LECTURE 34: APPLICATIONS OF NONLINEAR SYSTEMS
MINGFENG ZHAO
November 24, 2014
Predator-prey or Lotka-Volterra systems
Let
x =
number of the prey(e.g., hares)
y
number of the predator(e.g., foxes)
=
Recall the population model: the rate of growth of the population is proportional to the size of the
population.
In the predator-prey model, we have
x0
= ax − (by)x = (a − by)x
y0
=
(cx)y − dy = (cx − d)y
for some positive constants a, b, c, d > 0.
Critical points: Let’s solve
(a − by)x = 0,
(cx − d)y = 0.
and
Then we have two critical points:
(x, y) = (0, 0),

The Jacobian matrix of 
(a − by)x
(cx − d)y
and
(x, y) =

 is:

a − by

cy
−bx
cx − d
Then
1


d a
,
c b
.
2
MINGFENG ZHAO
I. At (0, 0), the linearization is:

u

0

 =
v

Since det 
a
0
0
−d
a
0
0
−d

u


.
v


 = −ad 6= 0, then the system is almost linear at (0, 0). Since 
a
0
0
−d

 has eigen-
values λ1 = a and λ2 = −d, then the solutions behaves like a saddle near (0, 0), which is unstable .
d a
II. At
,
, the linearization is:
c b

u

0

 =
v
0
ac
b
− bd
c

u

0

.
v


0 − bd
d
a
c 
 = −ad 6= 0, then the system is almost linear at
has
,
. Since 
Since det 
c
b
ac
ac
0
0
b
b
√
√
d a
eigenvalues λ1 = i ad and λ2 = −i ad, then the solutions behaves like a center near
,
. So we would
c b
d a
have a stable center, spiral sink, or a spiral source at
,
.
c b
Let’s study the trajectories for x, y > 0, tha is, we need to solve

0
− bd
c

dy
dx
(cx − d)y
y
cx − d
=
·
.
(a − by)x
a − by
x
=
This a separable equation, then
cx − d
a − by
dy =
dx.
y
x
So we get
a ln y − by = cx − d ln y + C.
That is, we have
y a e−by = ecx y −d eC .
So we get get an implicit equation for the trajectories:
C=
y a xd
= y a xd e−cx−by .
ecx+by
This constant C is called the constant of motion. That is, the trajectories for the predator-prey system are
y a xd
the level curves of cx+by .
e
LECTURE 34: APPLICATIONS OF NONLINEAR SYSTEMS
3
Figure 1. The phase portrait (left) and graphs of x and y for a sample trajectory (right) for x0 =
(0, 4 − 0.01y), y 0 = (0.003x − 0.3)y

 x0 = (a − by)x
Theorem 1. The solutions to the system
 y 0 = (cx − d)y
behaves like a center near
d a
,
, which is a
c b
stable center.
y a xd
Proof. We are going to show that the function g(x, y) = cx+by = y a xd e−cx−by will attain its maximum at
e
d a
,
for x, y > 0.
c b
Let f (x, y) = ln g(x, y) = a ln y + d ln x − cx − by, so it suffices to prove that f (x, y) will attain its maximum
d a
,
at
for x, y > 0. In fact, we know that
c b
fx
=
fy
=
d
−c
x
a
− b.
y
So only critical point for f (x, y) is:
d a
,
c b
.
By L’Hospital’s Rule, we know that
lim
x→∞
xd
ya
=
lim
= 0.
y→∞ eby
ecx
4
MINGFENG ZHAO
So we get
lim
|(x,y)|→∞
g(x, y) = 0,
which implies that
lim
|(x,y)|→∞
f (x, y) = −∞.
So we know that f (x, y) must attain its maximum point in x, y > 0, which must be a critical point of f . So
d a
f (x, y) will attain its maximum at
,
for x, y > 0
c b
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca