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The most basic model of the interaction between yeast and sugar must
account for two facts: sugar must be used for the yeast population to
grow, and the rate of population growth is proportional to the size of the
population. The first of these two facts basically means that for every
increase of the yeast population there is a corresponding decrease in the
sugar concentration, which may be written mathematically as
|
(1) |
(Recall that is the same as ). Here is the
amount of sugar required to produce a fuzzy baby yeast. The second effect
can be written
where is the per-capita growth rate of the yeast population, which
pretty obviously must depend on the sugar concentration. After all, if
there is no sugar there will be no growth. The simplest form of a function
which captures the effect of sugar on per-capita growth rate is
and substituting this in gives a second equation for rate of population
growth,
|
(2) |
The parameter can be interpreted as the rate of growth per sugar
concentration.
Next: Reduction to a Single
Up: The Logistic Model
Previous: The Logistic Model
James Powell
2000-07-31