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w1f - Douglass Houghton Workshop Section 1 Mon Worksheet...

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Douglass Houghton Workshop, Section 1, Mon 9/26/11 Worksheet Fifty Million Frenchmen 1. Last time we investigated a rule for how a population of fish might change. Let’s nail down the essential features of all similar rules. Here’s what we know: Rule Equilibrium Stable? P ( n + 1) = 1 . 5 P ( n ) - 200 400 An equilibrium is a population that will stay constant from year to year. An equilibrium ˆ P is stable if when the population starts a little above or below ˆ P , it moves toward ˆ P . Otherwise ˆ P is unstable . (a) Add rows to the table for these rules. You can reason either numerically, graphi- cally, algebraically, or with words. Note: these may be harder to explain in terms of fish, but it will be fun to try. P ( n + 1) = . 75 P ( n ) + 200 P ( n + 1) = . 4 P ( n ) + 600 P ( n + 1) = 1 . 1 P ( n ) - 330 P ( n + 1) = - . 5 P ( n ) + 1200 P ( n + 1) = - 1 . 3 P ( n ) + 460 P ( n + 1) = P ( n ) + 300 P ( n + 1) = - P ( n ) + 300 (b) Explain how to find the equilibrium and its stability for the general linear recur- rence P ( n + 1) = mP ( n ) + b. where m and b are constants.
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