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the stability ranges of the IHLC and the IHSS predicted by the bifurcation analysis
and shown by the direct calculation. Both regimes have a small basin of attraction.
3.2.2.3. Chaos provoked by repressive cell-to-cell communication
The bifurcation analysis (Fig. 3.5) predicts unstable anti-phase oscillations between
the torus bifurcation points TR
2
and TR
1
. To nd the stable solutions emerging
from those bifurcations, one can perform direct simulations starting with small cou-
pling Q, and trace the self-oscillatory regime up to strong coupling. The resulting
self-oscillations are stable and resistant to small perturbations in the initial con-
ditions and to dynamical noise. Interestingly, these stable self-oscillations display
very dierent dynamics with erratic amplitude and period, which is associated with
a positive maximal Lyapunov exponent, and thus corresponds to chaotic dynamics.
For a detailed description of the chaotic features of this regime and its validation
see Ullner et al. (2008).
3.2.2.4. Large system sizes
Typically, bacterial colonies consist of many cells and hence the results of the mini-
mal system with N = 2 repressilators have to be validated in large ensembles. Here
we show results for an ensemble of N = 100 coupled identical cells obtained from
direct calculations with random initial conditions. Figure 3.7 plots the resulting
frequency of stable regimes for increasing Q. The four main regimes HSS, IHSS,
IHLC and self-oscillations already observed in the minimal system can be detected
in the large systems too.
The results shown in Fig. 3.7 reveal a transition from self-oscillations to a single
stable xed point as the coupling Q increases.
This transition is gradual, and
1000
800
600
oscillatory
IHLC
IHSS
HSS
400
200
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Q
Fig. 3.7. Distribution of stable regimes for increasing coupling strength Q. The parameters are:
N = 100, n = 2:6, = 216,
a
= 0:85,
b
= 0:1,
c
= 0:1, = 25, k
s0
= 1:0, k
s1
= 0:01, and
= 2:0.
