You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
Discrete dynamical systems -- both real systems and mathematical models -- exhibit an extraordinary and surprizing variety of behavior. In this section we provide an overview -- looking at
We give examples of each of these kinds of behavior in this section. Click on the appropriate topic for a more detailed discussion of limits, cycles, or chaos.
The simplest population models exponential models
p(n + 1) = R p(n)
predict that population will rise or fall by a fixed percentage rate each generation. For example, the model
p(n + 1) = 1.20 p(n)
predicts that population will rise by 20% each generation. This model predicts that no matter how large the habitat is the population will eventually rise beyond its capacity. Mathematically, we say that for any number B there is an n such that |p(n)| > B. In words, we say that the sequence is unbounded.
If there is a B such that for every n, |p(n)| <= B then we say that the sequence is bounded and that B is a bound. In practice any real population sequence must be bounded because the earth is finite and any earthly habitat can only support a certain population.
The logistic model
p(1) = 50
p(n + 1) = 2.8 [1 - .001 p(n)] p(n)
behaves much more nicely as shown in the table, graph, and cobweb diagram below.
1 50.00 6 623.97 11 648.70 16 640.92
2 133.00 7 656.97 12 638.09 17 644.40
3 322.87 8 631.01 13 646.61 18 641.62
4 612.15 9 651.94 14 639.82 19 643.85
5 664.78 10 635.36 15 645.26 20 642.06
The population predicted by this model appears to be approaching the equilibrium point 642.86. In this situation we say that the population is approaching a limit.
The logistic model
p(1) = 50
p(n + 1) = 3.4 [1 - .001 p(n)] p(n)
behaves somewhat differently as shown in the table, graph, and cobweb diagram below.
1 50.00 6 840.11 11 454.66 16 842.65
2 161.50 7 456.69 12 843.01 17 450.81
3 460.42 8 843.62 13 449.97 18 841.77
4 844.67 9 448.54 14 841.49 19 452.85
5 446.08 10 841.00 15 453.51 20 842.44
In this case the population seems to be settling in a 2-cycle oscillating back-and-forth between two different populations -- roughly 450 and 840.
The logistic model
p(1) = 50
p(n + 1) = 4.0 [1 - .001 p(n)] p(n)
behaves very differently as shown in the table, graph, and cobweb diagram below.
1 50.00 6 645.70 11 999.65 16 316.37
2 190.00 7 915.09 12 1.39 17 865.11
3 615.60 8 310.82 13 5.57 18 466.77
4 946.55 9 856.84 14 22.14 19 995.58
5 202.38 10 490.67 15 86.59 20 17.59
In this case there is no apparent pattern at all. This situation is called chaos.
For each of the following models determine its long term behavior -- is it bounded, or unbounded, does it appear to be approaching a limit, settling down into a cycle, or behaving chaotically?
p(1) = 50
p(1) = 50
p(1) = 50
p(1) = 50
p(1) = 50
p(1) = 50
p(1) = 50