Unformatted text preview: Math 464, Worksheet 14
Consider the Population model from Homework 3:
• Birth rate is b = 0.1 adolescents per day per mature animal.
• Death rate for adolescents is dA = 0.01 per day per adolescent.
• Death rate for mature animals is dM = 0.05 per day per mature animal.
• Adolescents mature into adults at a rate of g = 0.1 per day per adolescent.
dA A g M dM 1. Write the diﬀerence equation in the form
where C is a constant matrix.
2. Rearrange your equation so that it has the form
3. Compute eigenvalues and associated eigenvectors for B . Use at least four signiﬁcant digits.
• The largest eigenvalue is called the dominant eigenvalue.
• The associated eigenvector is the dominant eigenvector.
4. Suppose the initial populations are
Write this as “steps along” each eigenvector.
Vocabulary Note: The mathematical way to say this is “express the initial population vector
in terms of the basis of eigenvectors.
1 5. Compute
Write your answer in terms of the basis of eigenvectors. You should be able to do this using
only your answers to Problems 3 and 4.
6. Graph both eigenvectors, the lines along the eigenvectors,
M1 and on one set of axes.
7. Is the move from A1
towards or away from the line along the dominant eigenvector?
M0 8. How about for the line along the other eigenvector?
9. How about from
10. From A2
? (Towards or away from each line.)
Mn−1 11. What property of eigenvalues causes your answers to Problems 7–10?
12. Using only your qualitative analysis so far, sketch of curve of all the points to
0 ≤ n ≤ 40.
13. Check you graph against a graph drawn by Excel.
14. Repeat all this (Problems 4–13) with initial populations
0 and 0
100 Draw all the curves on one graph.
15. Repeat this entire problem set (1–14) with the assumption that dA = 0.1.
Note: This computation is a little easier. Try to understand why. 2 An
Mn for ...
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