ws15 - Math 464, Worksheet 15 Consider a population model...

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Unformatted text preview: Math 464, Worksheet 15 Consider a population model in which each successive day’s population is given by xn+1 x =M n yn+1 yn You do not know the entries of the matrix M , but you do know that it’s eigenvalues and associated eigenvectors are 1 1 λ1 = 1.1, v1 = , and λ2 = 0.9, v2 = 3 −1 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 accuracy. 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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