Prove that the critical point
(
8
)
of the Volterra-Lotka system is a center; that is, the neighboring trajectories are periodic. [Hint: Observe that
(
9
)
is separable and show that its solution can be expressed as
[
x
2
A
e
−
B
x
2
]
⋅
[
x
1
C
e
−
D
x
1
]
=
K
]
(
26
)
.
Prove that the maximum of the function
x
p
e
−
q
x
is
(
p
/
q
e
)
p
, occurring at the unique value
x
=
p
/
q
(see Figure 5.24), so the critical values
(
8
)
maximize the factors on the left in
(
26
)
. Argue that if
K
takes the corresponding maximum value
(
A
/
B
e
)
A
(
C
/
D
e
)
C
, the critical point
(
8
)
is the (unique) solution of
(
26
)
, and it cannot be an endpoint of any trajectory for
(
26
)
with a lower value of
K
†
.
†
In fact, the periodic fluctuations predicted by the Volterra-Lotka model were observed in fish populations by Lotka’s son-in-law, Humberto D’Ancona.
x
2
(
t
)
≡
A
B
x
1
(
t
)
≡
C
D
(
8
)
d
x
2
d
x
1
=
−
C
x
2
+
D
x
1
x
2
A
x
1
−
B
x
1
x
2
=
x
2
x
1
×
−
C
+
D
x
1
A
−
B
x
2
(
9
)