Tuesday, March 6, 2007

The Beauty Of The Logistic Equation

The Logistic Map on the left, is derived from the Logistic Equation which was first introduced by Pierre Francois Verhulst to model the growth of populations. Briefly it states that a future population will depend on the rate of growth of the existing population, which itself is a function of the existing population. This equation thus describes how resources determine the rate of growth of a population of living creatures, and the changes in this rate of growth through time. The equation is:
Where, t=current time, and r=rate of growth of population and Xt the current Population is represented by a value from 0 to 1,
X t+1= rXt(1-Xt) .
** Notice how the bigger Xt is, then the smaller will be 1-Xt and therefore the smaller will be rXt(1-Xt).

This deceptively simple equation can give rise to many situations depending on the value of r. Therein lies its beauty. Most importantly this equation illustrates one of the main tenets of Chaos Theory that small initial differences can cause big unpredictable final results when r is at a value when Chaos ensues. Here then are the characteristics of the Logistic Map for different values of r:
  • For r=<1,> the population will die off.
  • For r from 2 to 3 the population grows slowly then dies.
  • At higher growth rates (3 to 3.45), the graphs start to get interesting. The population will settle into a pattern where it alternates between two populations.
  • At even higher growth rates (3.45 to 3.54), it will alternate between four populations, then eight, sixteen, thirty two , speed-doubling until it reaches Chaos.
  • From 3.55 to 4, the final output will depend on what the initial value of Xt was. Very small differences in Xt can lead to vast differences in the results.
  • For r=>4, the population will grow beyond its limit and die.
The image shows how x can go from one value, to two, four etc until it becomes unpredictable in the shaded area beyong 3.55. But still there are islands of stability shown for example by the white 'corridor' around r=3.8. I believe these are called 'Strange Attractors' in the language of Chaos Theory.
The Logistic Equation is related to the S-shaped growth curve in the top image which describes how many natural and social phenomenon in our Universe develop over time, be they businesses, plants, children, stock markets or a new technology
  • 1. A period of slow growth, until it reaches a critical threshold take-off stage.
  • 2. The growth is then at an exponential rate i.e. increasing at an increasing rate.
  • 3. Growth continues but the rate of growth tapers off i.e. increasing at a decreasing rate.
  • 4. Another stage not shown here is the absolute decline stage where the curve drops below it's maximum

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