Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 174

with

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 /

A 11 D 1 and A 12 D :

The characteristic polynomial of this matrix is the quadratic

2 .A 11 C ˇ S C NA 12 / C ˇ S A 11 ;

so the conditions for asymptotic stability are (see Appendix F)

.1 ˇ S /.1 A 11 / NA 12 >0;

(4.48)

.1 C ˇ S /.1 C A 11 / C NA 12 >0;

(4.49)

ˇ S A 11 <1; (4.50)

which reduce to the conditions,

.1 ˇ S / aN .N 1 C 2ˇ S /

aNˇ S .2.1 ˇ S / N/
2.N 1 C ˇ S / 2.N 1 C ˇ S /

>0; (4.51)

.1 C ˇ S / 2

aN .N 1 C 2ˇ S /
2.N 1 C ˇ S /

aNˇ S .2.1 ˇ S / N/
2.N 1 C ˇ S /

C

>0; (4.52)

aNˇ S .N 1 C 2ˇ S /
2.N 1 C ˇ S /

>ˇ S 1:

(4.53)

It is obvious that due to the algebraic complexity of these stability conditions
further analytical calculations will become quite involved. Therefore, instead we
give a brief numerical study that will give us a flavor of the results one might expect
to hold in general.
First of all we investigate the effect of the bifurcation that marks the exchange of
the equilibrium E 3 with the equilibrium E 1 occurring at

A.1 ˇ S /.N 1 C ˇ S /
cN 2

L bif D

.1/ . This is not a usual
transcritical bifurcation because E 1 and E 3 are fixed points of two different maps,
and the merging occurs along a line of non-differentiability. Indeed, this is a typical
border collision bifurcation, the effect of which is quite difficult to forecast. This is
shown by three different bifurcation diagrams obtained for increasing values of the
capacity limit L across the bifurcation value. The first bifurcation diagram, shown in
Fig. 4.13 is obtained with the set of parameters N D 4, A D 2, c D 0:15, ˇ S D 0:6,
and a D 0:5, with L in the range Œ0:5;2. For this set of parameters the bifurcation
value is L D 1:2 and, as it can be seen in Fig. 4.13, when L crosses the bifurcation

.3/

along the boundary that separates the regions D and D