Is this a mathematical exercise or does the student want to consider
practical solutions as well. Many have already commented on a purely
mathematical approach, but in reality, there are a great many number of
limiting factors to affect the growth curve. Living organisms do not
follow a mathematical model indefinitely. In the lab, the growth rate
depends on the volume of the container, the addition of nutrients, the
subtraction of wastes. With little outside involvement, you would soon
see an *S* curve an then a collapse, down to a very low sustainable
level or even to zero eventually. In nature continued growth would
also depend on the host if it is a disease germ, and you must consider
immune factors kicking in, and other bodily functions to contain the
infection, again the growth levels off to zero growth and eventually a
drop almost to zero. If it were in a pond or other natural environment
other factors include temperature, nutrients in the environment, pH of
the water supply, and a host of other considerations (including smaller
*bugs* feasting off the germ, so you might be able to see a microfood
chain taking place given time and proper equipment to observe such
events).
As part of a graduate school project in microbiology we did a study
of yeast colonies under differing controlled conditions where we added
and subtracted various nutrients and waste by-products and a number of
other factors in the dishes and did counts of the growth patterns.
That was many, many years ago, so I mostly had forgotten this until this
question came up.
If this student just wants to model a pure mathematical growth curve
as in a simple *what-if* computer game, well, forget everything I wrote.
But, if the student is interested in what really happens, then there
are all these considerations and more which fall well outside the realm
of the *pure* models.