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10.6 Phase Portraits 341

y

y(t)

60
II

40
III (k/c, a/b)
I
20
IV
x(t)

0 x
FIGURE 10.27 Trajectories for a typical competing
20 40 60 80 100 120 140
species model.

FIGURE 10.28 Trajectories for x = x −0.01xy, y =
4y − 0.1xy.

while the y population dies out with time (decreasing to zero in the limit as t →∞). If the initial
population is in II or III, then the y population wins and the x population dies out asymptotically.
The coefficients a,b,c,k play a crucial role in determining the asymptotes, hence the regions, so
just having a large initial population is not enough to guarantee survival.
As a specific example, Figure 10.28 shows a phase portrait for the model

x = 2x − 0.1xy

y = 4y − 0.1xy.

SECTION 10.6 PROBLEMS

In each of Problems 1 through 10, classify the origin of 4 −1
7. A =

the system X = AX for the given coefficient matrix. If 1 2
software is available, produce a phase portrait.
3 −5
8. A =

3 −5 8 −3
1. A =
5 −7
−2 −1
9. A =

1 4 3 −2
2. A =
3 0
−6 −7

10. A =
1 −5 7 −20
3. A =
1 −1
11. Derive a system of differential equations modeling

9 −7 the predator/prey relationship in an environ-
4. A =
6 −4 ment with indiscriminate harvesting. Do this by
assuming that there is some outside agent that

7 −17
5. A = removes numbers of both species from the sys-
2 1
tem at a rate proportional to the populations,
with the same constant of proportionality for both
2 −7
6. A = species.
5 −10
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October 14, 2010 20:32 THM/NEIL Page-341 27410_10_ch10_p295-342