# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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critical points are (0,0),(0,R)and (B,0).

It can be shown that (0,0) and (B,0) are unstable and (0,R) is asymptotically stable. Hence we conlcude, when coexistence is no longer possible, that x ^ 0 and y ^ R

and thus the bluegill population will die out.

12a. Setting each equation equal to zero, we obtain x = 0 or

(4 - x - y) = 0 and y = 0 or (2 + 2a - y - ax) = 0. Thus we

have (0,0), (4,0), (0,2 + 2a), and the intersection of

x + y = 4 and ax + y = 2 + 2a. If a Ï 1, this yields (2,2) as the fourth critical point.

Section 9.5

191

12b. For a = .75 the linear system is

Ã Y u

-2 -2 Yu

V-1.5 -2 Xvy

, which

has the characteristic equation r2 + 4r + 1 = 0 so that r = -2 ± ypi . Thus the critical point is an asymptotically

stable node. For a = 1.25, we have

Ã Y u

f ^ 'Ë´ Ë

¦2 -2 Y u

-2.5 -2Ëv

so

r2 + 4r -1 = 0 and r = -2 ± ä/~5 . Thus (2,2) is an unstable saddle point.

12c. Letting x = u+2 and y = v+2 yields

u' = (u+2)(4 - u-2 - v-2) = -2u - 2v - u2 - uv and v' = (v+2)(2 + 2a - v-2 - au - 2a) = -2au - 2v - v2 - auv. Thus the approximate linear system is u' = -2u - 2v and v' = -2au - 2v.

12d. The eigenvalues are given by -2-r -2

-2a -2-r

= r + 4r + 4

4a = 0, or r = -2 ± 2v a

Thus for 0 < a < 1 there are 2 negative real roots (asymptotially stable node) and for a > 1 the real roots differ in sign, yielding an unstable saddle point. a = 1 is the bifurcation point.

v

v

Section 9.5, Page 509

3b. We have x = 0 or (1 - .5x - .5y) = 0 and y = 0 or

(-.25 + .5x) = 0 and thus we have three critical points: (0,0), (2,0) and (1/2,3/2).

3c. For (0,0) the linear system is dx/dt = x and

f - ~ ë

dy/dt = -.25y and hence A =

which has

1 0 0 -1/4 _

eigenvalues r1 = 1 and r2 = -1/4 and corresponding

V 1 ^ r 0 ¯

eigenvectors and

v 0 v V 1 Ó

Thus (0,0) is an unstable

saddle point.

For (2,0), we let x = 2 + u and y = v in the given

du 1

equations and obtain — = -(u+v) - —u(u+v) and

dt

2

dv 3 1

— = —v + —uv. The linear portion of this has matrix

dt 4 2

192

Section 9.5

A =

' -i -i ë

0 3/4;

, which has the eigenvalues r1 = -1,

1 ¯ -4 ¯

r2 = 3/4 and corresponding eigenvectors and

v 0J V 7 ,

Thus (2,0) is also an unstable saddle point.

For

13

2 2

V 1

we let x = 1/2 + u and y = 3/2 + v in the

du

given equations, which yields ------------

dt

as the linear portion. Thus A

1 1 dv 3

- —u - —v, ---------- = —u

4 4 dt 4

1

4

3

— 0

4

, which has

eigenvalues r12 = (-1 ± y11i)/8. Thus

13

V 2 2 1

V1

is an

asymptotically stable spiral point since the eigenvalues are complex with negative real part. Using r1 = (-1 + \/11 i)/8 we find that one eigenvector is

2

and by Section 7.6 the second eigenvector is

1 + V11 i 2

1 - \hi i

3e.

3f. For (x,y) above the line x + y = 2 we see that x' < 0 and thus x must remain bounded. For (x,y) to the right of x = 1/2, y' > 0 so it appears that y could grow large asymptotic to x = constant. However, this implies a contradiction (x = constant implies x' = 0, but as y gets larger, x' gets increasingly negative) and hence we conclude y must remain bounded and hence (x,y) ^ (1/2,3/2) as t ^ •, again assuming they start in the first quadrant.

Section 9.5

193

7a. The amplitude ratio is (cK/g)/(^ac K/a) = a^J~c /g\pi .

7b. From Eq (2) a = .5, a = 1, g = .25 and c = .75, so the ratio is .5V .75 /.25^1 = 2\/ .75 = ä/3 = 1.732.

7c. A rough measurement of the amplitudes is (6.1 - 1)/2 = 2.55 and (3.8 -. 9)/2 = 1.45 and thus the ratio is approximately 1.76. In this case the linear approximation is a good predictor.

11. The presence of a trapping company actually would require a modification of the equations, either by altering the coefficients or by including nonhomogeneous terms on the right sides of the D.E. The effects of indiscreminate trapping could decrease the populations of both rabbits and fox significantly or decrease the fox population which could possibly lead to a large increase in the rabbit population. Over the long run it makes sense for a trapping company to operate in such a way that a consistent supply of pelts is available and to disturb the predator-prey system as little as possible. Thus, the company should trap fox only when their population is increasing, trap rabbits only when their population is increasing, trap rabbits and fox only during the time when both their populations are increasing, and trap neither during the time both their populations are decreasing. In this way the trapping company can have a

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