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Math 464, Worksheet 17
Consider the Population model from Worksheet 16:
Δ
x
= 0
.
02
x

0
.
01
x
2

0
.
01
xy
Δ
y
= 0
.
03
y

0
.
01
y
2

0
.
02
xy
Part I.
Here are some basic facts we already know:
•
If the population starts out at
x
(0) =
y
(0) = 1 then it never changes. In other words,
x
(
t
) = 1
and
y
(
t
) = 1 is an equilibrium solution.
•
If the population starts out elsewhere, it seems that one or the other species goes extinct.
•
The diFerence equations are nonlinear.
•
Therefore, we cannot write this equation as a matrix equation, and we can’t use eigenvectors
or eigenvalues to predict where solutions go.
•
Except that we can. Here’s how:
1. Con±rm that
x
=
y
= 1 is an equilibrium by plugging into the right hand side of the diFerence
equations.
2. Compute the
Jacobian
of the system. This is a twobytwo matrix
J
=
b
∂
Δ
x
∂x
∂
Δ
x
∂y
∂
Δ
y
∂x
∂
Δ
y
∂y
B
3. Evaluate
J
at the point
x
= 1,
y
= 1.
4. Compute eigenvalues and eigenvectors for
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This note was uploaded on 10/12/2011 for the course MATH 464 taught by Professor Dougbullock during the Fall '08 term at Boise State.
 Fall '08
 DougBullock
 Math

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