Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 173

If A.1 ˇ S /.N 1 C ˇ S /
cN 2

A
4c then

.3/

<L D is not empty, the unique

.1/ ) and all the trajectories starting in .3/

steady state is E 1 (since E 3 2 D D enter

.1/ . 2

D

A.1 ˇ S /.N 1 C ˇ S /
cN 2

If L D ,thenE 1 D E 3 , and the equilibrium is located

.3/ from .1/ .

along the boundary that separates D D

If L< A.1 ˇ S /.N 1 C ˇ S /
cN 2

, then the unique steady state is E 3 (since E 1 2

.3/ ).

D

With regard to the stability of the equilibria, it is easy to realize that whenever

.3/

E 3 2 D it is a stable equilibrium, because the Jacobian matrix of the map T j D .3/

is

1 a0
N.1 a/ ˇ S

:

J .3/ D

Hence its eigenvalues 1 a and ˇ S are always less than one. In contrast, when
E 1 2 D

.1/ , its stability is not as easily determined because this requires the study of
the eigenvalues of the Jacobian matrix

0
@

aˇ S 1

1
A

2c p c . .N 1/x C ˇ S S /

aA.N 1/

2c p c . .N 1/x C ˇ S S /

A

1 aN C

J .1/

N 1 aN C ˇ S C Naˇ S

2c p c . .N 1/x C ˇ S S / 1

D

aA.N 1/
2c p c . .N 1/x C ˇ S S /

A

evaluated at E 1 , which has the form

0
@

1
A

aN . N 1 C 2ˇ S /
2 . N 1 C ˇ S /

aˇ S . 2 . 1 ˇ S / N /
2 . N 1 C ˇ S /

1

N 1

J .1/ . E 1 / D :

aN . N 1 C 2ˇ S /
2 . N 1 C ˇ S /

N aˇ S . 2 . 1 ˇ S / N /
2 . N 1 C ˇ S /

ˇ S C

Here the equilibrium condition
2c r A

S 1 D 2c S 1

c .N 1/ x 1 C ˇ S

D 2A.N 1 C ˇ S /=N

has been used. We can see that the stability of E 1 depends only on the parameters
N, a and ˇ S . Moreover, the matrix J .1/ .E 1 / has the structure
A 11

;

A 12

NA 11 ˇ S C NA 12

A
4c the region D

2 Note that at L

D

.3/ reduce to the line b

D b 1 D b 2 , which is a set of measure

2 .

zero in R