735
SECTION 7.4
.
.
Systems of FirstOrder Differential Equations
599
the integral
x
n
+
1
x
n
f
(
x
,
y
(
x
))
dx
. One such estimate is a
Riemann
sum
using
leftendpoint
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,
x
n
+
1
x
n
f
(
x
,
y
(
x
))
dx
≈
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
) but
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
)
+
h f
(
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
+
h f
(
x
n
,
y
n
))]
.
Use the Improved Euler’s method for the IVP
y
=
y
,
y
(0)
=
1
with
h
=
0
.
1 to 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
predatorprey
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
)
=
bx
(
t
)
y
(
t
)
−
cy
(
t
)
for positive constants
a, b
and
c
. First, look carefully at the
equations. The term
bx
(
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 longrange effects of pesti
cides can be the exact opposite of the shortrange (and desired)
effects.
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 Spring '10
 atsumi
 Lotka–Volterra equation, 260 ft

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