Differential Equations Solutions 145

Differential Equations Solutions 145 - solutions, i the...

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Chapter 26 Solutions: Case Study: Beetles, Cannibalism, and Chaos: Analyzing a Dynamical System Model CHALLENGE 26.1. The results are shown in Figure 26.1. When μ A =0 . 1, the solution eventually settles into a cycle, oscillating between two diferent values: 18 . 7 and 321 . 6 larvae, 156 . 7 and 9 . 1 pupae, and 110 . 1 and 121 . 2 adults. Thus the population at 4 week intervals is constant. Note that the peak pupae population lags 2 weeks behind the peak larvae population, and that the oscillation o± the adult population is small compared to the larvae and pupae. For μ A =0 . 6, the population eventually approaches a ²xed point: 110 . 7 larvae, 54 . 0 pupae, and 42 . 3 adults. In the third case, μ A =0 . 9, there is no regular pattern ±or the solution, and it is called chaotic . The number o± larvae varies between 18 and 242, the number o± pupae between 8 and 117, and the number o± adults between 9 and 94. CHALLENGE 26.2. The results are shown in Figure 26.2. For the stable
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Unformatted text preview: solutions, i the model is initialized with population values near A fixed , L fixed , and P fixed , it will converge to these equilibrium values. CHALLENGE 26.3. The biurcation diagram is shown in Figure 26.3. The largest tested value o A that gives a stable solution is 0 . 58. I the computation were perormed in exact arithmetic, the graph would just be a plot o L fixed vs. A . When the solution is stable, rounding error in the computation produces a nearby point rom which the iteration tends to return to the xed point. When the solution is unstable, rounding error in the computation can cause the computed solution to drit away. Sometimes it produces a solution that oscillates between two values (or example, when A = 0 . 72) and sometimes the solution becomes chaotic or at least has a long cycle (or example, when A = 0 . 94). 155...
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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