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Date: 10-10-2016
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Date: 25-10-2016
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Date: 10-10-2016
70
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Deterministic Competition
Consider a simplified system, one that can be described by Nt objects at time t. For example, one could consider the number of grasshoppers on the plains of Africa, or on some small plot of land. Let there be competition between the growth processes and the decay processes so that the number of objects at time t + 1 is Nt + 1 = Nt exp[r (1 – Nt)], an exponential growth relationship. This equation is deterministic, for Nt determines Nt + 1 unambiguously. One can think of r as a measure of the ratio between growth and decay. Numerous mechanical, hydrodynamic, chemical, and electrical systems can be approximately modeled by this relationship.
How does the number of objects behave with elapsed time? If Nt = 1, then N remains 1 forever. In the general case, we can determine Nt as t→∞ to find out whether N approaches the equilibrium value 1. For instance, let r = 1 and begin with N0 = 0.5, and calculate with a calculator or personal computer. Now try different values for r. What behavior do you predict?
Answer
The time evolution here depends on the value of r. One finds that Nt = 1 is a stable equilibrium only when r lies between 0 and 2. If r = 2.3 with N0 = 0.5, then successive Nt will oscillate between about 1.59 and about 0.40 as a stable 2 cycle. For r > 3.102, no cycle is stable, all cycles are possible, etc.
In the chaotic regimes, the equation results are deterministic, but the time evolution is indistinguishable from that governed by probability laws. One really needs to see the calculations proceed to appreciate the amazing behavior of this simple-looking equation.
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