LotkaVolterra Rate Expressions
We can write the steps in the LotkaVolterra mechanism as
A + X > 2 X (Rate constant k_{1})
X + Y > 2 Y (Rate constant k_{2})
Y > B (Rate constant k_{3})
so that the rate laws for the intermediates are
d[X]/dt = k_{1}[A][X] – k_{2}[X][Y]
d[Y]/dt = k_{2}[X][Y] – k_{3}[Y]
If [A] is a constant ([A]_{0}) and if we assume steady state for X and Y, we can write
d[X]/dt = k_{1}[A]_{0}[X] – k_{2}[X][Y] = 0
d[Y]/dt = k_{2}[X][Y] – k_{3}[Y] = 0
or
[Y] = k_{1}[A]_{0}/k_{2} = constant
[X] = k_{3}/k_{2} = constant
which is what a steadystate assumption should give, but we know this assumption misses the key feature of this reaction mechanism: the oscillation in time of the intermediates' concentrations.
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