From:  Dave Rusin [rusin@math.niu.edu]
Date:  Wednesday, March 13, 1996 11:37 AM
To:  rmcgehee@TECLink.Net
Subject:  Re: reexpression

Someone, surely, will give you the lecture that it's inappropriate simply
to ask for any function which has a similar graph; what you ought to
do is make a model of the situation which explains why, for example,
the number of deaths is (like the number of births) proportional to a
product  N*(N_0 - N)  (N=number of individuals). I hope you will consider
such an approach -- if you have a sense of what drives your function in
the first place, it's easier to describe appropriate functions to fit
to your data.

Having said this, let me offer a suggestion. You know the function
2/(1+e^-t) increases from values nearly zero (for  t  < < 0 )  to the
value of  1  ( for  t > > 0 ). Of course you can add a time delay and
a time scaling factor and get the same general description; that is, you
can use any function of the form
	2/(1 + exp(-a(t-b)) ).

Here we have  a > 0  to get a curve whose graph has the increasing shape
described above, but of course we could look at functions with  a < 0
and get the mirror image of those curves: curves which decrease from
1  to  0  as  t  ranges over all real values. This suggests a way to
get a curve with your requirements: simply multiply two such functions
together -- one with  a > 0  and the other (corresponding to a larger
value of  b) with  a < 0.  For example, the function

4/ [ (1+exp(-(t+10))) * (1+exp(+(t-10))) ]

is nearly zero if  t  is large in magnitude; it rises near  t=-10  to
values just less than  1; stabilizes; drops near  t=+10 back to nearly zero.

This is just the kind of curve you would get if, during the extinction
phase, your population is dying off at a rate proportional to the
product  N*(N_0 - N)  as I suggested before.

dave