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 un...