CHAOS THEORY
and
Population Dynamics

        Consider a population that grows like many populations on the Earth. There is a birth rate and a death rate and there are some limiting factors of the environment which determine what a maximum value a population might be. Admittedly there could be a lot of debate on what might be a limiting or maximum population the Earth can sustain. Population planners deal and discuss this all the time. It may be food supply, energy or a number of things that contribute to this. The purpose of our experiment today is not to determine what those factors are but rather just to look at the mathematics of Chaos Theory and a very very simple application called the logistics equation. In modeling some population let a value between 0 and 1 represent the fraction of the population that exists at some given starting point. 0 represents an extinct population and 1 represents the maximum value it can have. The logistics equation is simply:

Xn+1 = a Xn ( 1 - Xn )
where a is a parameter of a dynamical system that dictates how a population responds to its environment. If it is high the population grows rapidly but if too high strange things may happen. There are favorable and unfavorable values of this parameter.  For some values of the parameter the population will reach an equilibrium value. Long before computeres the dog and the flea problem led to an equilibrium condition in what people could do by hand. But with computers different values of the parameter led to amazing results. That is your job today.

        Now, don't look to this as a human poipulation or necessarily even a living population. It is a model that describes to some degree how populations react to their environment. It could be fish in a tank, gypsy moths, rabbits on the prairie, etc. One interesting application by a student of mine was to use this to predict the flow of traffic on Twin Cities interstate highways. At some point things worked well and then boom, chaos and gridlock. So the mathematics can be applied in a lot of cases.

Your Job:

        Take the logistics equation with say a statring value of the population at .5 or .or .6. Make a spreadsheet using EXCEL to follow the population through, say 100 time intervals between successive n values. Vary the parameter in a dozen different columns with different population values. repeat this with different values of the parameter a. For reasons we don't go into here the parameter should be between 0 and 4. At what point does full blown chaos take place? What kinds of values does the population grow? When does it stabilize? Will it ever become extinct? What words of wisdom do you have for humanity?