James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

632 Chapter 9 Differential Equations
model. (Ask yourself what effect an increase in one species has
on the growth rate of the other.)
dx
(a) − 0.12x 2 0.0006x 2 1 0.00001xy
dt
dy
− 0.08x 1 0.00004xy
dt
dx
(b) − 0.15x 2 0.0002x 2 2 0.0006xy
dt
dy
− 0.2y 2 0.00008y 2 2 0.0002xy
dt
3. The system of differential equations
dx
dt
dy
dt
− 0.5x 2 0.004x 2 2 0.001xy
− 0.4y 2 0.001y 2 2 0.002xy
is a model for the populations of two species.
(a) Does the model describe cooperation, or competition,
or a predator-prey relationship?
(b) Find the equilibrium solutions and explain their
significance.
4. Lynx eat snowshoe hares and snowshoe hares eat woody plants
like willows. Suppose that, in the absence of hares, the willow
population will grow exponentially and the lynx population
will decay exponentially. In the absence of lynx and willow,
the hare population will decay exponentially. If Lstd, Hstd, and
Wstd represent the populations of these three species at time t,
write a system of differential equations as a model for their
dynamics. If the constants in your equation are all positive,
explain why you have used plus or minus signs.
6.
F
160
120
80
40
t=0
0 400 800 1200 1600 R
7–8 Graphs of populations of two species are shown. Use them to
sketch the corresponding phase trajectory.
7.
y
200
150
100
50
species 1
species 2
0 1
t
5–6 A phase trajectory is shown for populations of rabbits sRd and
foxes sFd.
(a) Describe how each population changes as time goes by.
(b) Use your description to make a rough sketch of the graphs of R
and F as functions of time.
5.
F
300
8.
y
1200
1000
800
600
400
200
species 1
species 2
200
0 5
10 15 t
100
t=0
9. In Example 1(b) we showed that the rabbit and wolf populations
satisfy the differential equation
0 400 800 1200 1600 2000 R
dW
dR
−
20.02W 1 0.00002RW
0.08R 2 0.001RW
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