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

Section 9.6 Predator-Prey Systems 633
By solving this separable differential equation, show that
R 0.02 W 0.08
e 0.00002R e 0.001W − C
where C is a constant.
It is impossible to solve this equation for W as an explicit
function of R (or vice versa). If you have a computer algebra
system that graphs implicitly defined curves, use this equation
and your CAS to draw the solution curve that passes through
the point s1000, 40d and compare with Figure 3.
10. Populations of aphids and ladybugs are modeled by the
equations
dA
− 2A 2 0.01AL
dt
dL
− 20.5L 1 0.0001AL
dt
(a) Find the equilibrium solutions and explain their
significance.
(b) Find an expression for dLydA.
(c) The direction field for the differential equation in part (b)
is shown. Use it to sketch a phase portrait. What do the
phase trajectories have in common?
L
400
300
200
100
0 5000 10000 15000 A
(d) Suppose that at time t − 0 there are 1000 aphids and
200 ladybugs. Draw the corresponding phase trajectory
and use it to describe how both populations change.
(e) Use part (d) to make rough sketches of the aphid and
lady bug populations as functions of t. How are the graphs
related to each other?
11. In Example 1 we used Lotka-Volterra equations to model
popu lations of rabbits and wolves. Let’s modify those equations
as follows:
dR
dt
dW
dt
− 0.08Rs1 2 0.0002Rd 2 0.001RW
− 20.02W 1 0.00002RW
CAS
(a) According to these equations, what happens to the
rabbit population in the absence of wolves?
(b) Find all the equilibrium solutions and explain their
significance.
(c) The figure shows the phase trajectory that starts at the
point s1000, 40d. Describe what eventually happens to
the rabbit and wolf populations.
W
70
60
50
40
800
1000 1200 1400
1600
(d) Sketch graphs of the rabbit and wolf populations as
functions of time.
12. In Exercise 10 we modeled populations of aphids and
ladybugs with a Lotka-Volterra system. Suppose we modify
those equations as follows:
dA
dt
dL
dt
− 2As1 2 0.0001Ad 2 0.01AL
− 20.5L 1 0.0001AL
(a) In the absence of ladybugs, what does the model predict
about the aphids?
(b) Find the equilibrium solutions.
(c) Find an expression for dLydA.
(d) Use a computer algebra system to draw a direction field
for the differential equation in part (c). Then use the
direction field to sketch a phase portrait. What do the
phase trajectories have in common?
(e) Suppose that at time t − 0 there are 1000 aphids and
200 ladybugs. Draw the corresponding phase trajectory
and use it to describe how both populations change.
(f) Use part (e) to make rough sketches of the aphid and
ladybug populations as functions of t. How are the
graphs related to each other?
R
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.