Unformatted text preview: Math 464, Worksheet 15
Consider a population model in which each successive day’s population is given by
You do not know the entries of the matrix M , but you do know that it’s eigenvalues and associated
λ1 = 1.1, v1 =
, and λ2 = 0.9, v2 =
1. If initial population is x0 = 4 and y0 = 4, graph the populations xn and yn in the x-y plane.
Pay particular attention to curvature and asymptotic behavior. Don’t worry about numerical
2. Graph xn as a function of n on n-x axes.
3. Graph yn as a function of n on n-y axes.
4. Check all of this with Excel. You will need to know the numerical values of M for this. Hint:
M = P DP −1.
5. Repeat 1–4 but with each of the following initial conditions:
(a) x0 = 0, y0 = 5.
(b) x0 = −3, y0 = 3.
(c) x0 = −5, y0 = 0.
6. Repeat 1–5 with λ1 = 0.8 and λ2 = 0.9. Keep the same eigenvectors.
7. Repeat 1–5 with λ1 = 1.1 and λ2 = 1.4. Keep the same eigenvectors.
8. Repeat 1–5 with λ1 = 1.1 and λ2 = −0.9. Keep the same eigenvectors.
9. Repeat 1–5 with λ1 = −1.1 and λ2 = 0.9. Keep the same eigenvectors. 1 ...
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