Information Technology Reference
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b)
a)
12
30
10
25
8
20
15
6
10
4
2
5
0
0
400
600
800
1000
0
200
400
600
time
time
c)
d)
1
12
10
0.8
8
0.6
6
0.4
4
0.2
2
0
0
0
100
200
300
400
0
200
400
600
800
1000
500
time
time
Fig. 3.3. Typical time series of the a
i
mRNA concentration for the four stable regimes: a) Q = 0:1
{ oscillatory, b) Q = 0:3 { inhomogeneous limit cycle, c) Q = 0:4 { inhomogeneous steady state,
and d) Q = 0:4 { homogeneous steady state. The common parameters are: N = 2, n = 2:6,
= 216,
a
= 0:85,
b
= 0:1,
c
= 0:1, = 25, k
s0
= 1:0, k
s1
= 0:01, = 2:0.
The basic continuation curve is characterized by two important properties: (1)
the presence of broken symmetry bifurcations (BP
1
and BP
2
in Fig. 3.4) where
inhomogeneous solutions arise, and (2) the stabilization of the homogeneous state
for large coupling values (Q > 0:129). The HSS solution is characterized by a
constant protein level concentration, stabilized through a saddle node bifurcation
(LP
1
in Fig. 3.4). A typical time series of this regime can be seen in Fig. 3.3(d).
Additionally, another HSS branch is found between LP
4
and HB
4
(Fig. 3.4), but it
is located outside the biologically relevant range (since Q > 1).
As a result of the symmetry breaking of the system through a pitchfork bifurca-
tion (BP
1
in Fig. 3.4), the unstable steady state splits in two additional branches,
giving rise to an inhomogeneous steady state (IHSS). This particular phenomenon
is model-independent, persisting for large parametric regions in several models of
diusively coupled chemical [Bar-Eli (1985); Dolnik and Marek (1988); Crowley and
Epstein (1989)] or biological oscillators [Kuznetsov et al. (2004); Tsaneva-Atanasova
et al. (2006)]. The IHSS in the present model is manifested through two distinct
steady protein concentration levels [Fig. 3.3(c)], gaining stability through a Hopf
bifurcation, denoted as HB
1
in Fig. 3.4, and thus leading to the so-called \oscillation
death" (OD) regime. This regime arises at a critical coupling Q
crit
= 0:3588 for
