The Logistic Difference Equation
The Logistic Difference Equation is discussed in detail in your text,
Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics.
It essentially relates a biology problem of a closed ecosystem such as
a fish pond with a mathematical model. In the model, there is a finite
number of fish
in the pond, and the equation predicts the population of the next
generation depending upon some reproduction rate. Interesting things happen
as this equation is iterated over time, and different population rates
are selected.
Depending upon the value of that growth rate, k, the following population dynamics
can be observed:
- The population converges to a single value, or attractor,
after a period of time.
- The population does not converge, but bounces back and forth between
two or more values. The point at which the split from one convergence
point into two attractors is called bifurcation.
- The population goes into chaos, where each generation has a different
number of fish than the previous, with little observable relationship to
the previous one.
Variables
x - the percentage of fish in the pond, greater than 0 but less than 1.
k - the growth rate, greater than 0 but lesss than or equal to 4.
The Equation
x_{n+1} = k * x_{n} * ( 1 - x_{n} )
Graph of 100 Iterations of the Logistic Difference Equation for 0 < k < 4