Chemistry Reference
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then for x
k
>x
k
.k
D
1;2/ we have a unique interior equilibrium. The quantities
at this interior equilibrium are given by
E
D
. x
1
; x
2
/
2.B
C
e
2
/.A
c
1
/
B.A
c
2
/
4.B
C
e
1
/.B
C
e
2
/
B
2
:
;
2.B
C
e
1
/.A
c
2
/
B.A
c
1
/
4.B
C
e
1
/.B
C
e
2
/
B
2
D
On the other hand, if
B<e
k
<0, x
k
<x
k
.k
D
1;2/ as before, but
B
2
>4.B
C
e
1
/.B
C
e
2
/;
(1.38)
then a situation of multiple equilibria might be obtained. This is the situation
depicted in Fig. 1.3, where in addition to the interior equilibrium there also appear
two boundary equilibria. The two coexisting boundary equilibria are given by
E
1
D
.x
1
;0/
I
E
2
D
.0;x
2
/;
where
A
c
1
2.B
C
e
1
/
I
x
2
A
c
2
2.B
C
e
2
/
;
x
1
D
D
are the monopoly quantities.
Let us first try to give conditions for the global asymptotic stability of an equi-
librium, which would also imply its uniqueness. We recall that an equilibrium is
globally asymptotically stable if any trajectory starting from an initial condition in
the strategy space converges to the equilibrium as t
!1
. In the case of the model
(1.36) the strategy space is given by the trapping region
D D
Œ0;L
1
Œ0;L
2
.How-
ever the map (1.36), whose iteration gives the time evolution of the duopoly game,
is not differentiable in the whole strategy space
D
because the reaction functions are
piecewise differentiable functions defined by
8
<
A
c
B
;
0
if Q
k
A
c
k
2.B
C
e
k
/L
k
R
k
.Q
k
/
D
L
k
if Q
k
;
:
B
.A
c
k
BQ
k
/=
2.B
C
e
k
/
otherwise:
Accordingly, the phase space
can be subdivided into nine regions defined by the
break points of the reaction functions (see Fig. 1.10), such that the map T
a
is dif-
ferentiable (indeed linear in this case) inside each of them, it is defined differently
in each region and it is not differentiable on the boundaries between the regions.
Depending on the possible combination of the reaction functions the different
components of the map are given by
D
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
a
1
.A
c
1
Bx
2
/=
2.B
C
e
1
/
;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
.A
c
2
Bx
1
/=
2.B
C
e
2
/
;
T
a
j
D
.1/
W
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