PHYSICS 202 – Physics and the Computer
Projects
Logistic Map
R. May wrote a seminal paper on the properties of a logistic map (see References). He was
motivated by trying to model interacting biological systems. In this simplified model the population of a
species in the next generation depends on the population in the current generation. The population can
grow if procreation occurs, but will shrink due to predators or insufficient resources.
Let xn be the population in the nth generation then:
xn 1  4rxn (1 x n ), 0  x  1, 0  r 1
Write a spreadsheet program that will allow you determine the population at each generation given the
population at the previous generation and parameters. Test the program by comparing the values in the
spreadsheet to a hand computation of the first 5 generations with r=0.2.
Now, let’s learn about population growth by exploring the results the program produces.
a) Let's examine the behavior of this system for several values of the parameter r and various initial
conditions. Let r = 0.2 and x0 = 0.6. Plot xn vs. n for at least 30 generations. What happens after many
generations? What about x0 = 0.2? Try a few other values of x0. Can you conclude anything about
the long-time value of x when r = 0.2?
b) Now let's change the control parameter. Choose r = 0.26 and x0 = 0.1, you might need to look for 50
or 100 generations. How is the long-time behavior different than in a)? Pick a few other values of x0.
c) Pick some other values of r, say 0.4, 0.6, what happens? Now look at 0.74, be sure to observe
enough generations to see what is happening.
d) Now let's look at a new regime, let r = 0.76. Note that the population fluctuates between two different
values in alternate generations.
e) Now try r = 0.88. How is this behavior different than d)?
f) Lastly, make a plot of the "final" value(s) of x for many different values of r in the range 0<r<1. You will
need to pick a starting value for x and r then go for many (200 or more?) generations to see what the
final value of x is. For some values of r there is more than one "final" value. Write down these "final"
values and then increase r a little and repeat.
References:
R. May, "Simple mathematical models with very complicated dynamics", Nature 261, 459 (1976)
The logistic map is also discussed in many books about chaos and nonlinear dynamical systems.
For example, see the discussion starting on p.128 of An Introduction to Computer Simulation Methods, 2 nd
edition, by H. Gould and J. Tobochnik (Addison-Wesley, 1996).
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