**Unformatted text preview: **(cos t − sin t). Y
P The system is unstable because Re (λi ) > 0 for each eigenvalue. Indeed, u1 (t)
and u2 (t) both become unbounded as t → ∞. However, a population cannot
become negative–once it’s zero, it’s extinct. Figure 7.4.2 shows that the graph
of u2 (t) will cross the horizontal axis before that of u1 (t). O
C
400
300 u1(t) 200 u2(t) 100 t 0
0.2 0.4 0.6 0.8 1 -100
-200
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Chapter 7
Eigenvalues and Eigenvectors
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Therefore, the prey species will become extinct at the value of t for which
u2 (t) = 0 —i.e., when
100et (cos t − sin t) = 0 =⇒ cos t = sin t =⇒ t = π
.
4 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] Exercises for section 7.4
7.4.1. Suppose that An×n is diagonalizable, and let P = [x1 | x2 | · · · | xn ]
be a matrix whose columns are a complete set of linearly independent
eigenvectors corresponding to eigenvalues λi . Show that the solution to
u = Au, u(0) = c, can be written as D
E T
H u(t) = ξ1 eλ1 t x1 + ξ2 eλ2 t x2 + · · · + ξn eλn t xn in which the coeﬃcients ξi satisfy the algebraic system Pξ = c. 7.4.2. Using only the eigenvalues, determine the long-run behavior of the solution to u = Au, u(0) = c for each of the following matrices.
−1 −2
1 −2
1 −2
(a) A =
. (b) A =
. (c) A =
.
0 −3
0
3
1 −1 IG
R 7.4.3. Competing Species. Consider two species that coexist in the same
environment but compete for the same resources. Suppose that the population of each species increases proportionally to the number of its
own kind but decreases proportionally to the number in the competing
species—say that the population of each species increases at a rate equal
to twice its existing number but decreases at a rate equal to the number
in the other population. Suppose that there are initially 100 of species I
and...

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