the larger a population grows, the less productive it becomes, perhaps because of lack of food or other overpopulation problems. So in creating his (abstract) model, he says, let's set the upper limit of a population at 1. (Think of it as 100% of the room available for growth). Then the room left over by the environment for a new generation is 1-x. This can be seen as a correction factor to unbounded growth. The Verhulst Model for limited population growth then becomes: xnew=a*x*(1-x). The population of a new generation is equal to the malthusian growth factor times the old population, and scaled down by the amount of room available for growth. In spite of its simplicity, it proves to be a fair model of what happens in nature. If the productivity factor a is 2, then starting the formula with a low seed value like 0.001, we see the population x rise and level off at 0.5. This is what we might expect in nature with animals with a healthy productivity. After an initial period of fast growth, the population stabilises.
If we set the productivity factor a to higher values, strange things happen. If a is 3.2, x grows rapidly at first, but then doesn't stabilise to one value; rather it alternates between two values endlessly. (See Fig. 4a.)
It doesn't matter what the seed value was, x ends up alternating between the same two values. If a is set a little bit larger than
3.4495, we find the values for x orbiting between four values eventually. Carefully increasing the value of a for still more trials, we find that the number of values that x seems to land on keeps bifurcating (to 8, and 16) until there is a value for a, 3.569946, just beyond which x fluctuates chaotically from one value to the next. Sometimes it bounces back and forth between a couple of values for a while, only to spin off again. (See fig. 4b.)This type of chaotic behaviour is also observed in nature, for example by an animal with a productivity so high that it overreaches the ability of the environment to support it. The population crashes, only to build up again. The interesting thing about the model is that it does show a kind of regularity, with x-values jumping up and down, but it never repeats itself exactly. This simple, deterministic mathematical formula can be just as erratic as measurements of real populations in nature!
There are more mysteries lurking here. While searching for the exact values of a where the behaviour of the model changed-where x values would settle down eventually to one, two, four or eight values-the physicist Mitchell Feigenbaum recently discovered a constant ratio between the a values. Still more astonishing was the discovery that other quite different mathematical formulas (still using an output-input loop to calculate a new value from an old value), and also experimental data exploring the onset of turbulent flow, also showed the doublings, and the same ratio between them, 4.6692... In short, a new universal constant was discovered by Feigenbaum, like the constant of gravity, the speed of light, or the weight of an electron.
We're not yet finished with Verhulst's model. If a is increased to
3.83, the chaotic behaviour eventually stops, and x circles eventually between only three values. (See fig. 4c.)Increasing a in small amounts for new trials results in period doubling of the values where x eventually settles down to- 6,12; and again chaotic behaviour sets in, up to a=4. (See fig. 4d.)
(We cannot set a to a number greater than 4, because that would produce x values greater than 1, or exceeding our original definition of the maximum population). With the help of a computer, a graph can be made of how Verhulst's formula behaves for all settings of the a value. (See fig. 4e.)
Fig. 4e Verhulst Model: all "a" values along x-axis
We see the doublings of x at so called bifurcation points, followed by chaotic regions, then windows, where x again has a low number of stable values. We get a shock of recognition when we magnify the region where x splits up again; the whole pattern reveals itself in miniature! (See fig. 4f.)
Fig. 4f Verhulst Model: all "a" values along x-axis (detail)
Indeed, it seems that the pattern contains nested within itself, its own replica! This kind of nested pattern is now called a "fractal". The Polish-born mathematician Benoit Mandelbrot derived the term from the Latin adjective fractus, meaning irregular or broken. Fractals are characterised by intricately nested patterns within patterns, with self-similarity on any scale. Fractals can be recognised in a wide range of natural phenomena and shapes, such as trees and clouds. Analysis of Indonesian Gamelan music reveals fractal structure. (See fig. 5).