Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 164

X

N

r k . a k 2ˇ S /
2 a k .1 C r k / <ˇ S C 1;

k D 1

and is unstable if

X

N

r k . a k 2ˇ S /
2 a k .1 C r k / >ˇ S

C 1:

kD1

Proof. In analyzing the local asymptotic stability of system (4.23)-(4.25) we first
have to determine its Jacobian matrix evaluated at the equilibrium, which turns out
to have the form

0 1

J 11 J 12 J 13

@ J 21 J 22 J 23 A ;

(4.35)

J 31 J 32 J 33

with

0 1

0 1 a 1 :::r 1 a 1
r 2 a 2 0 :::r 2 a 2
: : : : : : : : :
r N a N r N a N ::: 0

@ A

J 11 D

;

0 1 0 1

r 1 .1 a 1 / 0 ˇ S r 1

ˇ S r 2
: : :
ˇ S r N

@ A @ A

r 2 .1 a 2 /

J 12 D

; J 13 D ;

: : :

0 r N .1 a N /

0
@

1
A

0
@

1
A

0
@

1
A

0a 1 :::a 1
a 2 0 :::a 2
: : : : : : : : :
a N a N ::: 0

1 a 1 0 0

0
: : :
0

1 a 2

J 21 D

; J 22 ; J 23

D D ;

: : :

0 1 a N

0
@ X
l

1

r l a l ;:::; X
l

J 31 D A ;

r l a l

¤ 1 ¤ N

1 C ! ˇ S ;

N

X

J 32 D .r 1 .1 a 1 /;:::;r N .1 a N //; J 33 D

r l

l D 1