7-35
SECTION 7.4
.
.
Systems of First-Order Differential Equations
599
the integral
±
x
n
+
1
xn
f
(
x
,
y
(
x
))
dx
. One such estimate is a
Riemann sum using left-endpoint evaluation, given by
f
(
x
n
,
y
(
x
n
))
±
x
. Show that with this estimate you get Eu-
ler’s method. There are numerous ways of getting better es-
timates of the integral. One is to use the Trapezoidal Rule,
²
x
n
+
1
xn
f
(
x
,
y
(
x
))
≈
f
(
x
n
,
y
(
x
n
))
+
f
(
x
n
+
1
,
y
(
x
n
+
1
))
2
±
x
.
The drawback with this estimate is that you know
y
(
x
n
)bu
t
you do
not
know
y
(
x
n
+
1
). Brieﬂy explain why this state-
ment is correct. The way out is to use Euler’s method;
you do not know
y
(
x
n
+
1
)but you can approximate it by
y
(
x
n
+
1
)
≈
y
(
x
n
)
+
hf
(
x
n
,
y
(
x
n
)). Put all of this together to get
the Improved Euler’s method:
y
n
+
1
=
y
n
+
h
2
[
f
(
x
n
,
y
n
)
+
f
(
x
n
+
h
,
y
n
+
(
x
n
,
y
n
))]
.
Use the Improved Euler’s method for the IVP
y
±
=
y
,
y
(0)
=
1
with
h
=
0
.
1to compute
y
1
,
y
2
and
y
3
. Compare to the exact
values and the Euler’s method approximations given in exam-
ple 3.4.
2.
As in exercise 1, derive a numerical approximation method
based on (a) the Midpoint rule and (b) Simpson’s rule.
Compare your results to those obtained in example 3.4 and
exercise 1.
3.
In sections 7.1 and 7.2, you explored some differential equa-
tion models of population growth. An obvious ﬂaw in those
models was the consideration of a single species in isolation.
In this exercise, you will investigate a
predator-prey
model.
In this case, there are two species,
X
and
Y
, with populations
x
(
t
) and
y
(
t
), respectively. The general form of the model is
x
±
(
t
)
=
ax
(
t
)
−
bx
(
t
)
y
(
t
)
y
±
(
t
)
=
(
t
)
y
(
t
)
−
cy
(
t
)
for positive constants
a, b
and
c
. First, look carefully at the
equations. The term
(
t
)
y
(
t
)is included to represent the ef-
fects of encounters between the species. This effect is negative
on species
X
and positive on species
Y
.If
b
=
0, the species
don’t interact at all. In this case, show that species
Y
dies out
(with death rate
c
) and species
X
thrives (with growth rate
a
).
Given all of this, explain why
X
must be the prey and
Y
the
predator. Next, you should ﬁnd the equilibrium point for co-
existence. That is, ﬁnd positive values ¯
x
and ¯
y
such that both
x
±
(
t
)
=
0 and
y
±
(
t
)
=
0. For this problem, think of
X
as an in-
sect that damages farmers’ crops and
Y
as a natural predator
(e.g., a bat). A farmer might decide to use a pesticide to reduce
the damage caused by the
X
’s. Brieﬂy explain why the effects
of the pesticide might be to decrease the value of
a
and increase
the value of
c
.Now, determine how these changes affect the
equilibrium values. Show that the pest population
X
actually
increases and the predator population
Y
decreases. Explain, in
terms of the interaction between predator and prey, how this
could happen. The moral is that the long-range effects of pesti-
cides can be the exact opposite of the short-range (and desired)
effects.