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Homework 4
AMATH 383, Autumn 2009
Due: Wednesday, November 18, after class
1. (3+6+5+2 points)
Periodicity of the Volterra System:
Consider the Volterra equations
x
′
=
r x
−
axy y
′
=
−
k y
+
by x
for a predatorpreymodel with solutions
x
(
t
)
and
y
(
t
)
. In class we demonstrated that close
to the nontrivial stationary pont the solutions are periodic. Here, we want to proof that
all
positive solutions are periodic except those starting in the stationary point.
(a) Show that
x
,
y
,
t
can be scaled such that the equations for the scaled variables
˜
x
,
˜
y
,
˜
t
have
the form
x
′
=
x
−
xy,
y
′
=
γ
(
−
y
+
y x
)
where we dropped the tildas for simplicity and used the parameter
γ
=
k/r
. What are the
scales of
x
,
y
, and
t
?
(b) Show that along solution trajectories
(
x
(
t
)
,y
(
t
))
of the above systems the quantity
A
(
x,y
) =
y
−
ln
y
+
γ
(
x
−
ln
x
)
is constant, that is,
d
dt
A
(
x
(
t
)
,y
(
t
)) = 0
. This means that the solution curves in the
x

y

plane are given by the level lines
A
(
x,y
) =
c
for diFerent
c
.
(c) Show that
A
(
x,y
)
becomes in±nite at the axes
x
= 0
and
y
= 0
, is convex and has a global
minimum. At what point is the minimum?
(d) Try to imagine the shape of the function
A
(
x,y
)
and argue that the level lines
A
(
x,y
) =
c
for
c >
1 +
γ
are closed lines. Conclude that the solutions
(
x
(
t
)
,y
(
t
))
of the Volterra
equations are periodic for any positive initial conditions diFerent from the stationary point.
2. (3+5+6+6+1+2 points)
Carnivores in Australia:
Consider the system of ordinary diFerential
equations
dx
dt
=
rx
(1
−
x
K
0
)
−
axy
dy
dt
=
−
ky
+
bxy
describing the interaction of herbivores
x
(
t
)
and carnivores
y
(
t
)
. The environment has a carrying
capacity
K
0
for herbivores. This problem will walk you through what has been presented in class.
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This note was uploaded on 01/10/2010 for the course MATH 124 taught by Professor Walker during the Spring '08 term at University of Washington.
 Spring '08
 WAlker
 Calculus, Equations, Periodicity

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