The assumption of d1=d2 is feasible because aiiA and LuxR within the cells will be washed out at the same rate in chemostat.
As long as we can ensure the washing out rate is much more dominant than their natural half-life (easily achieved) the assumption should hold
Hypothesis 2 is not really essential
it is fortunate as it was hard to ensure
if d1=d2, the difference between W and V will decay to 0 exponentially (with a time constant 1/d1)
therefore after a little time we can assume V=W
the larger d1, the faster the assumption becomes valid
the larger the difference between initial values of V & W, the longer the settling time of reaching V=W only
In particular we are sure that the condition on the parameters for obtaining a limit cycle will still be identical in 2D and 3D despite of the initial concentrations of U V W.
Problem : in Theory , there is a Huge Difference Between 2D and 3D
Poincare-Bendixson Theorem works for 1D and 2D only, but not 3D
We only need simple requirements for a limit cycle in 2D
In 3D the requirement is more complex - or much more complex
So are our results in 2D worth anything ?
If our hypotheses are exactly met: Yes!
In practice hypotheses not exactly met, but we have a margin of error
A slight error on Hypothesis 2 is not important
Slight error on hypothesis 1 (d1 not strictly equal to d2): 2 Scenarii
Scenario 1: (the kind one)
For d1=d2 and a range of parameters well chosen we have oscillations
Because the system is well behaved , we still have oscillations at the vicinity of these parameters (hence for d1 slightly different from d2)
Scenario 2: (the not so nice one)
[aiiA] and [LuxR] get more and more out of synchronisation
However, if the hypotheses are almost met, we can hope to have a few synchronised cycles
The simulation above shows individual cycles of [aiiA] and [LuxR]
Frequencies are equal
Profiles very similar
Peak amplitudes different
Clearly for such cycles d1=d2 was not met and yet we have oscillations. We therefore have to study the 3D case in its entirety at some point
However for our current interest of whether the system can result in generation, 2D case of d1=d2 should be enough
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